a0
a1a2a3 a4a5a6a7a8a9a10a11
a12 a13 8–1
1,a14V a15a16a17[?1,1]a18a19a20a21a22a23a24a25a26a27a28a29a30a31a32a17,a33a34:
ψ, V?→ R
f(x) mapsto?→ integraltext1?1f(x)dx
a15V a18a29a35a36a30a31a24a25.
a37a38:
a39a40ψa15V a41Ra29a35a36a42a43,a44a45a46a47a29f(x),g(x) ∈V,k ∈R,a48
ψ(f(x) +g(x)) =
integraldisplay1
1
(f(x) +g(x))dx =
integraldisplay1
1
f(x)dx+
integraldisplay1
1
g(x)dx = ψ(f(x)) +ψ(g(x)),
ψ(kf(x)) =
integraldisplay1
1
kf(x)dx = k
integraldisplay1
1
f(x)dx = kψ(f(x)).
a26a49ψa15V a18a29a35a36a30a31a24a25.
2,a14V a15a25a50Ka18a29a35a363a51a30a31a32a17,η1,η2,η3a15a52a29a35a36a53,fa15V a18a29a35a36a30a31a24a25,a44
f(η1?2η2 +η3) = 2,f(η1 +η3) = 2,f(?η1 +η2 +η3) =?1.
a54f(x
1η1 +x2η2 +x3η3).
a55:
a56?
α1 = η1?2η2 +η3
α2 = η1 +η3
α3 =?η1 +η2 +η3
a57(α
1,α2,α3) = (η1,η2,η3)A,a58a59
A =
1 1?1
2 0 1
1 1 1
.
a57(η
1,η2,η3) = (α1,α2,α3)A?1,a26a49
f(x1η1 +x2η2 +x3η3) = (f(η1),f(η2),f(η3))
x1
x2
x3
= (f(α1),f(α2),f(α3))A?1
x1
x2
x3
= (2,2,?1)· 14
1?2 1
3 2 1
2 0 2
x1
x2
x3
= 3
2 x1 +
1
2 x3.
3,V a60η1,η2,η3
a61
a18a62,a63
a54
a35a30a31a24a25g,a64
g(3η1 +η2) = 2,g(η2?η3) = 1,g(2η1 +η3) = 2.
a55:
a14
g(η1) = a,g(η2) = b,g(η3) = c,
a57a65a66a67a68
3a+b = 2
b?c = 1
2a+c = 2.
· 1 ·
a69a68a =?1,b = 5,c = 4.
a70a71a26
a54
a29a30a31a24a25a72
g(x1η1 +x2η2 +x3η3) =?x1 + 5x2 + 4x3.
4,a14V a15a25a50Ka18a29na51a30a31a32a17,η1,···,ηna15a52a29a35a36a53,a1,···,ana15Ka59a46a47na36a25,a33
a34,a73a74V a18a75a35a29a30a31a24a25f,a64
f(ηi) = ai,i = 1,···,n.
a37a38,(
a73a74a31)a14α = x1η1 +x2η2 +···+xnηn ∈V,a56
f, V?→ K
α mapsto?→ f(α) =
nsummationtext
i=1
aixi
a76a77
a33a34fa15V a18a30a31a24a25,a44a78a79a26a80a81a82.
(a75a35a31)a14ga72V a29a30a31a24a25,a64
g(ηi) = ai,i = 1,···,n.
a57
a45a46a47a29α = x1η1 +x2η2 +···+xnηn ∈V a48
g(α) =
nsummationdisplay
i=1
xig(ηi) =
nsummationdisplay
i=1
xiai = f(α).
a83a84
a33a34
a85
a75a35a31.
5,a14V = K3,α = (x1,x2,x3),β = (y1,y2,y3),a86a87a88a89a90a91a24a25fa15a92a72V a18a29a93a30a31a24a25:
(1) f(α,β) = 2x1y1 +x1y2?3x2y1 +x2y2;
(2) f(α,β) = (x1?y2)2 +x2y1;
(3) f(α,β) = c,c∈K;
(4) f(α,β) = (2x1 +x2?3x3)(y1?y2 +y3).
a55,(1)
a15.
(2)a92.
(3)a94cnegationslash= 0a95,a92;a94c = 0a95,a15.
(4)a15.
6,a14fa72na51a30a31a32a17V a18a29a93a30a31a24a25,a56
W1 = {α∈V |f(α,β) = 0,?β ∈V},
W2 = {α∈V |f(β,α) = 0,?β ∈V}.
a33a34,W1
a96
W2
a97
a15V a29a30a31a98a32a17,a44dimW1 = dimW2.
a37a38,(1) a65a99
a45a46a47a29β ∈ V a48f(0,β) = 0,a100a1010 ∈ W1,W1
a102
a32,a103a45a46a47a29α1,α2 ∈ W1,
k ∈Ka49a60a46a47a29β ∈V a48
f(α1 +α2,β) = f(α1,β) +f(α2,β) = 0,
f(kα1,β) = kf(α1,β) = 0,
a100a101
α1 +α2 ∈W1,kα1 ∈W1.
a26a49W1a15V a29a30a31a98a32a17.
a61a104a105
a33W2
a106
a15V a29a30a31a98a32a17.
· 2 ·
(2)a14η1,···,ηna72V a29a53,fa74a53η1,···,ηna88a29a107a108a109a110a72B,a57a45a46a47a29a111a108
α = (x1 ··· xn)
η1
...
ηn
,β = (y
1 ··· yn)
η1
...
ηn
,
f(α,β) = (x1 ··· xn)B
y1
...
yn
.
a70a71
α =
nsummationtext
i=1
xiηi ∈ W1 (x1 ··· xn)B
y1
...
yn
= 0?(y
1,···,yn) ∈ Kn (x1 ··· xn)B =
0 (x1 ··· xn)a72a112a113a30a31a114a115a27XB = 0a29a69.
a26a49dimW1 =a112a113a30a31a114a115a27XB = 0a29
a69
a32a17a29a51a25= n?rankB.
a61a104a105
a33dimW2 = n?rankB,a26a49dimW1 = dimW2.
7,a14f a72Kn a18a29a35a36a90a91a24a25,a33a34,f a72Kn a18a29a93a30a31a24a25a29a116a117a118a119a81a82a15a73a74a109a110
A∈Mn(K),a64
f(X,Y) = XTAY,X,Y ∈Kn.
a37a38,(?)
a14fa72Kna18a93a30a31a24a25,a120fa29a107a108a109a110A,a57A∈Mn(K),a44
f(X,Y) = XTAY,?X,Y ∈Kn.
(?) a121a90a91a24a25a78a79
f(X,Y) = XTAY,?X,Y ∈Kn,
a57f
a39a40a15Kna18a93a30a31a24a25.
8,a45
a99a1225
a62a59a29a93a30a31a24a25,a63
a54a123a124
a29a107a108a109a110.
a55,(1)
2 1 0
3 1 0
0 0 0
.
(3)a94c = 0a95,a107a108a109a110= 0.
(4)
2?2 2
1?1 1
3 3?3
.
9,a14V = K4,a121a88a125a126V a29a90a91a24a25f:
f(α,β) = x1y1 +x2y2?x3y3?x4y4,
a58a59
α = (x1,x2,x3,x4),β = (y1,y2,y3,y4).
(1)a33a34,fa15V a18a29a35a36a93a30a31a24a25;
(2)a54fa74a53
η1 = (2,1,?1,1),η2 = (0,2,1,0),
η3 = (1,1,?2,1),η4 = (0,0,1,2)
a88a29a107a108a109a110;
(3)a127a128a35a36a78a79f(α,α) = 0a29a111a108αnegationslash= 0.
· 3 ·
a55,(1)
a129a130a131a33a132
a105
,a33a133.
(2)a134a135a48
(η1 η2 η3 η4) = (ε1 ε2 ε3 ε4)
2 0 1 0
1 2 1 0
1 1?2 1
1 0 1 2
a71fa74a53ε1,ε2,ε3,ε4a88a29a107a108a109a110a72?
1
1
1
1
,
a100a101fa74a53η1,η2,η3,η4a88a29a107a108a109a110a72?
2 1?1 1
0 2 1 0
1 1?2 1
0 0 1 2
1
1
1
1
2 0 1 0
1 2 1 0
1 1?2 1
1 0 1 2
=
3 3 0?1
3 3 4?1
0 4?3 0
1?1 0?5
.
(3)a120α = (1,1,1,1),a39a40a48f(α,α) = 0.
10,a14V = K4,α = (x1,x2,x3,x4),β = (y1,y2,y3,y4),
f(α,β) = 3x1y2?5x2y1 +x3y4?4x4y3.
(1)a54fa74a53
η1 = (2,1,?1,1),η2 = (1,2,1,?1),
η3 = (?1,1,2,1),η4 = (1,?1,1,2)
a88a29a107a108a109a110;
(2)a136a120V a29a53ε1,ε2,ε3,ε4:
(ε1,ε2,ε3,ε4) = (η1,η2,η3,η4)T,
a58a59
T =
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
,
a54f
a74ε1,ε2,ε3,ε4a88a29a107a108a109a110.
a55,(1)
a137fa74a138a40a53a88a29a107a108a109a110a139a72B,a137
a65
a138a40a53a41a53η1,η2,η3,η4a29a140a141a109a110a139a72A,a57
B =
0 3 0 0
5 0 0 0
0 0 0 1
0 0?4 0
,A =
2 1?1 1
1 2 1?1
1?1 2 1
1?1 1 2
,
a99
a15fa74a53η1,η2,η3,η4a88a29a107a108a109a110a72
C = ATBA =
1 4 2?17
20?1 22?7
7?17?4?2
22 2?17?4
.
· 4 ·
(2) fa74a53ε1,ε2,ε3,ε4a88a29a107a108a109a110a72
D = TTCT =
45 9 39?27
9?45 9?117
39?9 5 3
27 117 3 45
.
11,a14fa15na51a30a31a32a17V a18a29a93a30a31a24a25,a33a34,f
a102a142a143
a29a116a117a118a119a81a82a15,a70
f(α,β) = 0,a45a26a48a29α∈V,
a105
a49a144a128β = 0.
a37a38,(?)
a56
W1 = {α∈V |f(α,β) = 0,?β ∈V},
W2 = {α∈V |f(β,α) = 0,?β ∈V}.
a121f
a102a142a143
,a57a65a125a1261.3a60W1a29a125a126
a67W
1 = 0,a70a71
a65a145
a626a68W2 = 0,a100a101
a65f(α,β) = 0?α∈ V
a105
a49a144a128α = 0.
(?) a121f(α,β) = 0?α ∈ V
a105
a49a144a128α = 0,a57W2 = 0,
a61a104a105
a68W
1 = 0,
a57a65
a125a1261.3a60W1a29
a125a126
a67f
a102a142a143
.
12,a14A∈Mm(K),V = Mm,n(K),a125a126V a18a29a90a91a24a25fa121a88:
f(X,Y) = Tr(XTAY),X,Y ∈V.
(1)a33a34,fa15V a18a29a35a36a93a30a31a24a25;
(2)a54fa74a53E11,E12,···,E1n,···,Em1,···,Emn a88a29a107a108a109a110;
(3)a74a146a147a81a82a88,fa15
a102a142a143
a29.
a55,(1)
a14X = (xij)m×n,Y = (yij)m×n,A = (aij)m,a57
f(X,Y) =
nsummationdisplay
i=1
msummationdisplay
l=1
msummationdisplay
k=1
xlialkyki,
a70a71
a67f
a15a93a30a31a29.
(2) a65a99f(Est,Euv) = δtvasu,a100a101fa74a53E11,E12,···,E1n,···,Em1,···,Emna88a29a107a108a109a110a72
B =
a11E ··· a1mE
...,..,..
am1E ··· ammE
,
a58a59Ea15na148a149a150a114a110.
(3) a65a99|B| = |A|n,a26a49f
a102a142a143
|B| negationslash= 0 |A| negationslash= 0,a132f
a102a142a143
a29a116a117a118a119a81a82a15A
a15
a105a151
a109a110.
13,a33a34,Mn(K)a18a29a93a30a31a24a25
f(A,B) = TrAB,A,B ∈Mn(K)
a15
a102a142a143
a29.
a37a38:
a14A = (aij) ∈Mn(K),a121a152
f(A,B) = TrAB = 0?B ∈Mn(K)
a57f(A,E
ij) = 0?i,j = 1,···,n,a71
f(A,Eij) = TrAEij = aji,
· 5 ·
a26a49aji = 0a45i,j = 1,···,n,a132A = 0,a100a101f
a102a142a143
.
a136a33,a100a72
f(A,B) = TrAB = Tr((AT)TB) = Tr((AT)TEB),
a65a145
a6212(3)
a105
a67f
a102a142a143
.
a12 a13 8–2
1,a14fa15a30a31a32a17V a18a29a45a153a154a155a153a93a30a31a24a25,Wa15V a29a156a98a32a17.
a33a34,a45ξ /∈W,a118a48
a102a157
a111a108η ∈W +L(ξ),a64a45a26a48a29α∈W,
a97
a48f(η,α) = 0.
a37a38:
a121W = 0,a57a158a159a39a40a28a160,a161a14W negationslash= 0,a14α1,···,αsa72Wa29a53,a57a100ξ /∈W,ξ,α1,···,αs
a30a31a162a163,a164a165a30a31a114a115a27?
x0f(ξ,α1) +x1f(α1,α1) +···+xsf(αs,α1) = 0
x0f(ξ,α2) +x1f(α1,α2) +···+xsf(αs,α2) = 0
..........................................
x0f(ξ,αs) +x1f(α1,αs) +···+xsf(αs,αs) = 0
(*)
a101a112a113a30a31a114a115a27a29a114a115a36a25sa166
a99a167a67
a108a36a25s+ 1,a168(*)a48
a102a157
a69(a
0,a1,···,as),a56
η = a0ξ +a1α1 +···+asαs,
a57η ∈W +L(ξ),
a44η negationslash= 0 (a100α1,···,αsa30a31a162a163,a44a0,a1,···,as
a169
a19a72
a157
),a44
a65(*)a67
f(η,αi) = 0,i = 1,2,···,s.
a103a100α1,···,αsa72Wa29a53,a168a45a46a47a29α∈W
a97
a48f(η,α) = 0.
2,V
a96
f
a61
a18a62,Wa15V a29a30a31a98a32a17,a56
W⊥ = {α∈V |f(α,β) = 0,?β ∈W}.
a33a34,(1) W⊥a15V a29a30a31a98a32a17;
(2)a121a152W ∩W⊥ = {0},a57V = W ⊕W⊥.
a37a38,(1) a65f(0,β) = 0?β ∈W,
a105
a680 ∈W⊥,
a100a101W⊥
a102
a32.
a45a46a47a29α1,α2 ∈W⊥,k ∈K,a57?β ∈W,a48
f(α1 +α2,β) = f(α1,β) +f(α2,β) = 0,
f(kα1,β) = kf(α1,β) = 0,
a100a101α1 +α2 ∈W⊥,kα1 ∈W⊥,a168W⊥a15V a29a30a31a98a32a17.
(2) a45a46a47a29ξ /∈W,a65a18a62a26a33,a73a74η negationslash= 0 ∈W +L(ξ),a64
a68f(η,α) = 0?α ∈ W,
a132η ∈W⊥.
a139η = α+aξ,a57a100W ∩W⊥ = 0,a118a48anegationslash= 0,a26a49
ξ = a?1η?a?1α∈W⊥ +W.
a33
a68V?W⊥ +W.
3,a54
a105a151
a109a110T,a64TTATa72a45a170a171,a58a59Aa72a88a89a109a110:
(1)
1 1 0
1 2 2
0 2 5
; (2)
1?2 1
2 4 2
1 2 1
;
· 6 ·
(3)
0 1 1
1 0 1
1 1 0
; (4)
1 1 1
1 1 1
1 1 1
;
a55,(1)
a120T =
1?1 2
0 1?2
0 0 1
,a57TTAT =
1 0 0
0 1 0
0 0 1
.
(2)a120T =
1 0?1
0 14? 14
0 12 12
,a57TTAT =
1 0 0
0 1 0
0 0?1
.
(3)a120T =
1?1 1
0 1 0
1?1?1
,a57TTAT =
2 0 0
0?2 0
0 0?2
.
(4)a120T =
1?1?1
0 1 0
0 0 1
,a57TTAT =
1 0 0
0 0 0
0 0 0
.
4,a33a34:
λ1
λ2
...
λn
a96
λi1
λi2
...
λin
a123a172,
a58a59i1,···,ina151,···,na29a35a36a173a89.
a37a38:
a164a165na51a30a31a32a17V,a14fa72V a18a29a45a153a93a30a31a24a25,a52a74a53η1,···,ηna88a29a107a108a109a110a72
λ1
λ2
...
λn
,
a77a67η
i1,···,ηin
a174
a15V a29a53,a44fa74ηi1,···,ηina88a29a107a108a109a110a72
λi1
λi2
...
λin
,
a100a101
a83a175
a36a109a110
a123a172.
5,a33a34,a176a177a99ra29a45a153a109a110
a105
a49a178a72ra36a176a177
a991
a29a45a153a109a110a179a180.
a37a38:
a14Aa15a176a72ra29a45a153a109a110,a57a73a74
a105a151
a109a110T,a64
a68
TTAT =
a1
...
ar
0
...
0
,ai negationslash= 0.
· 7 ·
a56
Ai = T?T
0
...
ai
0
...
0
T?1,
a57A
ia106
a15a45a153a109a110,rankAi = 1a44A = A1 +A2 +···+Ar.
6,a14Aa72a23a109a110,a33a34,ATA
a96
Aa29a176
a123
a177.
a37a38,a77a67,ATA
a15a23a45a153a109a110,a164a165a23a25a50a18a29a112a113a30a31a114a115a27
ATAX = 0 (1)
a96
AX = 0,(2)
a39a40(2)a29
a69
a97
a15(1)a29
a69.
a14X ∈ Rna72(1)a29a35a36
a69.
a56
Y = AX =
y1
...
yn
.
a57
YTY = XTATAX = 0,
a70a71
y21 +y22 +···+y2n = 0.
a65a99y
ia181
a72a23a25,a100a101y1 = y2 = ··· = yn = 0,Y = 0,a132
AX = 0.
a70a71(1)a29
a69
a106a97
a15(2)a29
a69,(1)
a96
(2)
a61
a69,a65
a112a113a30a31a114a115a27
a69
a29a31a182
a67
rankATA = rankA.
7,a14Aa72a183a125a109a110,a33a34,A?1
a96
A?
a97
a15a183a125a109a110.
a37a38,a77a67A?1
a96
A?
a97
a15a23a45a153a109a110,a44A? = |A| ·A?1,a100Aa183a125,a73a74
a105a151
a23a109a110C a64
CTC = A,a70a71A?1 = C?TC?1
a106
a183a125,a65|A|> 0
a105
a67A? = |A|·A?1
a106
a183a125.
8,a33a34,a46a47a35a36a93a30a31a24a25
a97a105
a75a35a178a72a35a36a45a153a93a30a31a24a25a180a35a36a155a153a93a30a31a24a25a179a180.
a37a38,(1)
a14f(α,β)a15a35a36a93a30a31a24a25,a77a67
g(α,β) = 12 [f(α,β) +f(β,α)]
a15a45a153a93a30a31a24a25,
h(α,β) = 12 [f(α,β)?f(β,α)]
a72a155a153a93a30a31a24a25,a44
f(α,β) = g(α,β) +h(α,β).
(2)a103a14
f(α,β) = gprime(α,β) +hprime(α,β),
· 8 ·
a58a59gprime(α,β)a15a45a153a93a30a31a24a25,hprime(α,β)a15a155a153a93a30a31a24a25,a57
f(β,α) = gprime(β,α) +hprime(β,α) = gprime(α,β)?hprime(α,β).
a70a71
gprime(α,β) = 12 [f(α,β) +f(β,α)] = g(α,β),
hprime(α,β) = 12 [f(α,β)?f(β,α)] = h(α,β).
9.
a33a34,a93a30a31a24a25fa184a48a183a185a45a153a31a29a116a117a118a119a81a82a15fa72a45a153a154a155a153a93a30a31a24a25.
a37a38:
a116a117a31a15a39a40a29,a88a186a33a118a119a31.
(1)a121a45a46a47a29α∈V
a97
a48f(α,α) = 0,a57a45a46a47a29α,β ∈V,
0 = f(α+β,α+β) = f(α,α) +f(α,β) +f(β,β) +f(β,β) = f(α,β) +f(β,α).
a100a101f(α,β) =?f(β,α),fa15a155a153a93a30a31a24a25.
(2)a121a152a73a74γ ∈V a64f(γ,γ) negationslash= 0,a57a45a46a47a29α∈V,a65a99
f
parenleftbigg
α? f(α,γ)f(γ,γ) γ,γ
parenrightbigg
= f(α,γ)?f(α,γ) = 0,
a26a49f
parenleftbigg
γ,α? f(α,γ)f(γ,γ) γ
parenrightbigg
= 0,a100a101
f(α,γ) = f(γ,α),(*)
a45
a99
a46a47a29α,β ∈V,a49a88a187a117
a175a188a189a190a191a159:
(a)a121a152f(α,γ) negationslash= 0,a57
f
parenleftbigg
α,β? f(α,β)f(α,γ) γ
parenrightbigg
= f(α,β)?f(α,β) = 0,
a100a101f
parenleftbigg
β? f(α,β)f(α,γ) γ,α
parenrightbigg
= 0,a70a71
0 = f(β,α)? f(α,β)f(α,γ) f(γ,α)
= f(β,α)? f(α,β)f(α,γ) f(α,γ) a65(*)
= f(β,α)?f(α,β),
a132f(α,β) = f(β,α).
(b) a121a152f(α,γ) = 0,a57
f
parenleftbigg
α+γ,β? f(α,β) +f(γ,β)f(γ,γ) γ
parenrightbigg
= f(α,β) +f(γ,β)?f(α,β)?f(γ,β) = 0,
a100a101f
parenleftbigg
β? f(α,β) +f(γ,β)f(γ,γ) γ,α+γ
parenrightbigg
= 0,a70a71
f(β,α) +f(β,γ)?f(α,β)?f(γ,β) = 0.
a65(*)a67f(β,γ) = f(γ,β),
a100a101f(α,β) = f(β,α).
a65(a)
a180(b)
a105
a68f
a72a45a153a93a30a31a24a25.
10.
a14V a15a192a25a50a18a29a30a31a32a17,a58a51a25ngreaterorequalslant 2,fa15V a18a29a35a36a45a153a93a30a31a24a25,a33a34:
(1) V a59a48
a102a157
a111a108ξ,a64f(ξ,ξ) = 0;
(2)a94fa15
a102a142a143
a95,a118a48a30a31a162a163a29a111a108ξ,η,a78a79:
f(ξ,η) = 1,
f(ξ,ξ) = f(η,η) = 0.
· 9 ·
a37a38,(1) a65a99dimV greaterorequalslant 2.
a46a120V a29
a175
a36a30a31a162a163a29a111a108α,β,a121a152f(α,α) = 0,a57ξ = αa132a72a26
a54.
a161a14f(α,α) negationslash= 0,a572a113a114a115
t2f(α,α) + 2tf(α,β) +f(β,β) = 0 (*)
a74a192a25a193a194a195a48
a69.
a14t0 ∈Ca15ta29a35a36
a69.
a56
ξ = t0α+β,
a57ξ negationslash= 0 (
a100α,βa30a31a162a163),a44
f(ξ,ξ) = t20f(α,α) + 2t0f(α,β) +f(β,β) = 0.
a70a71ξ = t0α+βa132a72a26
a54.
(2) a65(1)a26a33,a73a74ξ negationslash= 0 ∈V a64f(ξ,ξ) = 0,a103a100f
a102a142a143
,a168a73a74α∈V a64f(ξ,α) negationslash= 0.
(a)a121f(α,α) = 0,a57a56η = 1f(ξ,α) α,a132a48
f(ξ,ξ) = f(η,η) = 0,f(ξ,η) = 1.
(b) a121f(α,α) negationslash= 0,a57a120
η = 1f(α,ξ) α? f(α,α)2(f(α,ξ))2 ξ,
a196a197
a131a33
a105
a67f(η,η) = 0,f(ξ,η) = 1,
a71ξ,ηa29a30a31a162a163a31a15a39a40a29,a168ξ,ηa132a72a26
a54.
11.
a33a34,a121a152a30a31a32a17V a18a29a45a153a93a30a31a24a25fa198a117
a69
a72
a175
a36a30a31a24a25a179a199:
f(α,β) = f1(α)f2(β),?α,β ∈V,
a57
a73a74
a102a157
a25λa60a30a31a24a25g,a64
f(α,β) = λg(α)g(β).
a37a38:
a121a152f = 0,a57a158a159a94a40a28a160,a161a14f negationslash= 0,a100a101a73a74α0,β0 ∈V,a64
a68f(α
0,β0) negationslash= 0,a125a126
g, V?→ K
γ mapsto?→ f(α0,γ)
a57g
a72V a18a30a31a24a25,a44g negationslash= 0,a45a46a47a29β ∈V,
g(β) = f(α0,β) = f1(α0)f2(β)
g(β) = f(α0,β) = f(β,α0) = f1(β)f2(α0)
a39a40f1(α0) negationslash= 0,f2(α0) negationslash= 0 (a92
a57g negationslash= 0),a65
a101
a67,
f1(β) = 1f
2(α0)
g(β)
f2(β) = 1f
1(α0)
g(β)?β ∈V.
a56λ = 1f
1(α0)f2(α0)
,a57
f(α,β) = f1(α)f2(β) = 1f
2(α0)
g(α)· 1f
1(α0)
g(β) = λg(α)g(β).
12.
a14Aa72a200a183a125a109a110,a33a34,A?
a106
a15a200a183a125a109a110.
a37a38:
a121a152rankA = n,a57Aa15a183a125a109a110,a145a627a66a33a34
a85A?
a183a125,a121a152rankAlessorequalslantn?2,a57A? = 0,
a70a71A?a200a183a125,a201a202a164a203rankA = n?1a29
a189
a171,a101a95rankA? = 1,a70a71A?a29a148a25greaterorequalslant 2a29a204a98a205
a97
a15
0,a71A?a291a148a204a98a205= Aii (i = 1,···,n) = Aa29aiia29a129a25a206a98a205(i = 1,···,n) = Aa29aiia29a206a98a205
(i = 1,···,n) = Aa29n?1a148a204a98a205greaterorequalslant 0 (a100Aa200a183a125),a26a49A?a200a183a125.
· 10 ·
13.
a33a34a125
a104
2.12.
a37a38,(1)?(2)
a14Aa200a183a125,a57a73a74
a105a151
a23a109a110T,a64
TTAT =
a1
...
ar
0
...
0
,ai negationslash= 0.
a65a99A
a200a183a125,TTAT
a106
a200a183a125,a168ai > 0,a26a49Aa29a183a207a31a208a25p = r = rankA;
(2)?(3) a65a209a14,a73a74
a105a151
a23a109a110T1,a64
TT1 AT1 =
a1
...
ar
0
...
0
,ai > 0.
a56
T2 =
1√
A1
...
1√
Ar
1
...
1
,T = T1T2,
a57
TTAT =
1
...
1
0
...
0
=
parenleftbiggE
r 0
0 0
parenrightbigg
.
(3)?(4) a65a209a14,a73a74
a105a151
a23a109a110T,a64
TTAT =
parenleftbiggE
r 0
0 0
parenrightbigg
.
a56
S =
parenleftbiggE
r 0
0 0
parenrightbigg
T,
a57
A = STS.
(4)?(1)a45a46a47a29X negationslash= 0 ∈ Rn,a56Y = SX,a57Y ∈Rn,a26a49
XTAX = XTSTSX = YTY greaterorequalslant 0,
Aa200a183a125.
· 11 ·
(1)?(5)a14Bk = A(i1,···,ik;i1,···,ik)a15Aa29a35a36a204a98a205.a57a45a46a47a29
Xk =
x1
...
xk
∈Rk,
a105
a49a210a35a36a89a111a108X ∈Rn,a64
a68
a52a29
a122i
ja89a29a91a211a177
a99x
j,a71a58a206a91a211a181
a177
a990,a57
0 lessorequalslantXTAX = XTk BkXk,
a100a101Bra15a200a183a125a29,a212a213(4),
a105
a68
a200a183a125a109a110a29a214a89a205
a102a215
,a132|Bk|greaterorequalslant 0.
(5)?(1)a45
a99
a46a47a29a183a23a25λ> 0,a164a165λE +Aa29ka148a204a98a110λEk +Ak,a83a36a98a109a110a29a214a89a205a72
fk(λ) = |λEk +Ak| = λk +a1λk?1 +···+ak.
a57
a212a213
a145
a627–3.8,(?1)iaia177
a99?A
ka29a19a216ia148a204a98a205a179a180,a71?Aka29a217a36ia148a204a98a205a177
a99A
ka29
a123a124i
a148a204a98a205a29(?1)ia218.a100a101aia177
a99A
ka29a26a48ia148a204a98a205a179a180,
a65a209
a14,ai greaterorequalslant 0,a70a71
fk(λ) > 0?λ> 0,i = 1,···,k.
a212a213a125
a104
2.11,λE +A(λ> 0)a15a183a125a109a110.
a46a120X negationslash= 0 ∈ Rn,a213a183a125a31,λa29a35a113a205
g(λ) = XT(λE +A)X = λXTX +XTAX > 0,?λ> 0.
a100a101XTAX greaterorequalslant 0 (a92
a57
a94λa116a117a166a95a219a48g(λ) < 0),a70a71Aa200a183a125.
14.
a204a45a170a30a18a19a151a29a18a220a170a171a109a110a153a72a221a222a18a220a170a171a109a110.
(1) a14Aa15a35a36a45a153a109a110,Ta72a221a222a18a220a170a171a109a110,a33a34,TTAT
a96
Aa29a45
a124a223a224
a204a98a205a48
a123
a61
a29
a225;
(2)a121a152a45a153a109a110a29
a223a224
a204a98a205a19
a169
a72
a157
,a57a73a74a35a221a222a18a220a170a171a109a110T,a64TTATa72a45a170a171.
a37a38,(1)
a14Ara72Aa29ra148
a223a224
a204a98a205(1 lessorequalslantrlessorequalslantn),
A =
parenleftbiggA
r?
parenrightbigg
.
a14Ta72a221a222a18a220a170a171a109a110,
T =
parenleftbiggT
11?
0 T22
parenrightbigg
,a58a59T11 =
1?
...
0 1
r
,
a57
TTAT =
parenleftbiggTT
11 0
TT22
parenrightbiggparenleftbiggA
r?
parenrightbiggparenleftbiggT
11?
0 T22
parenrightbigg
=
parenleftbiggTT
11ArT11?
parenrightbigg
.
a70a71TTATa29ra148
a223a224
a204a98a205a177
a99(
a226a47a41|T11| = 1)
|TT11AT11| = |TT11||A||T11| = |Ar|.
(2)a45Aa29a148a25a227a228a229a230,a120
T1 =
parenleftbiggE
n?1?A?1n?1B
0 1
parenrightbigg
,An?1 = A(1,···,n?1;1,···,n?1),B =
a1n
...
an?1,n
.
a83
a15a221a222a18a220a170a171a109a110,a57
TT1 AT1 =
parenleftbigg E 0
BTA?1n?1 1
parenrightbiggparenleftbiggA
n?1 B
BT ann
parenrightbiggparenleftbiggE?A?1
n?1B
0 1
parenrightbigg
=
parenleftbiggA
n?1 0
0 bn
parenrightbigg
,
· 12 ·
a58a59bn = ann?BTAn?1B,a65a99Aa29
a223a224
a204a98a205a19
a169
a720,a168An?1a29
a223a224
a204a98a205a19
a169
a720,a65a228a229
a209
a14,a73a74n?1a148a221a222a18a220a170a171a109a110T2a64
TT2 An?1T2 =
b1
...
bn?1
.
a56
T = T1
parenleftbiggT
2 0
0 1
parenrightbigg
,
a57T
a72a221a222a18a220a170a171a109a110,a44
TTAT =
b1
...
bn
.
a12 a13 8–3
1,a227
a102a142a143
a30a31a231a232
a143
a88a89a90a113a233a72a234a114a180:
(1) x21 + 5x22?4x23 + 2x1x2?4x1x3;
(2) 4x21 +x22 +x23?4x1x2 + 4x1x3?3x2x3;
(3) x1x2 +x1x3 +x2x3;
(4) 2x21 + 18x22 + 8x23?12x1x2 + 8x1x3?27x2x3;
(5) x21?2x1x2 + 2x1x3?2x1x4 +x22 + 2x2x3?4x2x4 +x23?2x24;
(6) x21 +x1x2 +x2x4.
a55,(1) x2
1 + 5x22?4x23 + 2x1x2?4x1x3 = (x1 +x2?2x3)2 + (2x2 +x3)2?(3x3)2,a56
y1 = x1 +x2?2x3
y2 = 2x2 +x3
y3 = 3x3
a132
x1 = y1? 12 y2 + 56 y3
x2 = 12 y2? 16 y3
x3 = 13 y3
a48
f(x1,x2,x3) = y21 +y22?y23.
(2)a235a205= (2x1?x2 +x3)2 +
parenleftBigx
2?x32
parenrightBig2
parenleftBigx
2 +x32
parenrightBig2
,a56
y1 = 2x1?x2 +x3
y2 = x2?x32
y3 = x2 +x32
a132
x1 = 12 y1 +y2
x2 = y2 +y3
x3 =?y2 +y3
a57
f(x1,x2,x3) = y21 +y22?y23.
(3)a56?
x1 = y1?y2?y3
x2 = y1 +y2?y3
x3 = y3
a48
f(x1,x2,x3) = y21?y22?y23.
· 13 ·
(4)a235a205= 2(x1?3x2 + 2x3)2 +
parenleftBig3x
2?x32
parenrightBig2
parenleftBig3x
2 +x32
parenrightBig2
,a56
y1 = x1?3x2 + 2x3
y2 = 3x2?x32
y3 = 3x2 +x32
a132
x1 = y1 + 3y2?y3
x2 = 13 y2 + 13 y3
x3 =?y2 +y3
a57
f(x1,x2,x3) = 2y21 +y22?y23.
(5)a56?
x1 = y1?y3?y4
x2 = 12 y2? 12 y3? 12 y4
x3 = 12 y2 + 12 y3 + 32 y4
x4 = y4
a57
a48
f(x1,x2,x3,x4) = y21 +y22?y23.
(6)a235a205= x21 +
parenleftBigx
2 +x1 +x42
parenrightBig2
parenleftBigx
2?x1?x42
parenrightBig2
,a56
y1 = x1
y2 = x2 +x1 +x42
y3 = x2?x1?x42
y4 = x3
a132
x1 = y1
x2 = y2 +y3
x3 = y4
x4 =?y1 +y2?y3
a57
f(x1,x2,x3,x4) = y21 +y22?y23.
2,λa120a236a225a95,a88a89a90a113a233a15a183a125a29:
(1) 5x21 +x22 +λx23 + 4x1x2?2x1x3?2x2x3;
(2) 2x21 +x22 + 3x23 + 2λx1x2 + 2x1x3;
(3) 2x21 + 2x22 +x23 + 2λx1x2 + 6x1x3 + 2x2x3.
a55,(1) A =
5 2?1
2 1?1
1?1 λ
,
a52a29
a223a224
a204a98a205D1 = 5 > 0,D2 = 1 > 0,D3 = λ? 2,a26a49a94
λ> 2a95a235a90a113a233a183a125.
(2)a90a113a233a109a110a29
a223a224
a204a98a205D1 = 2 > 0,D2 = 2?λ2,D3 = 5?3λ2.
a65D
2 > 0,
a68|λ|<√2;
a65D
3 > 0,
a68|λ|<
radicalBig5
3,
a26a49a94?
√15
3 <λ<
√15
3 a95a235a90a113a233a183a125.
(3) a90a113a233a109a110a29
a223a224
a204a98a205D1 = 2,D2 = 4?λ2,D3 =?λ2 + 6λ?16 =?(λ?3)2?7 < 0,a168
a169
a159λ
a120a236a23a25
a97a169
a198a64a101a90a113a233a183a125.
3,a88a89a90a113a233a15a92a183a125a154a200a183a125:
(1)
nsummationtext
i=1
x2i + summationtext
1lessorequalslanti<jlessorequalslantn
xixj; (2)
nsummationtext
i=1
x2i +
n?1summationtext
i=1
xixi+1;
(3) n
nsummationtext
i=1
x2i?
parenleftbigg nsummationtext
i=1
xi
parenrightbigg2
.
· 14 ·
a55,(1)
a90a113a233a109a110A =
1 12 ··· 12
1
2 1 ···
1
2.
..,..,..,..
1
2
1
2 ··· 1
,a52a29
a223a224
a204a98a205
Dr = |Ar| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 12 ··· 12
1
2 1 ···
1
2.
..,..,..,..
1
2
1
2 ··· 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
r
= 12r (r+ 1) > 0,r = 1,···,n.
a168a235a90a113a233a183a125.
(2) a235a205f = 12 x21 + 12 (x1 + x2)2 + 12 (x2 + x3)2 + ··· + 12 (xn?1 + xn)2 + 12 x2n greaterorequalslant 0,a100a101
f = 0 x1 = 0,x1 +x2 = 0,x2 +x3 = 0,···,xn?1 +xn = 0,xn = 0 x1 = x2 = ··· = xn = 0.
a168a235a90a113a233a183a125.
(3) a235a205= (?1)
nsummationtext
i=1
x2i?2 summationtext
1lessorequalslanti,jlessorequalslantn
xixj = summationtext
1lessorequalslanti,jlessorequalslantn
(xi?xj)2 greaterorequalslant 0,a120x1 = x2 = ··· = xn negationslash= 0
a105
a64a101
a90a113a233a120
a157
a225.
a100a101a235a90a113a233a200a183a125.
4,a14Aa72a23a45a153a109a110,a33a34:
(1)a94a23a25λa116a117a237a179a202,λE +Aa15a183a125a29;
(2) Aa200a183a125a94a44a238a94a45a46a236a29λ> 0,λE +A
a97
a183a125.
a37a38,(1)
a164a165A(λ) = λE +A,a52a29ra148
a223a224
a204a98a205
Dr(λ) = |λEr +Ar| = λr +a1λr?1 +···+ar.
a26a49a94λa116a117a237a95,a48Dr(λ) > 0,r = 1,···,n,a70a71a94λa116a117a237a95,λE +Aa183a125.
(2) (?)a239Aa200a183a125,a57a45a46a47a29X negationslash= 0 ∈Rn,XTAX greaterorequalslant 0,a70a71a45a46a47a29λ> 0a48
XT(λE +A)X = λXTX +XTAX > 0.
a168λE +Aa183a125.
(?) a45a46a47a29λ> 0a60X negationslash= 0 ∈Rn,a48
XT(λE +A)X = λXTX +XTAX > 0,
a70a71XTAX greaterorequalslant 0,a168Aa200a183a125.
5,a14A,B,Ca72a220a170a171a29a220a36a195a170,a33a34,a45a46a47a23a25x,y,za48
x2 +y2 +z2 greaterorequalslant 2xycosA+ 2xzcosB + 2yzcosC.
a37a38:
a164a165a90a113a233f(x,y,z) = x2 +y2 +z2?2xycosA?2xzcosB?2yzcosC.
f(x,y,z) = (x?ycosA?zcosB)2 +y2 sin2A+z2 sin2B?2yzcosAcosB?2yzcosC
= (x?ycosA?zcosB)2 +y2 sin2A+z2 sin2B?2yzsinAsinB
= (x?ycosA?zcosB)2 + (ysinA?zsinB)2.
a70a71fa200a183a125,a65a101
a67a158a159
a28a160.
· 15 ·
6,a33a34,a239
nsummationtext
i=1
nsummationtext
j=1
aijxixj (aij = aji)a15a183a125a90a113a233,a57
f(y1,y2,···,yn) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a11 a12 ··· a1n y1
a21 a22 ··· a2n y2
........................
an1 an2 ··· ann yn
y1 y2 ··· yn 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a15
a215
a125a90a113a233.
a37a38,a66a67
parenleftbigg E 0
YTA?1 1
parenrightbiggparenleftbigg A Y
YT 0
parenrightbigg
=
parenleftbiggA Y
0?YTA?1Y
parenrightbigg
,a26a49
f(y1,···,yn) =
vextendsinglevextendsingle
vextendsinglevextendsingle A Y
YT 0
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleA Y
0?YTA?1Y
vextendsinglevextendsingle
vextendsinglevextendsingle
= |A|(?YTA?1Y) = YT(?A?)Y.
a65A
a183a125
a105
a68A?
a183a125,a99a15?A?
a215
a125,a100a101f(y1,···,yn) = YT(?A?)Y a15
a215
a125a90a113a233.
7.
a14a48a23a240a25a90a113a24a25
f(x1,x2,···,xn) =
nsummationdisplay
i=1
nsummationdisplay
j=1
aijxixj +
nsummationdisplay
i=1
2bixi +c,aij = aji.
a56
A =
a11 a12 ··· a1n
a21 a22 ··· a2n
....................
an1 an2 ··· ann
,D =
a11 a12 ··· a1n b1
a21 a22 ··· a2n b2
........................
an1 an2 ··· ann bn
b1 b2 ··· bn c
.
(1)a33a34,a94A
a215
a125a95,fa48a201a237
a225,
a44fmax = |D||A| ;
(2)a14A
a215
a125,a63a241a125a94x1,···,xna72a236
a225
a95,fa120
a68
a201a237
a225.
a55,(1)
a120
T =
parenleftbiggE
n?A?1B
0 1
parenrightbigg
,B =
b1
...
bn
,
a56
y1
...
yn
yn+1
= T?1
x1
...
xn
1
,(*)
· 16 ·
a77a67y
n+1 = 1.
a57
f(x1,···,xn) = (x1 ··· xn 1)D
x1
...
xn
1
= (y1 ··· yn 1)TTDT
y1
...
yn
1
= (y1 ··· yn 1)
parenleftbiggA 0
0 d
parenrightbigg
y1
...
yn
1
= YTAY +d.
a65a99A
a215
a125,a168a45a46a47a29Y ∈ R
n
a48YTAY lessorequalslant 0,a26a49f lessorequalslantd.
a105a242
fa48a243a237
a225d,
a44a94Y = 0a95fa120a243a237
a225.a83a244
d = |A|d|A| = |T
TDT|
|A| =
|D|
|A|,
(2) a65(*),?
x1
...
xn
1
= T
y1
...
yn
1
=
parenleftbiggE
n?A?1B
0 1
parenrightbigg
y1
...
yn
1
,
a68X = Y?A?1B.
a94Y = 0a95X =?A?1B,a132a94?
x1
...
xn
=?A?1
b1
...
bn
a95,fa120a201a237
a225.
8,a245a246a247a248a249Aa188a249a250x(a251)a36a180Ba188a249a250y(a251)a36a29a252a28a253a24a25a72:
C(x,y) = x2 + 2xy+y2 + 100(a254a91).
a255a0a175a188
a249a250a29a80
a54
a24a25a72:
x = 26?pA,y = 10? 14pB,
a58a59pA,pBa72a249a250
a123a124
a29a1a2(a254a91/a251a36),a54a3a4a201a237a95a249a250a29a25a108a180
a3a4.
a55:
a213a62a47,a3a4a24a25a72
p(x,y) = xpa +ypb?C(x,y)
= x(26?x) +y(40?4y)?C(x,y)
=?2x2?2xy?5y2 + 26x+ 40y?100.
a253a62
a84
a15
a54
a90a113a24a25a29a201a237
a225.
a14
A =
parenleftbigg?2?1
1?5
parenrightbigg
,D =
2?1 13
1?5 20
13 20?100
,B =
parenleftbigg13
20
parenrightbigg
.
· 17 ·
a83a244A
a15
a215
a125a109a110,a212a213
a145
a627,a94parenleftbigg
x
y
parenrightbigg
=?A?1B =?A?1
parenleftbigg13
20
parenrightbigg
=? 19
parenleftbigg?5 1
1?2
parenrightbiggparenleftbigg13
20
parenrightbigg
=
parenleftbigg5
3
parenrightbigg
a95,a3a4a201a237,a44a201a237
a3a4
a72
pmax = |D||A| = 25a254a91.
a168a94
a175a188
a249a250a117a5a1a128500a36
a96
300a36a95,
a105a6
a201a237
a3a425
a254a91.
a12 a13 8–4
1,a54a183a185a109a110T,a64T?1ATa72a45a170a171,a14Aa72a88a89a109a110:
(1)
2?2 0
2 1?2
0?2 0
; (2)
2 2?2
2 5?4
2?4 5
;
(3)
2?2 0 1
2 2 1 0
0 1 2?2
1 0?2 2
; (4)
1?1 3?2
1 1?2 3
3?2 1?1
2 3?1 1
;
(5)
1?3 3?3
3?1?3 3
3?3?1?3
3 3?3?1
; (6)
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
.
a55,(1) T = 1
3
2 2 1
1?2 2
2 1 2
,T?1AT =
1 0 0
0 4 0
0 0?2
.
(2) T = 13
2 2 1
2 1 2
1 2?2
,T?1AT =
1 0 0
0 1 0
0 0 10
.
(3) T = 12
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
,T?1AT =
1 0 0 0
0?1 0 0
0 0 3 0
0 0 0 5
.
(4) T = 12
1 1 1 1
1 1?1?1
1?1?1 1
1?1 1?1
,T?1AT =
1 0 0 0
0?1 0 0
0 0?3 0
0 0 0 7
.
(5) T = 12
1 1 1 1
1?1 1?1
1?1?1 1
1 1?1?1
,T?1AT =
4 0 0 0
0?4 0 0
0 0?4 0
0 0 0 8
.
(6) T = 12
1 1 1 1
1?1 1?1
1?1?1 1
1 1?1?1
,T?1AT =
4 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
.
2,a227a183a185a30a31a231a232
a143
a88a89a23a90a113a233a72a7a193a171:
(1) x21 + 2x22 + 3x23?4x1x2?4x2x3;
(2) x21?2x22?2x23?4x1x2 + 4x1x3 + 8x2x3;
· 18 ·
(3) 3x21 + 6x22 + 3x23?4x1x2?8x1x3?4x2x3;
(4) 2x1x2 + 2x3x4;
(5) x21 +x22 +x23 +x24 + 2x1x2?2x1x4?2x2x3 + 2x3x4;
(6) 2x1x2 + 2x1x3 + 2x1x4 + 2x2x3 + 2x2x4 + 2x3x4.
a55,(1)
a210a183a185a30a31a231a232?
x1 = 23 y1? 23 y2 + 13 y3
x2 = 23 y1 + 13 y2? 23 y3
x3 = 13 y1 + 23 y2 + 23 y3
a105a8
a90a113a233
a143
a72a7a193a171
y21 + 2y22 + 5y23.
(2)a210a183a185a30a31a231a232?
x1 =? 2
√5
5 y1?
2√5
15 y2 +
1
3 y3
x2 =
√5
5 y1?
4√5
15 y2 +
2
3 y3
x3 =
√5
3 y2 +
2
3 y3
a105a8
a90a113a233
a143
a72a7a193a171
2y21 + 2y22?7y23.
(3)a210a183a185a30a31a231a232?
x1 = 23 y1? 23 y2 + 13 y3
x2 = 13 y1 + 23 y2 + 23 y3
x3 = 23 y1 + 13 y2? 23 y3
a105a8
a90a113a233
a143
a72a7a193a171
2y21 + 7y22 + 7y23.
(4)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1 + 12 y2? 12 y3? 12 y4
x3 = 12 y1? 12 y2 + 12 y3? 12 y4
x4 = 12 y1? 12 y2? 12 y3 + 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
y21 +y22?y23?y24.
(5)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1? 12 y2? 12 y3 + 12 y4
x3 = 12 y1 + 12 y2? 12 y3? 12 y4
x4 = 12 y1? 12 y2 + 12 y3? 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
y21 +y22?y23 + 3y24.
· 19 ·
(6)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1 + 12 y2? 12 y3? 12 y4
x3 = 12 y1? 12 y2 + 12 y3? 12 y4
x4 = 12 y1? 12 y2? 12 y3 + 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
3y21?y22?y23?y24.
3,a14Aa72a35a36na148a23a109a110,a44|A| negationslash= 0,a33a34,A
a105
a117
a69
a28
A = QT,
a58a59Qa15a183a185a109a110,Ta15a18a220a170a171a109a110.
a37a38:
a120na51a9a10
a244a68
a32a17V,a14η1,···,ηna15a52a29a35a36a11a193a183a185a53,a56
(α1,···,αn) = (η1,···,ηn)A (*)
a65a99|A| negationslash= 0,A
a105a151
,a100a101α1,···,αna15V a29a53,a124a227a12a13a14–a15a16a17a183a185
a143
a114a230,
a105
a68V
a29a35a36a11a193a183
a185a53β1,···,βn,a56
(α1,···,αn) = (β1,···,βn)T,
a65a122a18a19
a125
a104
3.4a29a33a34
a105
a67,T
a72a18a220a170a171a109a110,a56
(β1,···,βn) = (η1,···,ηn)Q,
a57Q
a72a183a185a109a110,a44
(α1,···,αn) = (β1,···,βn)T = (η1,···,ηn)QT,
a96
(*)a20
a21,a68A = QT.
4,a14Aa72a35a36na148a183a125a109a110,a33a34,a73a74a18a220a170a171a109a110T,a64
A = TTT.
a37a38,a65A
a183a125
a105
a67
a73a74
a105a151
a23a109a110Ba64
a68A = BTB,a65
a18a62,a73a74a183a185a109a110Q
a96
a18a220a170a171a109a110
T,a64
a68
B = QT.
a70a71
A = BTB = TTQTQT = TTT.
5,a14Aa72a23a45a153a109a110,a33a34,Aa183a125(a200a183a125)a29a116a117a118a119a81a82a15Aa29a17a22
a225
a19a237
a99(
a237
a99
a177
a99)
a157
.
a37a38:
a73a74a183a185a109a110T,a64
a68
T?1AT = TTAT =
λ1
λ2
...
λn
,
a58a59λ1,λ2,···,λn a72Aa29a19a216a17a22
a225,a99
a15Aa183a125(a200a183a125) TTAT a183a125(a200a183a125) λi >
0 (λi greaterorequalslant 0),i = 1,···,n.
6,a33a34,a175a36a23a45a153a109a110
a123a23
a29a116a117a118a119a81a82a15a52a135a48
a123
a61
a29a17a22a24a25a205.
· 20 ·
a37a38:
a118a119a31a39a40,a88a33a116a117a31,a14a23a45a153a109a110A,Ba48
a123
a61
a29a17a22a24a25a205,a70a71a52a135a48
a123
a61
a29a17
a22
a225λ
1,λ2,···,λn.
a99
a15a73a74a183a185a109a110T
a96
Q,a64
a68
T?1AT = TTAT =
λ1
λ2
...
λn
,
Q?1BQ = QTBQ =
λ1
λ2
...
λn
.
a39a40A
a96
Ba123a23.
7,a14Aa72na148a23a109a110,a33a34,a73a74a183a185a109a110T,a64T?1ATa72a220a170a171a109a110a29a116a117a118a119a81a82a15Aa29a17
a22
a225
a19a15a23a25.
a37a38,(?)
a14a48a183a185a109a110Ta64
T?1AT =
λ1?
λ2
...
0 λn
,
a57λ
1,···,λna26
a72Aa29na36a17a22
a225,a65a99A,T
a181
a72a23a109a110,T?1AT
a106
a15a23a109a110,a168a17a22
a225λ
1,···,λna97
a15a23a25.
(?) a45na227a228a229a230,n = 1a95a158a159a39a40a28a160,a161a74a209a14a158a159a45n?1a148a78a79a81a82a29a23a109a110a28a160,a164
a165na148a23a109a110A.
a14V a72na51a9a10
a244a68
a32a17,η1,···,ηna72V a29a11a193a183a185a53,a56a27 ∈ End(V),a64
a68
(a27η1,···,a27ηn) = (η1,···,ηn)A.
a14λ1 ∈ Ra72a27a29a46a47a17a22
a225,a57λ
1a106
a15a27a29a17a22
a225.
a56α1a72a27a29a28
a99
a17a22
a225λ
1a29a149a150a17a22a111a108,
a57
α1
a105a29
a116a72V a29a11a193a183a185a53α1,α2,···,αn,a56
(a27α1,···,a27αn) = (α1,···,αn)
λ1?
0.
.,A1
0
,
(α1,···,αn) = (η1,···,ηn)T1,
a57T
1a72a183a185a109a110,a44 parenleftbigg
λ1?
0 A1
parenrightbigg
= T?11 AT1.
a77a67A
1a72n?1a148a23a109a110,a44a58a17a22
a225
a19a15Aa29a17a22
a225,
a70a71
a106a97
a15a23a25,a65a228a229
a209
a14,a73a74a183a185a109a110
T2,a64
T?12 A1T2 =
λ2?
...
0 λn
,
a56
T = T1
parenleftbigg1 0
0 T2
parenrightbigg
,
· 21 ·
a57T
a72a183a185a109a110,a44
T?1AT =
λ1?
λ2
...
0 λn
,
a65
a228a229a230a235
a104
a67a158a159
a28a160.
8,a33a34,a17a22
a225
a19a15a23a25a29a183a185a110a118a15a45a153a109a110.
a37a38:
a14Aa72a17a22
a225
a19a15a23a25a29a183a185a110,a65a18a62,a73a74a183a185a109a110T,a64
TTAT = T?1AT = D
a72a18a220a170a171a110,a103a100a72Da15a183a185a110,a168D?1 = DT
a106
a15a18a220a170a171a109a110,a30a52a103a15a88a220a170a171a110,a168Da15a45
a170a110,a70a71Da72a45a153a109a110,a168
A = TDTT
a106
a72a45a153a109a110.
9.
a14A,Ba15
a175
a36na148a23a45a153a109a110,a44Ba183a125,a33a34,a73a74
a105a151
a109a110T,a64
TTAT
a96
TTBT
a61
a95a72a45a170a171.
a37a38,a65a99B
a183a125,a100a101a73a74
a105a151
a109a110Sa64
a68
STBS = E.
a71STAS
a174
a72a23a45a153a109a110,a168a73a74a183a185a109a110Qa64D = QT(STAS)Qa72a45a170a110,a56T = SQ,a57T
a105a151
,a44
TTAT = D,TTBT = E,
a181
a72a45a170a110.
10.
a14Aa72a183a125a109a110,a33a34,a73a74a183a125a109a110B,a64B2 = A.
a37a38:
a73a74a183a185a110T,a64
A = T?1
λ1
λ2
...
λn
T = TT
λ1
λ2
...
λn
T.
a100a72Aa183a125,a168λi > 0,a56
B = TT
√λ
1 √
λ2
...
√λ
n
T,
a57B
a183a125,a44
B2 = A.
11.
a14A = (aij)
a96
B = (bij)
a97
a15na148a183a125(a200a183a125)a109a110,a56C = (aijbij),a33a34,C
a106
a15a183a125(a200
a183a125)a109a110.
a37a38:
a73a74a23a109a110P,a64
PTP = B.
· 22 ·
a139P = (pij),a57
bij =
nsummationdisplay
k=1
pkipkj.
a99
a15a45a46a47a29X = (xi) ∈ Rn,
XTCX =
nsummationdisplay
i=1
nsummationdisplay
j=1
aijbijxixj
=
nsummationdisplay
i=1
nsummationdisplay
j=1
aij
parenleftBigg nsummationdisplay
k=1
pkipkj
parenrightBigg
xixj
=
nsummationdisplay
k=1
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj)
.
a65a99A
a183a125(a200a183a125),a26a49
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj) greaterorequalslant 0,
a70a71XTCX greaterorequalslant 0,a168Ca200a183a125,a103a239A
a96
Ba31a183a125,a57P
a105a151
,a56XTCX greaterorequalslant 0,a57
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj) = 0,
a71Aa183a125,a168
pk1x1 = pk2x2 = ··· = pknxn = 0,k = 1,2,···,n.
a103a100P
a105a151
,Pa29a46a236a35a89a18a29a91a211
a169a105
a198a19a72
a157
,a239
pijj negationslash= 0,j = 1,2,···,n,
a57x
j = 0,j = 1,2,···,n,a168
a65XTCX = 0
a105
a144a128X = 0,a70a71Ca183a125.
a12 a13 8–5
1,a33a34,a23a155a153a109a110a29a17a22
a225
a19a15
a157
a154a32a33a25.
a37a38:
a14Aa72a23a155a153a109a110,λa15Aa29a35a36a17a22
a225,a77a67?λ2
a15?A2a29a35a36a17a22
a225.
a71?A2 = ATA,
a168?A2a200a183a125,
a105
a67?λ2 lessorequalslant 0 (a145
a628–4.5),a70a71λa72
a157
a154a32a33a25.
2,a33a34,a121a152Aa15a35a36a23a155a153a109a110,a57B = (E?A)(E +A)?1a15a35a36a183a185a109a110.
a37a38,a65
a18a62
a67,E +A
a105a151
,a70a71
BTB = [(E?A)(E +A)?1]T[(E?A)(E +A)?1]
= (E?A)?1(E +A)(E?A)(E +A)?1
= (E?A)?1(E?A)(E +A)(E +A)?1 = E,
a168Ba15a183a185a109a110.
3.
a14fa15na51a9a10
a244a68
a32a17V a29
a102a157
a155a153a93a30a31a24a25,a33a34,a73a74
a102a157
a111a108α,β ∈V a60a> 0,a64
a68
a45a46a47a29ξ ∈V a48
f(α,ξ) = a(β,ξ),f(β,ξ) =?a(α,ξ).
a37a38:
a14η1,···,ηna15Va29a35a36a11a193a183a185a53,fa74a101a53a88a29a107a108a109a110a72A,a57Aa72a23a155a153a109a110,a44a45
a46a47a29ξ =
nsummationtext
i=1
xiηi,η =
nsummationtext
i=1
yiηi,a48
f(ξ,η) = XTAY.
· 23 ·
a100Aa15a23a155a153a109a110,a168ATAa72a200a183a125a109a110,a71f negationslash= 0,a168Anegationslash= 0,a70a71ATAnegationslash= 0,a26a49ATAa48
a102a157
a17a22
a225.
a46a120ATAa29a35a36
a102a157
a17a22
a225λ,a57λ> 0.
a56a = √λ,a14?
a1
...
an
a72ATAa29a28
a99
a17a22
a225λ
a29a17a22a111a108,a14?
b1
...
bn
=? 1
aA
a1
...
an
.
a56
α =
nsummationdisplay
i=1
aiηi,β =
nsummationdisplay
i=1
biηi,
a57αnegationslash= 0.
a88a33α,β,aa78a79a119
a54.
a45a46a47a29ξ =
nsummationtext
i=1
xiηi,a48
f(α,ξ) = (a1 ··· an)A
x1
...
xn
=?
A
a1
...
an
T?
x1
...
xn
= a(b1 ··· bn)
x1
...
xn
= a(β,ξ),
f(β,ξ) = (b1 ··· bn)A
x1
...
xn
=? 1
a (a1 ··· an)A
TA
x1
...
xn
=? 1a
ATA
a1
...
an
T?
x1
...
xn
=?a(a
1 ··· an)
x1
...
xn
=?a(α,ξ),
a103f(β,α) =?a(α,α) negationslash= 0,a26a49β negationslash= 0.
4.
a14fa15na51a9a34a32a17V a18a29a155a153a93a30a31a24a25.
a33a34,a73a74a11a193a183a185a53η1,ξ1,η2,ξ2,···,ηr,ξr,ζ1,···,ζn?2r,a64fa163
a99a83
a36a53a29a107a108a109a110a184a48a121a88
a117a35a109a110a29a171a205:
diag
parenleftbiggparenleftbigg 0 a
1
a1 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
,ai > 0.
a37a38:
a45V a29a51a25na227a25a36a228a229a230,a94n = 1a95,f = 0,a158a159a39a40a28a160,a161
a209
a14
a158a159
a45m<n
a97
a28
a160,a33a34a94dimV = n
a106
a28a160.
a121f = 0,a158a159a39a40a28a160,a121f negationslash= 0,a65a145a623,a73a74
a102a157
a111a108η1,ξ1a60a25a1 > 0,a64a45a46a47a29ξ ∈Va48
f(η1,ξ) = a1(ξ1,ξ),f(ξ1,ξ) =?a1(η1,ξ).
a65a99η
1,ξ1a29a46a35a218a25kη1,kξ1a106
a78a79a18a37a177a205,a168
a105
a14η1,ξ1
a97
a15V a59a149a150a111a108.
a103,0 = f(ξ1,ξ1) =?a(η1,ξ1),a168η1,ξ1a183a185,a70a71η1,ξ1a72V a29a11a193a183a185a111a108a27.
a56
L = L(η1,ξ1),W = L⊥,
· 24 ·
a57V = L ⊥ W,dimL = 2,dimW = n?2,f
a105a38
a210a15Wa18a29a155a153a93a30a31a24a25,a65a228a229
a209
a14,a73a74W
a29a11a193a183a185a53
η2,ξ2,···,ηr,ξr,ζ1,···,ζn?2r
a60ai > 0,i = 2,···,r,a64f|W a163
a99a83
a36a53a29a107a108a109a110a72a117a35a45a170a110:
diag
parenleftbiggparenleftbigg 0 a
2
a2 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
.
a77a67η
1,ξ1,···,ηr,ξr,ζ1,···,ζn?2ra39
a28V a29a11a193a183a185a53,a65a99a94igreaterorequalslant 2a95a48
f(η1,ξi) = a(ξ1,ξi) = 0,f(η1,ηi) = a(ξ1,ηi) = 0,
f(ξ1,ξi) =?a(η1,ξi) = 0,f(ξ1,ηi) =?a(η1,ηi) = 0,
a100a71fa74a53η1,ξ1,···,ηr,ξr,ζ1,···,ζn?2ra88a29a107a108a109a110a72
diag
parenleftbiggparenleftbigg 0 a
1
a1 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
.
a70a71
a65
a25a36a228a229a230a235
a104
a67a158a159
a28a160.
a12 a13 8–6
1,a14a40a109a110
A = 19
4 + 3i 4i?6?2i
4i 4?3i?2?6i
6 + 2i?2?6i 1
,
a54
a45a170a109a110Ba60a40a109a110U,a64
B = U?1AU.
a55,A
a29a17a22
a225
a72λ1 = 1,λ2 = i,λ3 =?i,a123a124a29a17a22a111a108a72
i
1
12
,
i
12
1
,
12 i
1
1
.
a52a135a41
a123
a183a185,a149a150
a143
a202
a68
α1 = 13
2i
2
1
,α2 = 13
2i
1
2
,α3 = 13
i
2
2
.
a56
U = 13
2i 2i?i
2?1 2
1 2 2
,
a57U
a72a40a109a110,a44
B = U?1AU =
1 0 0
0 i 0
0 0?i
.
2,a14a42a43a44a17a109a110
A =
3?i 0
i 3 0
0 0 4
,
a54
a45a170a109a110Ba60a40a109a110U,a64
B = U?1AU.
· 25 ·
a55,A
a29a17a22
a225
a72λ1 = 2,λ2 = λ3 = 4,a28
a99
a17a22
a2252
a29a17a22a111a108a72
α1 =
i
1
0
,
a28
a99
a17a22
a2254
a29a17a22a111a108a72
α2 =
i
1
0
,α3 =
0
0
1
.
a52a135a41
a123
a183a185,a149a150
a143
a202
a68
η1 = 1√2
i
1
0
,η2 = 1√
2
i
1
0
,η3 =
0
0
1
.
a56
U =
√2
2 i?
√2
2 i 0√
2
2
√2
2 00 0 1
,
a57U
a72a40a109a110,a44
B = U?1AU =
2 0 0
0 4 0
0 0 4
.
3,a33a34,a40a109a110a29a17a22a225a29a45a721.
a37a38:
a14λ0a72a40a109a110Aa46a35a17a22
a225,
α =
a1
...
an
∈Cn
a72Aa29a28
a99
a17a22
a225λ
0a29a17a22a111a108.
a57
Aα = λ0α.
a70a71
αTα = αT(ATA)α = (Aα)T(Aα) = λ0αT ·λ0α = λ0λ0αTα,
a65a99αnegationslash= 0,αTα> 0,
a168λ0λ0 = 1.
4,a14Aa72a35a36
a105a151
a192a109a110,a33a34,A
a105
a117
a69
a72
A = UT,
a58a59,Ua15a40a109a110,Ta15a35a36a45a170a30a18a91a211a19a72a183a23a25a29a18a220a170a171a109a110,a46a33a34
a83
a36a117
a69
a15a75a35a29.
a37a38,(a)
a47a48a227a228a229a230a33a34:
a121Ba72a35n×ra89a78a176a109a110,a57a73a74a45a170a30a18a91a211a19a72a183a29ra148a18a220a170a171a110T,a64C = BTa29a89a111
a108a27a72Cn a59a149a150a183a185a111a108a27.
a45ra227a228a229a230,a94r = 1a95
a158a159
a39a40a28a160,a161
a209
a125
a158a159
a45a89a25<ra29a89a78a176a109a110a28a160,a164a165n×ra89
a78a176a109a110.
· 26 ·
a14Ba29a89a72α1,···,αr,a57α1,···,αra30a31a162a163,a56a1 = 1|α
1|
,a1i =? (αi,α1)|α
1|
,i = 2,···,r,
T1 =
a1 a12 ··· a1r
1 0
...
0 1
,
C1 = BT1 = (β1,β2,···,βr),
a57C
1
a174
a72a89a78a176,a44
|β1| = 1,(β1,βi) = 0,i = 2,···,r.
a56
B1 = (β2,···,βr).
a57B
1a72n×(r?1)a29a89a78a176a109a110,
a65
a228a229a230
a209
a14,a73a74r?1a148a18a220a170a171a109a110
T2 =
a2?
...
0 ar
,a
i > 0,igreaterorequalslant 2,
a64B1T2a29a89a111a108a72a149a150a183a185a111a108a27,a56
T = T1
parenleftbigg1 0
0 T2
parenrightbigg
,
a57
T =
a1?
a2
...
0 an
a72a18a220a170a171a29,a44ai > 0,a56C = BT = (β1 |B1T1),a57Ca29a49a89
a97
a15a149a150a111a108,a103a100β1
a96
B1a29a49a89a183
a185,a71B1Ta29a49a89a72B1a29a30a31a27
a172,
a168Ca29a89a111a108a27a72a149a150a183a185a111a108a27.
(b) a14Aa72na148
a105a151
a192a109a110,a57a65(a)a67,a73a74a45a170a30a18a91a211a19a72a183a29a18a220a170a171a109a110S,a64ASa29a89
a111a108a27a72a149a150a183a185a111a108a27,a70a71
U = AS
a72a40a109a110,a56T = S?1,a57Ta72a18a220a170a171a109a110,a103a100Sa29a45a170a30a18a91a211a19a183,a168Ta29a45a170a30a18a91a211a19a183,
a44
A = UT.
(c)a14a136a48a40a109a110U1a60a45a170a30a18a91a211a19a183a29a18a220a170a171a109a110T1,a64A = U1T1,a57
UT = U1T1,
a70a71
TT?11 = U?1U1.
a18a205a50a51a15a18a220a170a171a110,a52a51a72a183a185a110,a70a71U?1U1a72a45a170a110,a103a100a101a109a110a29a45a170a30a18a91a211a19a183,a168
U?1U1 = E,a99a15
U = U1,T = T1.
a75a35a31
a68
a33.
5,a33a34,a45a46a35a192a109a110A,a118a73a74a40a109a110U,a64U?1AUa72a18a220a170a171a109a110.
· 27 ·
a37a38:
a45Aa29a148a25na227a228a229a230,n = 1a95
a158a159
a39a40a28a160,a161
a209
a125
a158a159
a45a148a25a166
a99n
a29a109a110a28a160,a164
a165na148a109a110A.
a14λ1a72Aa29a46a35a17a22
a225,α
1 ∈ Cna72Aa29a28
a99
a17a22
a225λ
1a29a149a150a17a22a111a108,a8
α1
a29
a116a72a40a32a17Cn
a29a11a193a183a185a53α1,···,αn,a56
U1 = (α1,α2,···,αn),
a57U
1a72a40a109a110,a44
AU1 = U1
parenleftbiggλ
1?
0 A1
parenrightbigg
,
a57
U?11 AU1 =
parenleftbiggλ
1?
0 A1
parenrightbigg
.
a65
a228a229
a209
a14,a73a74n?1a148a40a109a110U2,a64
U?12 A1U2 =
λ2?
...
0 λn
.
a56
U = U1
parenleftbigg1 0
0 U2
parenrightbigg
,
a57U
a72a40a109a110,a44
U?1AU =
λ1?
λ2
...
0 λn
.
6,a33a34,a45a46a35a40a109a110A,a118a48a40a109a110U,a64U?1AUa72a45a170a110.
a37a38,a65
a18a62,a73a74a40a109a110Ua64
U?1AU = B
a72a18a220a170a171a109a110,a100a18a205a50a51a72a40a109a110,a168Ba72a40a109a110,a99a15Ba53a15a40a109a110a103a15a18a220a170a171a109a110,a118a72a45
a170a110.
7,a33a34,a42a43a44a17a109a110a29a17a22
a225
a19a15a23a25,a44a52a29a28
a99
a169a61
a17a22
a225
a29a17a22a111a108
a123
a41a183a185.
a37a38,(a)
a14λa72a42a43a44a17a109a110Ha29a35a36a17a22
a225,α∈ Cn
a72Ha29a28
a99
a17a22
a225λ
a29a17a22a111a108,a57
λαTα = αTAα = αTATα = AαTα = λαTα.
a65a99αTα> 0,
a26a49λ = λ,λ∈ R.
(b) a14α,βa117a5a72Ha29a28
a99
a169a61
a17a22
a225λ
1,λ2a29a17a22a111a108,
a57
λ2αTβ = αTAβ = αTATβ = AαTβ = λ1αTβ.
(a226a47,λ1 ∈R)a99a15(λ1?λ2)αTβ = 0,a65λ1 negationslash= λ2
a105
a68αTβ = 0,
a132α⊥β.
8,a33a34,a45a46a35a42a43a44a17a109a110H,a118a48a40a109a110U,a64U?1HUa72a45a170a171.
a37a38,a65a145
a625,a73a74a40a109a110U,a64T = U
1HU
a15a18a220a170a171a109a110,a103
TT = (U?1HU)T = UTHU
T
= UTHU = U?1HU = T.
a100a101Ta15a45a170a110.
9.
a14Aa72a192a109a110,a121a152ATA = AAT,a57a153Aa72a11a193a114a110,a33a34,a45a46a35a11a193a114a110,a118a48a40a109a110U,
a64U?1AUa72a45a170a171.
· 28 ·
a37a38,a65a145
a625,a73a74a40a109a110U,a64T = U?1AUa15a18a220a170a171a109a110,a103
TTT = (U?1AU)T ·U?1AU = UTAU
T
·UTAU
= UTATAU = UTAATU
= U?1AU ·(U?1AU)T = TTT.
a100a101T
a106
a15a11a193a114a110,a65a109a110a29a54a230
a76a77
a33a34,a18a220a170a171a29a11a193a114a110a118a72a45a170a110,a100a101
a158a159
a28a160.
a12 a13 8–7
1,a74K3a59,a54a53(1,0,2),(1,2,1),(0,2,1)a29a45a55a53.
a55:
a14K3a29a138a40a53a72ε1 = (1,0,0),ε2 = (0,1,0),ε3 = (0,0,1),f1,f2,f3a72ε1,ε2,ε3a29a45a55a53,
g1,g2,g3a72α1 = (1,0,2),α2 = (1,2,1),α3 = (0,2,1)a29a45a55a53,a56(α1,α2,α3) = (ε1,ε2,ε3)A,a57
A =
1 1 0
0 2 2
2 1 1
.
a65a56
a627.1a67,
(g1,g2,g3) = (f1,f2,f3)A?T,
a83a244
A?T = 14
0 4?4
1 1 1
2?2 2
.
a100a101a45a46a47a29α = (x,y,z) ∈K3,a48
g1(x,y,z) = 14 (?f2 + 2f3)(x,y,z) =? 12 y+ 12 z,
g2(x,y,z) = 14 (4f1 +f2?2f3)(x,y,z) = x+ 14 y? 12 z,
g3(x,y,z) = 14 (?4f1 +f2 + 2f3)(x,y,z) =?x+ 14 y+ 12 z.
2,a14η1,η2,η3a15a30a31a32a17V a29a35a36a53,f1,f2,f3a15a52a29a45a55a53,
α1 = η1 + 2η2 + 3η2,α2 = η1 +η2?η3,α3 = η1 +η2.
a63a33α1,α2,α3a15a52a29a35a36a53a46
a54
a58a45a55a53(a227f1,f2,f3a178a128).
a55:
a14
(α1,α2,α3) = (η1,η2,η3)A,
a57
A =
1 1 1
2 1 1
3?1 0
.
a77a67A
a105a151
,a44
A?1 =
1 1 0
3 3?1
5?4 1
.
a100a101α1,α2,α3a15V a29a35a36a53,a14fprime1,fprime2,fprime3a15α1,α2,α3a29a45a55a53,a56
(fprime1,fprime2,fprime3) = (f1,f2,f3)S,
· 29 ·
a57a65a56
a627.1a68
S = A?T =
1?3 5
1 3?4
0?1 1
,
(fprime1,fprime2,fprime3) = (f1,f2,f3)
1?3 5
1 3?4
0?1 1
.
3,a14Va15a25a50Ka18a29a35a36a30a31a32a17,f1,···,fsa15V a29sa36
a102a157
a30a31a24a25,a33a34,a73a74a111a108α∈V,
a64
fi(α) negationslash= 0,i = 1,···,s.
a37a38:
a14
Wi = {α∈V |f(α) = 0},i = 1,2,···,s.
a57W
ia15V a29a98a32a17,a103a100a72fi negationslash= 0,Wi negationslash= V,a56
W = W1 ∪W2 ∪···∪Ws,
a57W
a169
a15Va29a30a31a98a32a17(a122a220a19a145a623–4.4),a100a101W negationslash= V,a103W?V,a118a48α∈V,αnegationslash∈W,a99a15a45a26
a48a29i = 1,···,sa48αnegationslash∈Wi,a132fi(α) negationslash= 0.
4,a14α1,···,αsa15a30a31a32a17V a59a29sa36
a102a157
a111a108,a33a34,a73a74V a18a29a30a31a24a25f,a64
f(αi) negationslash= 0,i = 1,···,s.
a37a38:
a164a165a45a55a32a17V?,a57α1,···,αs
a105a38
a210V?a18a29sa36a30a31a24a25,a168
a65
a18a62,a73a74f ∈V?,a64
α?i (f) = f(αi) negationslash= 0,i = 1,···,s.
5,a14V a15a25a50Ka18a29a35a36a30a31a32a17,f1,···,fsa15V a29sa36a30a31a24a25,a57
a172
W = {α∈V |fi(α) = 0,i = 1,···,s}.
a33a34,(1) Wa15V a29a35a36a30a31a98a32a17(a153a72a30a31a24a25f1,···,fsa29
a157a143
a98a32a17);
(2) V a29a46a47a30a31a98a32a17
a97
a15a245a58a30a31a24a25a29
a157a143
a98a32a17.
a37a38,(1) a65f
i(0) = 0,i = 1,···,s,
a680 ∈W,W
a102
a32,a14α,β ∈W,k ∈K,a57a45i = 1,···,s,
fi(α+β) = fi(α) +fi(β) = 0,
fi(kα) = kfi(α) = 0,
a26a49α+β ∈W,kα∈W,Wa15V a29a30a31a98a32a17.
(2)a14Wa72Va29a35a36a30a31a98a32a17,a14α1,···,αra15Wa29a53,
a8
a52
a29
a116a72Va29a53α1,···,αn,a45a46a47
a29
α = x1α1 +···+xrαr +xr+1αr+1 +···+xnαn,
a125a126
f1(α) = xr+1,f2(α) = xr+2,···fn?r(α) = xr+n,
a57a77a67f
1,···,fn?r a97
a15V a29a30a31a24a25,a39a40a45a46a47a29α ∈ W a48fi(α) = 0,i = 1,···,n?r,a103a239
α =
nsummationtext
i=1
xiαia78a79
fi(α) = 0,i = 1,···,n?r,
a57
a48xr+1 = ··· = xn = 0,a70a71
α = x1α1 +···+xrαr ∈W.
· 30 ·
a100a101Wa15f1,···,fn?ra29
a157a143
a98a32a17.
6,a14fa72na51a30a31a32a17V a18a29
a102a157
a30a31a24a25,a33a34,a73a74V a29a53η1,···,ηn,a64
α =
nsummationdisplay
i=1
xiηi,
a97
a48f(α) = x1.
a37a38,a65a99f
a102a157
,a168a73a74γ ∈V a64
a68
f(γ) = cnegationslash= 0 ∈K.
a56α = γc,a57αnegationslash= 0,a44f(α) = 1.
a8
α
a29
a116a72V a29a53α1 = α,α2,···,αn,a56
η1 = α1,η2 = α2?f(α2)α1,···,ηi = αi?f(αi)α1,···,ηn = αn?f(αn)α1,
a57η
1,···,ηna106
a15V a29a53,a44
f(η1) = 1,f(ηi) = 0,i = 2,···,n.
a70a71a45a46a47a29α =
nsummationtext
i=1
xiηi,a48
f(α) =
nsummationdisplay
i=1
xif(ηi) = x1.
7,a14a27a72a25a50Ka18na51a30a31a32a17V a29a30a31a59a232,η1,···,ηna72V a29a53,f1,···,fn a72η1,···,ηna29
a45a55a53.
(1)a33a34,a45V a29a46a35a30a31a24a25f,fa27
a174
a15V a29a30a31a24a25;
(2)a125a126V?a41a138a60a29a42a43a27?a72:
a27
, f mapsto?→f
a27
a33a34,a27
a15V?a29a30a31a59a232;
(3)a121a27a74a53η1,···,ηna88a29a109a110a15A,a63a54a27?a74a53f1,···,fna88a29a109a110.
a37a38,(1)
a39a40fa27a15V a41Ka29a42a43,a45a46a47a29α,β ∈V,k ∈K,a48
(fa27)(α+β) = f(a27(α+β)) = f(a27α+a27β) = f(a27α) +f(a27β) = (fa27)(α) + (fa27)(β),
(fa27)(kα) = f(a27(kα)) = f(ka27α) = kf(a27α) = k(fa27)(α),
a26a49fa27a15V a18a29a30a31a24a25.
(2) a65(1)a67,a27
a15V?a29a35a36a59a232,a45a46a47a29f,g ∈V?,k∈K,α∈V,a48
(a27
(f +g))(α) = (f +g)(
a27α) = f(a27α) +g(a27α) = (fa27)(α) + (ga27)(α) = (a27
f)(α) + (
a27
g)(α),
a65α
a29a46a47a31
a105
a68
a27
(f +g) =
a27
f +
a27
g.
a103
a65
(a27
(kf))(α) = (kf)(
a27α) = k(fa27)(α) = k(a27
f)(α),
a65α
a29a46a47a31
a105
a68
a27
(kf) = k
a27
f.
a100a101a27
a15V?a29a30a31a59a232.
(3) a65a66a67,
(a27η1,···,a27ηn) = (η1,···,ηn)A,
a14
(a27
f1,···,
a27
fn) = (f1,···,fn)S,
· 31 ·
a57
a27
fj =
nsummationdisplay
k=1
skjfk,j = 1,···,n.
a70a71
(a27
fj)(ηi) =
nsummationdisplay
k=1
skjfk(ηi) = sij,i,j = 1,···,n.
a136a35a114a186,
(a27
fj)(ηi) = fj(
a27ηi) = fj
parenleftBigg nsummationdisplay
l=1
aljηl
parenrightBigg
=
nsummationdisplay
l=1
aljfj(ηl) = aji,i,j = 1,···,n.
a99
a15
aji = sij,i,j = 1,···,n.
a65
a101
a68
S = AT.
· 32 ·
a1a2a3 a4a5a6a7a8a9a10a11
a12 a13 8–1
1,a14V a15a16a17[?1,1]a18a19a20a21a22a23a24a25a26a27a28a29a30a31a32a17,a33a34:
ψ, V?→ R
f(x) mapsto?→ integraltext1?1f(x)dx
a15V a18a29a35a36a30a31a24a25.
a37a38:
a39a40ψa15V a41Ra29a35a36a42a43,a44a45a46a47a29f(x),g(x) ∈V,k ∈R,a48
ψ(f(x) +g(x)) =
integraldisplay1
1
(f(x) +g(x))dx =
integraldisplay1
1
f(x)dx+
integraldisplay1
1
g(x)dx = ψ(f(x)) +ψ(g(x)),
ψ(kf(x)) =
integraldisplay1
1
kf(x)dx = k
integraldisplay1
1
f(x)dx = kψ(f(x)).
a26a49ψa15V a18a29a35a36a30a31a24a25.
2,a14V a15a25a50Ka18a29a35a363a51a30a31a32a17,η1,η2,η3a15a52a29a35a36a53,fa15V a18a29a35a36a30a31a24a25,a44
f(η1?2η2 +η3) = 2,f(η1 +η3) = 2,f(?η1 +η2 +η3) =?1.
a54f(x
1η1 +x2η2 +x3η3).
a55:
a56?
α1 = η1?2η2 +η3
α2 = η1 +η3
α3 =?η1 +η2 +η3
a57(α
1,α2,α3) = (η1,η2,η3)A,a58a59
A =
1 1?1
2 0 1
1 1 1
.
a57(η
1,η2,η3) = (α1,α2,α3)A?1,a26a49
f(x1η1 +x2η2 +x3η3) = (f(η1),f(η2),f(η3))
x1
x2
x3
= (f(α1),f(α2),f(α3))A?1
x1
x2
x3
= (2,2,?1)· 14
1?2 1
3 2 1
2 0 2
x1
x2
x3
= 3
2 x1 +
1
2 x3.
3,V a60η1,η2,η3
a61
a18a62,a63
a54
a35a30a31a24a25g,a64
g(3η1 +η2) = 2,g(η2?η3) = 1,g(2η1 +η3) = 2.
a55:
a14
g(η1) = a,g(η2) = b,g(η3) = c,
a57a65a66a67a68
3a+b = 2
b?c = 1
2a+c = 2.
· 1 ·
a69a68a =?1,b = 5,c = 4.
a70a71a26
a54
a29a30a31a24a25a72
g(x1η1 +x2η2 +x3η3) =?x1 + 5x2 + 4x3.
4,a14V a15a25a50Ka18a29na51a30a31a32a17,η1,···,ηna15a52a29a35a36a53,a1,···,ana15Ka59a46a47na36a25,a33
a34,a73a74V a18a75a35a29a30a31a24a25f,a64
f(ηi) = ai,i = 1,···,n.
a37a38,(
a73a74a31)a14α = x1η1 +x2η2 +···+xnηn ∈V,a56
f, V?→ K
α mapsto?→ f(α) =
nsummationtext
i=1
aixi
a76a77
a33a34fa15V a18a30a31a24a25,a44a78a79a26a80a81a82.
(a75a35a31)a14ga72V a29a30a31a24a25,a64
g(ηi) = ai,i = 1,···,n.
a57
a45a46a47a29α = x1η1 +x2η2 +···+xnηn ∈V a48
g(α) =
nsummationdisplay
i=1
xig(ηi) =
nsummationdisplay
i=1
xiai = f(α).
a83a84
a33a34
a85
a75a35a31.
5,a14V = K3,α = (x1,x2,x3),β = (y1,y2,y3),a86a87a88a89a90a91a24a25fa15a92a72V a18a29a93a30a31a24a25:
(1) f(α,β) = 2x1y1 +x1y2?3x2y1 +x2y2;
(2) f(α,β) = (x1?y2)2 +x2y1;
(3) f(α,β) = c,c∈K;
(4) f(α,β) = (2x1 +x2?3x3)(y1?y2 +y3).
a55,(1)
a15.
(2)a92.
(3)a94cnegationslash= 0a95,a92;a94c = 0a95,a15.
(4)a15.
6,a14fa72na51a30a31a32a17V a18a29a93a30a31a24a25,a56
W1 = {α∈V |f(α,β) = 0,?β ∈V},
W2 = {α∈V |f(β,α) = 0,?β ∈V}.
a33a34,W1
a96
W2
a97
a15V a29a30a31a98a32a17,a44dimW1 = dimW2.
a37a38,(1) a65a99
a45a46a47a29β ∈ V a48f(0,β) = 0,a100a1010 ∈ W1,W1
a102
a32,a103a45a46a47a29α1,α2 ∈ W1,
k ∈Ka49a60a46a47a29β ∈V a48
f(α1 +α2,β) = f(α1,β) +f(α2,β) = 0,
f(kα1,β) = kf(α1,β) = 0,
a100a101
α1 +α2 ∈W1,kα1 ∈W1.
a26a49W1a15V a29a30a31a98a32a17.
a61a104a105
a33W2
a106
a15V a29a30a31a98a32a17.
· 2 ·
(2)a14η1,···,ηna72V a29a53,fa74a53η1,···,ηna88a29a107a108a109a110a72B,a57a45a46a47a29a111a108
α = (x1 ··· xn)
η1
...
ηn
,β = (y
1 ··· yn)
η1
...
ηn
,
f(α,β) = (x1 ··· xn)B
y1
...
yn
.
a70a71
α =
nsummationtext
i=1
xiηi ∈ W1 (x1 ··· xn)B
y1
...
yn
= 0?(y
1,···,yn) ∈ Kn (x1 ··· xn)B =
0 (x1 ··· xn)a72a112a113a30a31a114a115a27XB = 0a29a69.
a26a49dimW1 =a112a113a30a31a114a115a27XB = 0a29
a69
a32a17a29a51a25= n?rankB.
a61a104a105
a33dimW2 = n?rankB,a26a49dimW1 = dimW2.
7,a14f a72Kn a18a29a35a36a90a91a24a25,a33a34,f a72Kn a18a29a93a30a31a24a25a29a116a117a118a119a81a82a15a73a74a109a110
A∈Mn(K),a64
f(X,Y) = XTAY,X,Y ∈Kn.
a37a38,(?)
a14fa72Kna18a93a30a31a24a25,a120fa29a107a108a109a110A,a57A∈Mn(K),a44
f(X,Y) = XTAY,?X,Y ∈Kn.
(?) a121a90a91a24a25a78a79
f(X,Y) = XTAY,?X,Y ∈Kn,
a57f
a39a40a15Kna18a93a30a31a24a25.
8,a45
a99a1225
a62a59a29a93a30a31a24a25,a63
a54a123a124
a29a107a108a109a110.
a55,(1)
2 1 0
3 1 0
0 0 0
.
(3)a94c = 0a95,a107a108a109a110= 0.
(4)
2?2 2
1?1 1
3 3?3
.
9,a14V = K4,a121a88a125a126V a29a90a91a24a25f:
f(α,β) = x1y1 +x2y2?x3y3?x4y4,
a58a59
α = (x1,x2,x3,x4),β = (y1,y2,y3,y4).
(1)a33a34,fa15V a18a29a35a36a93a30a31a24a25;
(2)a54fa74a53
η1 = (2,1,?1,1),η2 = (0,2,1,0),
η3 = (1,1,?2,1),η4 = (0,0,1,2)
a88a29a107a108a109a110;
(3)a127a128a35a36a78a79f(α,α) = 0a29a111a108αnegationslash= 0.
· 3 ·
a55,(1)
a129a130a131a33a132
a105
,a33a133.
(2)a134a135a48
(η1 η2 η3 η4) = (ε1 ε2 ε3 ε4)
2 0 1 0
1 2 1 0
1 1?2 1
1 0 1 2
a71fa74a53ε1,ε2,ε3,ε4a88a29a107a108a109a110a72?
1
1
1
1
,
a100a101fa74a53η1,η2,η3,η4a88a29a107a108a109a110a72?
2 1?1 1
0 2 1 0
1 1?2 1
0 0 1 2
1
1
1
1
2 0 1 0
1 2 1 0
1 1?2 1
1 0 1 2
=
3 3 0?1
3 3 4?1
0 4?3 0
1?1 0?5
.
(3)a120α = (1,1,1,1),a39a40a48f(α,α) = 0.
10,a14V = K4,α = (x1,x2,x3,x4),β = (y1,y2,y3,y4),
f(α,β) = 3x1y2?5x2y1 +x3y4?4x4y3.
(1)a54fa74a53
η1 = (2,1,?1,1),η2 = (1,2,1,?1),
η3 = (?1,1,2,1),η4 = (1,?1,1,2)
a88a29a107a108a109a110;
(2)a136a120V a29a53ε1,ε2,ε3,ε4:
(ε1,ε2,ε3,ε4) = (η1,η2,η3,η4)T,
a58a59
T =
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
,
a54f
a74ε1,ε2,ε3,ε4a88a29a107a108a109a110.
a55,(1)
a137fa74a138a40a53a88a29a107a108a109a110a139a72B,a137
a65
a138a40a53a41a53η1,η2,η3,η4a29a140a141a109a110a139a72A,a57
B =
0 3 0 0
5 0 0 0
0 0 0 1
0 0?4 0
,A =
2 1?1 1
1 2 1?1
1?1 2 1
1?1 1 2
,
a99
a15fa74a53η1,η2,η3,η4a88a29a107a108a109a110a72
C = ATBA =
1 4 2?17
20?1 22?7
7?17?4?2
22 2?17?4
.
· 4 ·
(2) fa74a53ε1,ε2,ε3,ε4a88a29a107a108a109a110a72
D = TTCT =
45 9 39?27
9?45 9?117
39?9 5 3
27 117 3 45
.
11,a14fa15na51a30a31a32a17V a18a29a93a30a31a24a25,a33a34,f
a102a142a143
a29a116a117a118a119a81a82a15,a70
f(α,β) = 0,a45a26a48a29α∈V,
a105
a49a144a128β = 0.
a37a38,(?)
a56
W1 = {α∈V |f(α,β) = 0,?β ∈V},
W2 = {α∈V |f(β,α) = 0,?β ∈V}.
a121f
a102a142a143
,a57a65a125a1261.3a60W1a29a125a126
a67W
1 = 0,a70a71
a65a145
a626a68W2 = 0,a100a101
a65f(α,β) = 0?α∈ V
a105
a49a144a128α = 0.
(?) a121f(α,β) = 0?α ∈ V
a105
a49a144a128α = 0,a57W2 = 0,
a61a104a105
a68W
1 = 0,
a57a65
a125a1261.3a60W1a29
a125a126
a67f
a102a142a143
.
12,a14A∈Mm(K),V = Mm,n(K),a125a126V a18a29a90a91a24a25fa121a88:
f(X,Y) = Tr(XTAY),X,Y ∈V.
(1)a33a34,fa15V a18a29a35a36a93a30a31a24a25;
(2)a54fa74a53E11,E12,···,E1n,···,Em1,···,Emn a88a29a107a108a109a110;
(3)a74a146a147a81a82a88,fa15
a102a142a143
a29.
a55,(1)
a14X = (xij)m×n,Y = (yij)m×n,A = (aij)m,a57
f(X,Y) =
nsummationdisplay
i=1
msummationdisplay
l=1
msummationdisplay
k=1
xlialkyki,
a70a71
a67f
a15a93a30a31a29.
(2) a65a99f(Est,Euv) = δtvasu,a100a101fa74a53E11,E12,···,E1n,···,Em1,···,Emna88a29a107a108a109a110a72
B =
a11E ··· a1mE
...,..,..
am1E ··· ammE
,
a58a59Ea15na148a149a150a114a110.
(3) a65a99|B| = |A|n,a26a49f
a102a142a143
|B| negationslash= 0 |A| negationslash= 0,a132f
a102a142a143
a29a116a117a118a119a81a82a15A
a15
a105a151
a109a110.
13,a33a34,Mn(K)a18a29a93a30a31a24a25
f(A,B) = TrAB,A,B ∈Mn(K)
a15
a102a142a143
a29.
a37a38:
a14A = (aij) ∈Mn(K),a121a152
f(A,B) = TrAB = 0?B ∈Mn(K)
a57f(A,E
ij) = 0?i,j = 1,···,n,a71
f(A,Eij) = TrAEij = aji,
· 5 ·
a26a49aji = 0a45i,j = 1,···,n,a132A = 0,a100a101f
a102a142a143
.
a136a33,a100a72
f(A,B) = TrAB = Tr((AT)TB) = Tr((AT)TEB),
a65a145
a6212(3)
a105
a67f
a102a142a143
.
a12 a13 8–2
1,a14fa15a30a31a32a17V a18a29a45a153a154a155a153a93a30a31a24a25,Wa15V a29a156a98a32a17.
a33a34,a45ξ /∈W,a118a48
a102a157
a111a108η ∈W +L(ξ),a64a45a26a48a29α∈W,
a97
a48f(η,α) = 0.
a37a38:
a121W = 0,a57a158a159a39a40a28a160,a161a14W negationslash= 0,a14α1,···,αsa72Wa29a53,a57a100ξ /∈W,ξ,α1,···,αs
a30a31a162a163,a164a165a30a31a114a115a27?
x0f(ξ,α1) +x1f(α1,α1) +···+xsf(αs,α1) = 0
x0f(ξ,α2) +x1f(α1,α2) +···+xsf(αs,α2) = 0
..........................................
x0f(ξ,αs) +x1f(α1,αs) +···+xsf(αs,αs) = 0
(*)
a101a112a113a30a31a114a115a27a29a114a115a36a25sa166
a99a167a67
a108a36a25s+ 1,a168(*)a48
a102a157
a69(a
0,a1,···,as),a56
η = a0ξ +a1α1 +···+asαs,
a57η ∈W +L(ξ),
a44η negationslash= 0 (a100α1,···,αsa30a31a162a163,a44a0,a1,···,as
a169
a19a72
a157
),a44
a65(*)a67
f(η,αi) = 0,i = 1,2,···,s.
a103a100α1,···,αsa72Wa29a53,a168a45a46a47a29α∈W
a97
a48f(η,α) = 0.
2,V
a96
f
a61
a18a62,Wa15V a29a30a31a98a32a17,a56
W⊥ = {α∈V |f(α,β) = 0,?β ∈W}.
a33a34,(1) W⊥a15V a29a30a31a98a32a17;
(2)a121a152W ∩W⊥ = {0},a57V = W ⊕W⊥.
a37a38,(1) a65f(0,β) = 0?β ∈W,
a105
a680 ∈W⊥,
a100a101W⊥
a102
a32.
a45a46a47a29α1,α2 ∈W⊥,k ∈K,a57?β ∈W,a48
f(α1 +α2,β) = f(α1,β) +f(α2,β) = 0,
f(kα1,β) = kf(α1,β) = 0,
a100a101α1 +α2 ∈W⊥,kα1 ∈W⊥,a168W⊥a15V a29a30a31a98a32a17.
(2) a45a46a47a29ξ /∈W,a65a18a62a26a33,a73a74η negationslash= 0 ∈W +L(ξ),a64
a68f(η,α) = 0?α ∈ W,
a132η ∈W⊥.
a139η = α+aξ,a57a100W ∩W⊥ = 0,a118a48anegationslash= 0,a26a49
ξ = a?1η?a?1α∈W⊥ +W.
a33
a68V?W⊥ +W.
3,a54
a105a151
a109a110T,a64TTATa72a45a170a171,a58a59Aa72a88a89a109a110:
(1)
1 1 0
1 2 2
0 2 5
; (2)
1?2 1
2 4 2
1 2 1
;
· 6 ·
(3)
0 1 1
1 0 1
1 1 0
; (4)
1 1 1
1 1 1
1 1 1
;
a55,(1)
a120T =
1?1 2
0 1?2
0 0 1
,a57TTAT =
1 0 0
0 1 0
0 0 1
.
(2)a120T =
1 0?1
0 14? 14
0 12 12
,a57TTAT =
1 0 0
0 1 0
0 0?1
.
(3)a120T =
1?1 1
0 1 0
1?1?1
,a57TTAT =
2 0 0
0?2 0
0 0?2
.
(4)a120T =
1?1?1
0 1 0
0 0 1
,a57TTAT =
1 0 0
0 0 0
0 0 0
.
4,a33a34:
λ1
λ2
...
λn
a96
λi1
λi2
...
λin
a123a172,
a58a59i1,···,ina151,···,na29a35a36a173a89.
a37a38:
a164a165na51a30a31a32a17V,a14fa72V a18a29a45a153a93a30a31a24a25,a52a74a53η1,···,ηna88a29a107a108a109a110a72
λ1
λ2
...
λn
,
a77a67η
i1,···,ηin
a174
a15V a29a53,a44fa74ηi1,···,ηina88a29a107a108a109a110a72
λi1
λi2
...
λin
,
a100a101
a83a175
a36a109a110
a123a172.
5,a33a34,a176a177a99ra29a45a153a109a110
a105
a49a178a72ra36a176a177
a991
a29a45a153a109a110a179a180.
a37a38:
a14Aa15a176a72ra29a45a153a109a110,a57a73a74
a105a151
a109a110T,a64
a68
TTAT =
a1
...
ar
0
...
0
,ai negationslash= 0.
· 7 ·
a56
Ai = T?T
0
...
ai
0
...
0
T?1,
a57A
ia106
a15a45a153a109a110,rankAi = 1a44A = A1 +A2 +···+Ar.
6,a14Aa72a23a109a110,a33a34,ATA
a96
Aa29a176
a123
a177.
a37a38,a77a67,ATA
a15a23a45a153a109a110,a164a165a23a25a50a18a29a112a113a30a31a114a115a27
ATAX = 0 (1)
a96
AX = 0,(2)
a39a40(2)a29
a69
a97
a15(1)a29
a69.
a14X ∈ Rna72(1)a29a35a36
a69.
a56
Y = AX =
y1
...
yn
.
a57
YTY = XTATAX = 0,
a70a71
y21 +y22 +···+y2n = 0.
a65a99y
ia181
a72a23a25,a100a101y1 = y2 = ··· = yn = 0,Y = 0,a132
AX = 0.
a70a71(1)a29
a69
a106a97
a15(2)a29
a69,(1)
a96
(2)
a61
a69,a65
a112a113a30a31a114a115a27
a69
a29a31a182
a67
rankATA = rankA.
7,a14Aa72a183a125a109a110,a33a34,A?1
a96
A?
a97
a15a183a125a109a110.
a37a38,a77a67A?1
a96
A?
a97
a15a23a45a153a109a110,a44A? = |A| ·A?1,a100Aa183a125,a73a74
a105a151
a23a109a110C a64
CTC = A,a70a71A?1 = C?TC?1
a106
a183a125,a65|A|> 0
a105
a67A? = |A|·A?1
a106
a183a125.
8,a33a34,a46a47a35a36a93a30a31a24a25
a97a105
a75a35a178a72a35a36a45a153a93a30a31a24a25a180a35a36a155a153a93a30a31a24a25a179a180.
a37a38,(1)
a14f(α,β)a15a35a36a93a30a31a24a25,a77a67
g(α,β) = 12 [f(α,β) +f(β,α)]
a15a45a153a93a30a31a24a25,
h(α,β) = 12 [f(α,β)?f(β,α)]
a72a155a153a93a30a31a24a25,a44
f(α,β) = g(α,β) +h(α,β).
(2)a103a14
f(α,β) = gprime(α,β) +hprime(α,β),
· 8 ·
a58a59gprime(α,β)a15a45a153a93a30a31a24a25,hprime(α,β)a15a155a153a93a30a31a24a25,a57
f(β,α) = gprime(β,α) +hprime(β,α) = gprime(α,β)?hprime(α,β).
a70a71
gprime(α,β) = 12 [f(α,β) +f(β,α)] = g(α,β),
hprime(α,β) = 12 [f(α,β)?f(β,α)] = h(α,β).
9.
a33a34,a93a30a31a24a25fa184a48a183a185a45a153a31a29a116a117a118a119a81a82a15fa72a45a153a154a155a153a93a30a31a24a25.
a37a38:
a116a117a31a15a39a40a29,a88a186a33a118a119a31.
(1)a121a45a46a47a29α∈V
a97
a48f(α,α) = 0,a57a45a46a47a29α,β ∈V,
0 = f(α+β,α+β) = f(α,α) +f(α,β) +f(β,β) +f(β,β) = f(α,β) +f(β,α).
a100a101f(α,β) =?f(β,α),fa15a155a153a93a30a31a24a25.
(2)a121a152a73a74γ ∈V a64f(γ,γ) negationslash= 0,a57a45a46a47a29α∈V,a65a99
f
parenleftbigg
α? f(α,γ)f(γ,γ) γ,γ
parenrightbigg
= f(α,γ)?f(α,γ) = 0,
a26a49f
parenleftbigg
γ,α? f(α,γ)f(γ,γ) γ
parenrightbigg
= 0,a100a101
f(α,γ) = f(γ,α),(*)
a45
a99
a46a47a29α,β ∈V,a49a88a187a117
a175a188a189a190a191a159:
(a)a121a152f(α,γ) negationslash= 0,a57
f
parenleftbigg
α,β? f(α,β)f(α,γ) γ
parenrightbigg
= f(α,β)?f(α,β) = 0,
a100a101f
parenleftbigg
β? f(α,β)f(α,γ) γ,α
parenrightbigg
= 0,a70a71
0 = f(β,α)? f(α,β)f(α,γ) f(γ,α)
= f(β,α)? f(α,β)f(α,γ) f(α,γ) a65(*)
= f(β,α)?f(α,β),
a132f(α,β) = f(β,α).
(b) a121a152f(α,γ) = 0,a57
f
parenleftbigg
α+γ,β? f(α,β) +f(γ,β)f(γ,γ) γ
parenrightbigg
= f(α,β) +f(γ,β)?f(α,β)?f(γ,β) = 0,
a100a101f
parenleftbigg
β? f(α,β) +f(γ,β)f(γ,γ) γ,α+γ
parenrightbigg
= 0,a70a71
f(β,α) +f(β,γ)?f(α,β)?f(γ,β) = 0.
a65(*)a67f(β,γ) = f(γ,β),
a100a101f(α,β) = f(β,α).
a65(a)
a180(b)
a105
a68f
a72a45a153a93a30a31a24a25.
10.
a14V a15a192a25a50a18a29a30a31a32a17,a58a51a25ngreaterorequalslant 2,fa15V a18a29a35a36a45a153a93a30a31a24a25,a33a34:
(1) V a59a48
a102a157
a111a108ξ,a64f(ξ,ξ) = 0;
(2)a94fa15
a102a142a143
a95,a118a48a30a31a162a163a29a111a108ξ,η,a78a79:
f(ξ,η) = 1,
f(ξ,ξ) = f(η,η) = 0.
· 9 ·
a37a38,(1) a65a99dimV greaterorequalslant 2.
a46a120V a29
a175
a36a30a31a162a163a29a111a108α,β,a121a152f(α,α) = 0,a57ξ = αa132a72a26
a54.
a161a14f(α,α) negationslash= 0,a572a113a114a115
t2f(α,α) + 2tf(α,β) +f(β,β) = 0 (*)
a74a192a25a193a194a195a48
a69.
a14t0 ∈Ca15ta29a35a36
a69.
a56
ξ = t0α+β,
a57ξ negationslash= 0 (
a100α,βa30a31a162a163),a44
f(ξ,ξ) = t20f(α,α) + 2t0f(α,β) +f(β,β) = 0.
a70a71ξ = t0α+βa132a72a26
a54.
(2) a65(1)a26a33,a73a74ξ negationslash= 0 ∈V a64f(ξ,ξ) = 0,a103a100f
a102a142a143
,a168a73a74α∈V a64f(ξ,α) negationslash= 0.
(a)a121f(α,α) = 0,a57a56η = 1f(ξ,α) α,a132a48
f(ξ,ξ) = f(η,η) = 0,f(ξ,η) = 1.
(b) a121f(α,α) negationslash= 0,a57a120
η = 1f(α,ξ) α? f(α,α)2(f(α,ξ))2 ξ,
a196a197
a131a33
a105
a67f(η,η) = 0,f(ξ,η) = 1,
a71ξ,ηa29a30a31a162a163a31a15a39a40a29,a168ξ,ηa132a72a26
a54.
11.
a33a34,a121a152a30a31a32a17V a18a29a45a153a93a30a31a24a25fa198a117
a69
a72
a175
a36a30a31a24a25a179a199:
f(α,β) = f1(α)f2(β),?α,β ∈V,
a57
a73a74
a102a157
a25λa60a30a31a24a25g,a64
f(α,β) = λg(α)g(β).
a37a38:
a121a152f = 0,a57a158a159a94a40a28a160,a161a14f negationslash= 0,a100a101a73a74α0,β0 ∈V,a64
a68f(α
0,β0) negationslash= 0,a125a126
g, V?→ K
γ mapsto?→ f(α0,γ)
a57g
a72V a18a30a31a24a25,a44g negationslash= 0,a45a46a47a29β ∈V,
g(β) = f(α0,β) = f1(α0)f2(β)
g(β) = f(α0,β) = f(β,α0) = f1(β)f2(α0)
a39a40f1(α0) negationslash= 0,f2(α0) negationslash= 0 (a92
a57g negationslash= 0),a65
a101
a67,
f1(β) = 1f
2(α0)
g(β)
f2(β) = 1f
1(α0)
g(β)?β ∈V.
a56λ = 1f
1(α0)f2(α0)
,a57
f(α,β) = f1(α)f2(β) = 1f
2(α0)
g(α)· 1f
1(α0)
g(β) = λg(α)g(β).
12.
a14Aa72a200a183a125a109a110,a33a34,A?
a106
a15a200a183a125a109a110.
a37a38:
a121a152rankA = n,a57Aa15a183a125a109a110,a145a627a66a33a34
a85A?
a183a125,a121a152rankAlessorequalslantn?2,a57A? = 0,
a70a71A?a200a183a125,a201a202a164a203rankA = n?1a29
a189
a171,a101a95rankA? = 1,a70a71A?a29a148a25greaterorequalslant 2a29a204a98a205
a97
a15
0,a71A?a291a148a204a98a205= Aii (i = 1,···,n) = Aa29aiia29a129a25a206a98a205(i = 1,···,n) = Aa29aiia29a206a98a205
(i = 1,···,n) = Aa29n?1a148a204a98a205greaterorequalslant 0 (a100Aa200a183a125),a26a49A?a200a183a125.
· 10 ·
13.
a33a34a125
a104
2.12.
a37a38,(1)?(2)
a14Aa200a183a125,a57a73a74
a105a151
a23a109a110T,a64
TTAT =
a1
...
ar
0
...
0
,ai negationslash= 0.
a65a99A
a200a183a125,TTAT
a106
a200a183a125,a168ai > 0,a26a49Aa29a183a207a31a208a25p = r = rankA;
(2)?(3) a65a209a14,a73a74
a105a151
a23a109a110T1,a64
TT1 AT1 =
a1
...
ar
0
...
0
,ai > 0.
a56
T2 =
1√
A1
...
1√
Ar
1
...
1
,T = T1T2,
a57
TTAT =
1
...
1
0
...
0
=
parenleftbiggE
r 0
0 0
parenrightbigg
.
(3)?(4) a65a209a14,a73a74
a105a151
a23a109a110T,a64
TTAT =
parenleftbiggE
r 0
0 0
parenrightbigg
.
a56
S =
parenleftbiggE
r 0
0 0
parenrightbigg
T,
a57
A = STS.
(4)?(1)a45a46a47a29X negationslash= 0 ∈ Rn,a56Y = SX,a57Y ∈Rn,a26a49
XTAX = XTSTSX = YTY greaterorequalslant 0,
Aa200a183a125.
· 11 ·
(1)?(5)a14Bk = A(i1,···,ik;i1,···,ik)a15Aa29a35a36a204a98a205.a57a45a46a47a29
Xk =
x1
...
xk
∈Rk,
a105
a49a210a35a36a89a111a108X ∈Rn,a64
a68
a52a29
a122i
ja89a29a91a211a177
a99x
j,a71a58a206a91a211a181
a177
a990,a57
0 lessorequalslantXTAX = XTk BkXk,
a100a101Bra15a200a183a125a29,a212a213(4),
a105
a68
a200a183a125a109a110a29a214a89a205
a102a215
,a132|Bk|greaterorequalslant 0.
(5)?(1)a45
a99
a46a47a29a183a23a25λ> 0,a164a165λE +Aa29ka148a204a98a110λEk +Ak,a83a36a98a109a110a29a214a89a205a72
fk(λ) = |λEk +Ak| = λk +a1λk?1 +···+ak.
a57
a212a213
a145
a627–3.8,(?1)iaia177
a99?A
ka29a19a216ia148a204a98a205a179a180,a71?Aka29a217a36ia148a204a98a205a177
a99A
ka29
a123a124i
a148a204a98a205a29(?1)ia218.a100a101aia177
a99A
ka29a26a48ia148a204a98a205a179a180,
a65a209
a14,ai greaterorequalslant 0,a70a71
fk(λ) > 0?λ> 0,i = 1,···,k.
a212a213a125
a104
2.11,λE +A(λ> 0)a15a183a125a109a110.
a46a120X negationslash= 0 ∈ Rn,a213a183a125a31,λa29a35a113a205
g(λ) = XT(λE +A)X = λXTX +XTAX > 0,?λ> 0.
a100a101XTAX greaterorequalslant 0 (a92
a57
a94λa116a117a166a95a219a48g(λ) < 0),a70a71Aa200a183a125.
14.
a204a45a170a30a18a19a151a29a18a220a170a171a109a110a153a72a221a222a18a220a170a171a109a110.
(1) a14Aa15a35a36a45a153a109a110,Ta72a221a222a18a220a170a171a109a110,a33a34,TTAT
a96
Aa29a45
a124a223a224
a204a98a205a48
a123
a61
a29
a225;
(2)a121a152a45a153a109a110a29
a223a224
a204a98a205a19
a169
a72
a157
,a57a73a74a35a221a222a18a220a170a171a109a110T,a64TTATa72a45a170a171.
a37a38,(1)
a14Ara72Aa29ra148
a223a224
a204a98a205(1 lessorequalslantrlessorequalslantn),
A =
parenleftbiggA
r?
parenrightbigg
.
a14Ta72a221a222a18a220a170a171a109a110,
T =
parenleftbiggT
11?
0 T22
parenrightbigg
,a58a59T11 =
1?
...
0 1
r
,
a57
TTAT =
parenleftbiggTT
11 0
TT22
parenrightbiggparenleftbiggA
r?
parenrightbiggparenleftbiggT
11?
0 T22
parenrightbigg
=
parenleftbiggTT
11ArT11?
parenrightbigg
.
a70a71TTATa29ra148
a223a224
a204a98a205a177
a99(
a226a47a41|T11| = 1)
|TT11AT11| = |TT11||A||T11| = |Ar|.
(2)a45Aa29a148a25a227a228a229a230,a120
T1 =
parenleftbiggE
n?1?A?1n?1B
0 1
parenrightbigg
,An?1 = A(1,···,n?1;1,···,n?1),B =
a1n
...
an?1,n
.
a83
a15a221a222a18a220a170a171a109a110,a57
TT1 AT1 =
parenleftbigg E 0
BTA?1n?1 1
parenrightbiggparenleftbiggA
n?1 B
BT ann
parenrightbiggparenleftbiggE?A?1
n?1B
0 1
parenrightbigg
=
parenleftbiggA
n?1 0
0 bn
parenrightbigg
,
· 12 ·
a58a59bn = ann?BTAn?1B,a65a99Aa29
a223a224
a204a98a205a19
a169
a720,a168An?1a29
a223a224
a204a98a205a19
a169
a720,a65a228a229
a209
a14,a73a74n?1a148a221a222a18a220a170a171a109a110T2a64
TT2 An?1T2 =
b1
...
bn?1
.
a56
T = T1
parenleftbiggT
2 0
0 1
parenrightbigg
,
a57T
a72a221a222a18a220a170a171a109a110,a44
TTAT =
b1
...
bn
.
a12 a13 8–3
1,a227
a102a142a143
a30a31a231a232
a143
a88a89a90a113a233a72a234a114a180:
(1) x21 + 5x22?4x23 + 2x1x2?4x1x3;
(2) 4x21 +x22 +x23?4x1x2 + 4x1x3?3x2x3;
(3) x1x2 +x1x3 +x2x3;
(4) 2x21 + 18x22 + 8x23?12x1x2 + 8x1x3?27x2x3;
(5) x21?2x1x2 + 2x1x3?2x1x4 +x22 + 2x2x3?4x2x4 +x23?2x24;
(6) x21 +x1x2 +x2x4.
a55,(1) x2
1 + 5x22?4x23 + 2x1x2?4x1x3 = (x1 +x2?2x3)2 + (2x2 +x3)2?(3x3)2,a56
y1 = x1 +x2?2x3
y2 = 2x2 +x3
y3 = 3x3
a132
x1 = y1? 12 y2 + 56 y3
x2 = 12 y2? 16 y3
x3 = 13 y3
a48
f(x1,x2,x3) = y21 +y22?y23.
(2)a235a205= (2x1?x2 +x3)2 +
parenleftBigx
2?x32
parenrightBig2
parenleftBigx
2 +x32
parenrightBig2
,a56
y1 = 2x1?x2 +x3
y2 = x2?x32
y3 = x2 +x32
a132
x1 = 12 y1 +y2
x2 = y2 +y3
x3 =?y2 +y3
a57
f(x1,x2,x3) = y21 +y22?y23.
(3)a56?
x1 = y1?y2?y3
x2 = y1 +y2?y3
x3 = y3
a48
f(x1,x2,x3) = y21?y22?y23.
· 13 ·
(4)a235a205= 2(x1?3x2 + 2x3)2 +
parenleftBig3x
2?x32
parenrightBig2
parenleftBig3x
2 +x32
parenrightBig2
,a56
y1 = x1?3x2 + 2x3
y2 = 3x2?x32
y3 = 3x2 +x32
a132
x1 = y1 + 3y2?y3
x2 = 13 y2 + 13 y3
x3 =?y2 +y3
a57
f(x1,x2,x3) = 2y21 +y22?y23.
(5)a56?
x1 = y1?y3?y4
x2 = 12 y2? 12 y3? 12 y4
x3 = 12 y2 + 12 y3 + 32 y4
x4 = y4
a57
a48
f(x1,x2,x3,x4) = y21 +y22?y23.
(6)a235a205= x21 +
parenleftBigx
2 +x1 +x42
parenrightBig2
parenleftBigx
2?x1?x42
parenrightBig2
,a56
y1 = x1
y2 = x2 +x1 +x42
y3 = x2?x1?x42
y4 = x3
a132
x1 = y1
x2 = y2 +y3
x3 = y4
x4 =?y1 +y2?y3
a57
f(x1,x2,x3,x4) = y21 +y22?y23.
2,λa120a236a225a95,a88a89a90a113a233a15a183a125a29:
(1) 5x21 +x22 +λx23 + 4x1x2?2x1x3?2x2x3;
(2) 2x21 +x22 + 3x23 + 2λx1x2 + 2x1x3;
(3) 2x21 + 2x22 +x23 + 2λx1x2 + 6x1x3 + 2x2x3.
a55,(1) A =
5 2?1
2 1?1
1?1 λ
,
a52a29
a223a224
a204a98a205D1 = 5 > 0,D2 = 1 > 0,D3 = λ? 2,a26a49a94
λ> 2a95a235a90a113a233a183a125.
(2)a90a113a233a109a110a29
a223a224
a204a98a205D1 = 2 > 0,D2 = 2?λ2,D3 = 5?3λ2.
a65D
2 > 0,
a68|λ|<√2;
a65D
3 > 0,
a68|λ|<
radicalBig5
3,
a26a49a94?
√15
3 <λ<
√15
3 a95a235a90a113a233a183a125.
(3) a90a113a233a109a110a29
a223a224
a204a98a205D1 = 2,D2 = 4?λ2,D3 =?λ2 + 6λ?16 =?(λ?3)2?7 < 0,a168
a169
a159λ
a120a236a23a25
a97a169
a198a64a101a90a113a233a183a125.
3,a88a89a90a113a233a15a92a183a125a154a200a183a125:
(1)
nsummationtext
i=1
x2i + summationtext
1lessorequalslanti<jlessorequalslantn
xixj; (2)
nsummationtext
i=1
x2i +
n?1summationtext
i=1
xixi+1;
(3) n
nsummationtext
i=1
x2i?
parenleftbigg nsummationtext
i=1
xi
parenrightbigg2
.
· 14 ·
a55,(1)
a90a113a233a109a110A =
1 12 ··· 12
1
2 1 ···
1
2.
..,..,..,..
1
2
1
2 ··· 1
,a52a29
a223a224
a204a98a205
Dr = |Ar| =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 12 ··· 12
1
2 1 ···
1
2.
..,..,..,..
1
2
1
2 ··· 1
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
r
= 12r (r+ 1) > 0,r = 1,···,n.
a168a235a90a113a233a183a125.
(2) a235a205f = 12 x21 + 12 (x1 + x2)2 + 12 (x2 + x3)2 + ··· + 12 (xn?1 + xn)2 + 12 x2n greaterorequalslant 0,a100a101
f = 0 x1 = 0,x1 +x2 = 0,x2 +x3 = 0,···,xn?1 +xn = 0,xn = 0 x1 = x2 = ··· = xn = 0.
a168a235a90a113a233a183a125.
(3) a235a205= (?1)
nsummationtext
i=1
x2i?2 summationtext
1lessorequalslanti,jlessorequalslantn
xixj = summationtext
1lessorequalslanti,jlessorequalslantn
(xi?xj)2 greaterorequalslant 0,a120x1 = x2 = ··· = xn negationslash= 0
a105
a64a101
a90a113a233a120
a157
a225.
a100a101a235a90a113a233a200a183a125.
4,a14Aa72a23a45a153a109a110,a33a34:
(1)a94a23a25λa116a117a237a179a202,λE +Aa15a183a125a29;
(2) Aa200a183a125a94a44a238a94a45a46a236a29λ> 0,λE +A
a97
a183a125.
a37a38,(1)
a164a165A(λ) = λE +A,a52a29ra148
a223a224
a204a98a205
Dr(λ) = |λEr +Ar| = λr +a1λr?1 +···+ar.
a26a49a94λa116a117a237a95,a48Dr(λ) > 0,r = 1,···,n,a70a71a94λa116a117a237a95,λE +Aa183a125.
(2) (?)a239Aa200a183a125,a57a45a46a47a29X negationslash= 0 ∈Rn,XTAX greaterorequalslant 0,a70a71a45a46a47a29λ> 0a48
XT(λE +A)X = λXTX +XTAX > 0.
a168λE +Aa183a125.
(?) a45a46a47a29λ> 0a60X negationslash= 0 ∈Rn,a48
XT(λE +A)X = λXTX +XTAX > 0,
a70a71XTAX greaterorequalslant 0,a168Aa200a183a125.
5,a14A,B,Ca72a220a170a171a29a220a36a195a170,a33a34,a45a46a47a23a25x,y,za48
x2 +y2 +z2 greaterorequalslant 2xycosA+ 2xzcosB + 2yzcosC.
a37a38:
a164a165a90a113a233f(x,y,z) = x2 +y2 +z2?2xycosA?2xzcosB?2yzcosC.
f(x,y,z) = (x?ycosA?zcosB)2 +y2 sin2A+z2 sin2B?2yzcosAcosB?2yzcosC
= (x?ycosA?zcosB)2 +y2 sin2A+z2 sin2B?2yzsinAsinB
= (x?ycosA?zcosB)2 + (ysinA?zsinB)2.
a70a71fa200a183a125,a65a101
a67a158a159
a28a160.
· 15 ·
6,a33a34,a239
nsummationtext
i=1
nsummationtext
j=1
aijxixj (aij = aji)a15a183a125a90a113a233,a57
f(y1,y2,···,yn) =
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a11 a12 ··· a1n y1
a21 a22 ··· a2n y2
........................
an1 an2 ··· ann yn
y1 y2 ··· yn 0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
a15
a215
a125a90a113a233.
a37a38,a66a67
parenleftbigg E 0
YTA?1 1
parenrightbiggparenleftbigg A Y
YT 0
parenrightbigg
=
parenleftbiggA Y
0?YTA?1Y
parenrightbigg
,a26a49
f(y1,···,yn) =
vextendsinglevextendsingle
vextendsinglevextendsingle A Y
YT 0
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingleA Y
0?YTA?1Y
vextendsinglevextendsingle
vextendsinglevextendsingle
= |A|(?YTA?1Y) = YT(?A?)Y.
a65A
a183a125
a105
a68A?
a183a125,a99a15?A?
a215
a125,a100a101f(y1,···,yn) = YT(?A?)Y a15
a215
a125a90a113a233.
7.
a14a48a23a240a25a90a113a24a25
f(x1,x2,···,xn) =
nsummationdisplay
i=1
nsummationdisplay
j=1
aijxixj +
nsummationdisplay
i=1
2bixi +c,aij = aji.
a56
A =
a11 a12 ··· a1n
a21 a22 ··· a2n
....................
an1 an2 ··· ann
,D =
a11 a12 ··· a1n b1
a21 a22 ··· a2n b2
........................
an1 an2 ··· ann bn
b1 b2 ··· bn c
.
(1)a33a34,a94A
a215
a125a95,fa48a201a237
a225,
a44fmax = |D||A| ;
(2)a14A
a215
a125,a63a241a125a94x1,···,xna72a236
a225
a95,fa120
a68
a201a237
a225.
a55,(1)
a120
T =
parenleftbiggE
n?A?1B
0 1
parenrightbigg
,B =
b1
...
bn
,
a56
y1
...
yn
yn+1
= T?1
x1
...
xn
1
,(*)
· 16 ·
a77a67y
n+1 = 1.
a57
f(x1,···,xn) = (x1 ··· xn 1)D
x1
...
xn
1
= (y1 ··· yn 1)TTDT
y1
...
yn
1
= (y1 ··· yn 1)
parenleftbiggA 0
0 d
parenrightbigg
y1
...
yn
1
= YTAY +d.
a65a99A
a215
a125,a168a45a46a47a29Y ∈ R
n
a48YTAY lessorequalslant 0,a26a49f lessorequalslantd.
a105a242
fa48a243a237
a225d,
a44a94Y = 0a95fa120a243a237
a225.a83a244
d = |A|d|A| = |T
TDT|
|A| =
|D|
|A|,
(2) a65(*),?
x1
...
xn
1
= T
y1
...
yn
1
=
parenleftbiggE
n?A?1B
0 1
parenrightbigg
y1
...
yn
1
,
a68X = Y?A?1B.
a94Y = 0a95X =?A?1B,a132a94?
x1
...
xn
=?A?1
b1
...
bn
a95,fa120a201a237
a225.
8,a245a246a247a248a249Aa188a249a250x(a251)a36a180Ba188a249a250y(a251)a36a29a252a28a253a24a25a72:
C(x,y) = x2 + 2xy+y2 + 100(a254a91).
a255a0a175a188
a249a250a29a80
a54
a24a25a72:
x = 26?pA,y = 10? 14pB,
a58a59pA,pBa72a249a250
a123a124
a29a1a2(a254a91/a251a36),a54a3a4a201a237a95a249a250a29a25a108a180
a3a4.
a55:
a213a62a47,a3a4a24a25a72
p(x,y) = xpa +ypb?C(x,y)
= x(26?x) +y(40?4y)?C(x,y)
=?2x2?2xy?5y2 + 26x+ 40y?100.
a253a62
a84
a15
a54
a90a113a24a25a29a201a237
a225.
a14
A =
parenleftbigg?2?1
1?5
parenrightbigg
,D =
2?1 13
1?5 20
13 20?100
,B =
parenleftbigg13
20
parenrightbigg
.
· 17 ·
a83a244A
a15
a215
a125a109a110,a212a213
a145
a627,a94parenleftbigg
x
y
parenrightbigg
=?A?1B =?A?1
parenleftbigg13
20
parenrightbigg
=? 19
parenleftbigg?5 1
1?2
parenrightbiggparenleftbigg13
20
parenrightbigg
=
parenleftbigg5
3
parenrightbigg
a95,a3a4a201a237,a44a201a237
a3a4
a72
pmax = |D||A| = 25a254a91.
a168a94
a175a188
a249a250a117a5a1a128500a36
a96
300a36a95,
a105a6
a201a237
a3a425
a254a91.
a12 a13 8–4
1,a54a183a185a109a110T,a64T?1ATa72a45a170a171,a14Aa72a88a89a109a110:
(1)
2?2 0
2 1?2
0?2 0
; (2)
2 2?2
2 5?4
2?4 5
;
(3)
2?2 0 1
2 2 1 0
0 1 2?2
1 0?2 2
; (4)
1?1 3?2
1 1?2 3
3?2 1?1
2 3?1 1
;
(5)
1?3 3?3
3?1?3 3
3?3?1?3
3 3?3?1
; (6)
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
.
a55,(1) T = 1
3
2 2 1
1?2 2
2 1 2
,T?1AT =
1 0 0
0 4 0
0 0?2
.
(2) T = 13
2 2 1
2 1 2
1 2?2
,T?1AT =
1 0 0
0 1 0
0 0 10
.
(3) T = 12
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
,T?1AT =
1 0 0 0
0?1 0 0
0 0 3 0
0 0 0 5
.
(4) T = 12
1 1 1 1
1 1?1?1
1?1?1 1
1?1 1?1
,T?1AT =
1 0 0 0
0?1 0 0
0 0?3 0
0 0 0 7
.
(5) T = 12
1 1 1 1
1?1 1?1
1?1?1 1
1 1?1?1
,T?1AT =
4 0 0 0
0?4 0 0
0 0?4 0
0 0 0 8
.
(6) T = 12
1 1 1 1
1?1 1?1
1?1?1 1
1 1?1?1
,T?1AT =
4 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
.
2,a227a183a185a30a31a231a232
a143
a88a89a23a90a113a233a72a7a193a171:
(1) x21 + 2x22 + 3x23?4x1x2?4x2x3;
(2) x21?2x22?2x23?4x1x2 + 4x1x3 + 8x2x3;
· 18 ·
(3) 3x21 + 6x22 + 3x23?4x1x2?8x1x3?4x2x3;
(4) 2x1x2 + 2x3x4;
(5) x21 +x22 +x23 +x24 + 2x1x2?2x1x4?2x2x3 + 2x3x4;
(6) 2x1x2 + 2x1x3 + 2x1x4 + 2x2x3 + 2x2x4 + 2x3x4.
a55,(1)
a210a183a185a30a31a231a232?
x1 = 23 y1? 23 y2 + 13 y3
x2 = 23 y1 + 13 y2? 23 y3
x3 = 13 y1 + 23 y2 + 23 y3
a105a8
a90a113a233
a143
a72a7a193a171
y21 + 2y22 + 5y23.
(2)a210a183a185a30a31a231a232?
x1 =? 2
√5
5 y1?
2√5
15 y2 +
1
3 y3
x2 =
√5
5 y1?
4√5
15 y2 +
2
3 y3
x3 =
√5
3 y2 +
2
3 y3
a105a8
a90a113a233
a143
a72a7a193a171
2y21 + 2y22?7y23.
(3)a210a183a185a30a31a231a232?
x1 = 23 y1? 23 y2 + 13 y3
x2 = 13 y1 + 23 y2 + 23 y3
x3 = 23 y1 + 13 y2? 23 y3
a105a8
a90a113a233
a143
a72a7a193a171
2y21 + 7y22 + 7y23.
(4)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1 + 12 y2? 12 y3? 12 y4
x3 = 12 y1? 12 y2 + 12 y3? 12 y4
x4 = 12 y1? 12 y2? 12 y3 + 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
y21 +y22?y23?y24.
(5)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1? 12 y2? 12 y3 + 12 y4
x3 = 12 y1 + 12 y2? 12 y3? 12 y4
x4 = 12 y1? 12 y2 + 12 y3? 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
y21 +y22?y23 + 3y24.
· 19 ·
(6)a210a183a185a30a31a231a232?
x1 = 12 y1 + 12 y2 + 12 y3 + 12 y4
x2 = 12 y1 + 12 y2? 12 y3? 12 y4
x3 = 12 y1? 12 y2 + 12 y3? 12 y4
x4 = 12 y1? 12 y2? 12 y3 + 12 y4
a105a8
a90a113a233
a143
a72a7a193a171
3y21?y22?y23?y24.
3,a14Aa72a35a36na148a23a109a110,a44|A| negationslash= 0,a33a34,A
a105
a117
a69
a28
A = QT,
a58a59Qa15a183a185a109a110,Ta15a18a220a170a171a109a110.
a37a38:
a120na51a9a10
a244a68
a32a17V,a14η1,···,ηna15a52a29a35a36a11a193a183a185a53,a56
(α1,···,αn) = (η1,···,ηn)A (*)
a65a99|A| negationslash= 0,A
a105a151
,a100a101α1,···,αna15V a29a53,a124a227a12a13a14–a15a16a17a183a185
a143
a114a230,
a105
a68V
a29a35a36a11a193a183
a185a53β1,···,βn,a56
(α1,···,αn) = (β1,···,βn)T,
a65a122a18a19
a125
a104
3.4a29a33a34
a105
a67,T
a72a18a220a170a171a109a110,a56
(β1,···,βn) = (η1,···,ηn)Q,
a57Q
a72a183a185a109a110,a44
(α1,···,αn) = (β1,···,βn)T = (η1,···,ηn)QT,
a96
(*)a20
a21,a68A = QT.
4,a14Aa72a35a36na148a183a125a109a110,a33a34,a73a74a18a220a170a171a109a110T,a64
A = TTT.
a37a38,a65A
a183a125
a105
a67
a73a74
a105a151
a23a109a110Ba64
a68A = BTB,a65
a18a62,a73a74a183a185a109a110Q
a96
a18a220a170a171a109a110
T,a64
a68
B = QT.
a70a71
A = BTB = TTQTQT = TTT.
5,a14Aa72a23a45a153a109a110,a33a34,Aa183a125(a200a183a125)a29a116a117a118a119a81a82a15Aa29a17a22
a225
a19a237
a99(
a237
a99
a177
a99)
a157
.
a37a38:
a73a74a183a185a109a110T,a64
a68
T?1AT = TTAT =
λ1
λ2
...
λn
,
a58a59λ1,λ2,···,λn a72Aa29a19a216a17a22
a225,a99
a15Aa183a125(a200a183a125) TTAT a183a125(a200a183a125) λi >
0 (λi greaterorequalslant 0),i = 1,···,n.
6,a33a34,a175a36a23a45a153a109a110
a123a23
a29a116a117a118a119a81a82a15a52a135a48
a123
a61
a29a17a22a24a25a205.
· 20 ·
a37a38:
a118a119a31a39a40,a88a33a116a117a31,a14a23a45a153a109a110A,Ba48
a123
a61
a29a17a22a24a25a205,a70a71a52a135a48
a123
a61
a29a17
a22
a225λ
1,λ2,···,λn.
a99
a15a73a74a183a185a109a110T
a96
Q,a64
a68
T?1AT = TTAT =
λ1
λ2
...
λn
,
Q?1BQ = QTBQ =
λ1
λ2
...
λn
.
a39a40A
a96
Ba123a23.
7,a14Aa72na148a23a109a110,a33a34,a73a74a183a185a109a110T,a64T?1ATa72a220a170a171a109a110a29a116a117a118a119a81a82a15Aa29a17
a22
a225
a19a15a23a25.
a37a38,(?)
a14a48a183a185a109a110Ta64
T?1AT =
λ1?
λ2
...
0 λn
,
a57λ
1,···,λna26
a72Aa29na36a17a22
a225,a65a99A,T
a181
a72a23a109a110,T?1AT
a106
a15a23a109a110,a168a17a22
a225λ
1,···,λna97
a15a23a25.
(?) a45na227a228a229a230,n = 1a95a158a159a39a40a28a160,a161a74a209a14a158a159a45n?1a148a78a79a81a82a29a23a109a110a28a160,a164
a165na148a23a109a110A.
a14V a72na51a9a10
a244a68
a32a17,η1,···,ηna72V a29a11a193a183a185a53,a56a27 ∈ End(V),a64
a68
(a27η1,···,a27ηn) = (η1,···,ηn)A.
a14λ1 ∈ Ra72a27a29a46a47a17a22
a225,a57λ
1a106
a15a27a29a17a22
a225.
a56α1a72a27a29a28
a99
a17a22
a225λ
1a29a149a150a17a22a111a108,
a57
α1
a105a29
a116a72V a29a11a193a183a185a53α1,α2,···,αn,a56
(a27α1,···,a27αn) = (α1,···,αn)
λ1?
0.
.,A1
0
,
(α1,···,αn) = (η1,···,ηn)T1,
a57T
1a72a183a185a109a110,a44 parenleftbigg
λ1?
0 A1
parenrightbigg
= T?11 AT1.
a77a67A
1a72n?1a148a23a109a110,a44a58a17a22
a225
a19a15Aa29a17a22
a225,
a70a71
a106a97
a15a23a25,a65a228a229
a209
a14,a73a74a183a185a109a110
T2,a64
T?12 A1T2 =
λ2?
...
0 λn
,
a56
T = T1
parenleftbigg1 0
0 T2
parenrightbigg
,
· 21 ·
a57T
a72a183a185a109a110,a44
T?1AT =
λ1?
λ2
...
0 λn
,
a65
a228a229a230a235
a104
a67a158a159
a28a160.
8,a33a34,a17a22
a225
a19a15a23a25a29a183a185a110a118a15a45a153a109a110.
a37a38:
a14Aa72a17a22
a225
a19a15a23a25a29a183a185a110,a65a18a62,a73a74a183a185a109a110T,a64
TTAT = T?1AT = D
a72a18a220a170a171a110,a103a100a72Da15a183a185a110,a168D?1 = DT
a106
a15a18a220a170a171a109a110,a30a52a103a15a88a220a170a171a110,a168Da15a45
a170a110,a70a71Da72a45a153a109a110,a168
A = TDTT
a106
a72a45a153a109a110.
9.
a14A,Ba15
a175
a36na148a23a45a153a109a110,a44Ba183a125,a33a34,a73a74
a105a151
a109a110T,a64
TTAT
a96
TTBT
a61
a95a72a45a170a171.
a37a38,a65a99B
a183a125,a100a101a73a74
a105a151
a109a110Sa64
a68
STBS = E.
a71STAS
a174
a72a23a45a153a109a110,a168a73a74a183a185a109a110Qa64D = QT(STAS)Qa72a45a170a110,a56T = SQ,a57T
a105a151
,a44
TTAT = D,TTBT = E,
a181
a72a45a170a110.
10.
a14Aa72a183a125a109a110,a33a34,a73a74a183a125a109a110B,a64B2 = A.
a37a38:
a73a74a183a185a110T,a64
A = T?1
λ1
λ2
...
λn
T = TT
λ1
λ2
...
λn
T.
a100a72Aa183a125,a168λi > 0,a56
B = TT
√λ
1 √
λ2
...
√λ
n
T,
a57B
a183a125,a44
B2 = A.
11.
a14A = (aij)
a96
B = (bij)
a97
a15na148a183a125(a200a183a125)a109a110,a56C = (aijbij),a33a34,C
a106
a15a183a125(a200
a183a125)a109a110.
a37a38:
a73a74a23a109a110P,a64
PTP = B.
· 22 ·
a139P = (pij),a57
bij =
nsummationdisplay
k=1
pkipkj.
a99
a15a45a46a47a29X = (xi) ∈ Rn,
XTCX =
nsummationdisplay
i=1
nsummationdisplay
j=1
aijbijxixj
=
nsummationdisplay
i=1
nsummationdisplay
j=1
aij
parenleftBigg nsummationdisplay
k=1
pkipkj
parenrightBigg
xixj
=
nsummationdisplay
k=1
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj)
.
a65a99A
a183a125(a200a183a125),a26a49
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj) greaterorequalslant 0,
a70a71XTCX greaterorequalslant 0,a168Ca200a183a125,a103a239A
a96
Ba31a183a125,a57P
a105a151
,a56XTCX greaterorequalslant 0,a57
nsummationdisplay
i=1
nsummationdisplay
j=1
aij(pkixi)(pkjxj) = 0,
a71Aa183a125,a168
pk1x1 = pk2x2 = ··· = pknxn = 0,k = 1,2,···,n.
a103a100P
a105a151
,Pa29a46a236a35a89a18a29a91a211
a169a105
a198a19a72
a157
,a239
pijj negationslash= 0,j = 1,2,···,n,
a57x
j = 0,j = 1,2,···,n,a168
a65XTCX = 0
a105
a144a128X = 0,a70a71Ca183a125.
a12 a13 8–5
1,a33a34,a23a155a153a109a110a29a17a22
a225
a19a15
a157
a154a32a33a25.
a37a38:
a14Aa72a23a155a153a109a110,λa15Aa29a35a36a17a22
a225,a77a67?λ2
a15?A2a29a35a36a17a22
a225.
a71?A2 = ATA,
a168?A2a200a183a125,
a105
a67?λ2 lessorequalslant 0 (a145
a628–4.5),a70a71λa72
a157
a154a32a33a25.
2,a33a34,a121a152Aa15a35a36a23a155a153a109a110,a57B = (E?A)(E +A)?1a15a35a36a183a185a109a110.
a37a38,a65
a18a62
a67,E +A
a105a151
,a70a71
BTB = [(E?A)(E +A)?1]T[(E?A)(E +A)?1]
= (E?A)?1(E +A)(E?A)(E +A)?1
= (E?A)?1(E?A)(E +A)(E +A)?1 = E,
a168Ba15a183a185a109a110.
3.
a14fa15na51a9a10
a244a68
a32a17V a29
a102a157
a155a153a93a30a31a24a25,a33a34,a73a74
a102a157
a111a108α,β ∈V a60a> 0,a64
a68
a45a46a47a29ξ ∈V a48
f(α,ξ) = a(β,ξ),f(β,ξ) =?a(α,ξ).
a37a38:
a14η1,···,ηna15Va29a35a36a11a193a183a185a53,fa74a101a53a88a29a107a108a109a110a72A,a57Aa72a23a155a153a109a110,a44a45
a46a47a29ξ =
nsummationtext
i=1
xiηi,η =
nsummationtext
i=1
yiηi,a48
f(ξ,η) = XTAY.
· 23 ·
a100Aa15a23a155a153a109a110,a168ATAa72a200a183a125a109a110,a71f negationslash= 0,a168Anegationslash= 0,a70a71ATAnegationslash= 0,a26a49ATAa48
a102a157
a17a22
a225.
a46a120ATAa29a35a36
a102a157
a17a22
a225λ,a57λ> 0.
a56a = √λ,a14?
a1
...
an
a72ATAa29a28
a99
a17a22
a225λ
a29a17a22a111a108,a14?
b1
...
bn
=? 1
aA
a1
...
an
.
a56
α =
nsummationdisplay
i=1
aiηi,β =
nsummationdisplay
i=1
biηi,
a57αnegationslash= 0.
a88a33α,β,aa78a79a119
a54.
a45a46a47a29ξ =
nsummationtext
i=1
xiηi,a48
f(α,ξ) = (a1 ··· an)A
x1
...
xn
=?
A
a1
...
an
T?
x1
...
xn
= a(b1 ··· bn)
x1
...
xn
= a(β,ξ),
f(β,ξ) = (b1 ··· bn)A
x1
...
xn
=? 1
a (a1 ··· an)A
TA
x1
...
xn
=? 1a
ATA
a1
...
an
T?
x1
...
xn
=?a(a
1 ··· an)
x1
...
xn
=?a(α,ξ),
a103f(β,α) =?a(α,α) negationslash= 0,a26a49β negationslash= 0.
4.
a14fa15na51a9a34a32a17V a18a29a155a153a93a30a31a24a25.
a33a34,a73a74a11a193a183a185a53η1,ξ1,η2,ξ2,···,ηr,ξr,ζ1,···,ζn?2r,a64fa163
a99a83
a36a53a29a107a108a109a110a184a48a121a88
a117a35a109a110a29a171a205:
diag
parenleftbiggparenleftbigg 0 a
1
a1 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
,ai > 0.
a37a38:
a45V a29a51a25na227a25a36a228a229a230,a94n = 1a95,f = 0,a158a159a39a40a28a160,a161
a209
a14
a158a159
a45m<n
a97
a28
a160,a33a34a94dimV = n
a106
a28a160.
a121f = 0,a158a159a39a40a28a160,a121f negationslash= 0,a65a145a623,a73a74
a102a157
a111a108η1,ξ1a60a25a1 > 0,a64a45a46a47a29ξ ∈Va48
f(η1,ξ) = a1(ξ1,ξ),f(ξ1,ξ) =?a1(η1,ξ).
a65a99η
1,ξ1a29a46a35a218a25kη1,kξ1a106
a78a79a18a37a177a205,a168
a105
a14η1,ξ1
a97
a15V a59a149a150a111a108.
a103,0 = f(ξ1,ξ1) =?a(η1,ξ1),a168η1,ξ1a183a185,a70a71η1,ξ1a72V a29a11a193a183a185a111a108a27.
a56
L = L(η1,ξ1),W = L⊥,
· 24 ·
a57V = L ⊥ W,dimL = 2,dimW = n?2,f
a105a38
a210a15Wa18a29a155a153a93a30a31a24a25,a65a228a229
a209
a14,a73a74W
a29a11a193a183a185a53
η2,ξ2,···,ηr,ξr,ζ1,···,ζn?2r
a60ai > 0,i = 2,···,r,a64f|W a163
a99a83
a36a53a29a107a108a109a110a72a117a35a45a170a110:
diag
parenleftbiggparenleftbigg 0 a
2
a2 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
.
a77a67η
1,ξ1,···,ηr,ξr,ζ1,···,ζn?2ra39
a28V a29a11a193a183a185a53,a65a99a94igreaterorequalslant 2a95a48
f(η1,ξi) = a(ξ1,ξi) = 0,f(η1,ηi) = a(ξ1,ηi) = 0,
f(ξ1,ξi) =?a(η1,ξi) = 0,f(ξ1,ηi) =?a(η1,ηi) = 0,
a100a71fa74a53η1,ξ1,···,ηr,ξr,ζ1,···,ζn?2ra88a29a107a108a109a110a72
diag
parenleftbiggparenleftbigg 0 a
1
a1 0
parenrightbigg
,···,
parenleftbigg 0 a
r
ar 0
parenrightbigg
,0,···,0
parenrightbigg
.
a70a71
a65
a25a36a228a229a230a235
a104
a67a158a159
a28a160.
a12 a13 8–6
1,a14a40a109a110
A = 19
4 + 3i 4i?6?2i
4i 4?3i?2?6i
6 + 2i?2?6i 1
,
a54
a45a170a109a110Ba60a40a109a110U,a64
B = U?1AU.
a55,A
a29a17a22
a225
a72λ1 = 1,λ2 = i,λ3 =?i,a123a124a29a17a22a111a108a72
i
1
12
,
i
12
1
,
12 i
1
1
.
a52a135a41
a123
a183a185,a149a150
a143
a202
a68
α1 = 13
2i
2
1
,α2 = 13
2i
1
2
,α3 = 13
i
2
2
.
a56
U = 13
2i 2i?i
2?1 2
1 2 2
,
a57U
a72a40a109a110,a44
B = U?1AU =
1 0 0
0 i 0
0 0?i
.
2,a14a42a43a44a17a109a110
A =
3?i 0
i 3 0
0 0 4
,
a54
a45a170a109a110Ba60a40a109a110U,a64
B = U?1AU.
· 25 ·
a55,A
a29a17a22
a225
a72λ1 = 2,λ2 = λ3 = 4,a28
a99
a17a22
a2252
a29a17a22a111a108a72
α1 =
i
1
0
,
a28
a99
a17a22
a2254
a29a17a22a111a108a72
α2 =
i
1
0
,α3 =
0
0
1
.
a52a135a41
a123
a183a185,a149a150
a143
a202
a68
η1 = 1√2
i
1
0
,η2 = 1√
2
i
1
0
,η3 =
0
0
1
.
a56
U =
√2
2 i?
√2
2 i 0√
2
2
√2
2 00 0 1
,
a57U
a72a40a109a110,a44
B = U?1AU =
2 0 0
0 4 0
0 0 4
.
3,a33a34,a40a109a110a29a17a22a225a29a45a721.
a37a38:
a14λ0a72a40a109a110Aa46a35a17a22
a225,
α =
a1
...
an
∈Cn
a72Aa29a28
a99
a17a22
a225λ
0a29a17a22a111a108.
a57
Aα = λ0α.
a70a71
αTα = αT(ATA)α = (Aα)T(Aα) = λ0αT ·λ0α = λ0λ0αTα,
a65a99αnegationslash= 0,αTα> 0,
a168λ0λ0 = 1.
4,a14Aa72a35a36
a105a151
a192a109a110,a33a34,A
a105
a117
a69
a72
A = UT,
a58a59,Ua15a40a109a110,Ta15a35a36a45a170a30a18a91a211a19a72a183a23a25a29a18a220a170a171a109a110,a46a33a34
a83
a36a117
a69
a15a75a35a29.
a37a38,(a)
a47a48a227a228a229a230a33a34:
a121Ba72a35n×ra89a78a176a109a110,a57a73a74a45a170a30a18a91a211a19a72a183a29ra148a18a220a170a171a110T,a64C = BTa29a89a111
a108a27a72Cn a59a149a150a183a185a111a108a27.
a45ra227a228a229a230,a94r = 1a95
a158a159
a39a40a28a160,a161
a209
a125
a158a159
a45a89a25<ra29a89a78a176a109a110a28a160,a164a165n×ra89
a78a176a109a110.
· 26 ·
a14Ba29a89a72α1,···,αr,a57α1,···,αra30a31a162a163,a56a1 = 1|α
1|
,a1i =? (αi,α1)|α
1|
,i = 2,···,r,
T1 =
a1 a12 ··· a1r
1 0
...
0 1
,
C1 = BT1 = (β1,β2,···,βr),
a57C
1
a174
a72a89a78a176,a44
|β1| = 1,(β1,βi) = 0,i = 2,···,r.
a56
B1 = (β2,···,βr).
a57B
1a72n×(r?1)a29a89a78a176a109a110,
a65
a228a229a230
a209
a14,a73a74r?1a148a18a220a170a171a109a110
T2 =
a2?
...
0 ar
,a
i > 0,igreaterorequalslant 2,
a64B1T2a29a89a111a108a72a149a150a183a185a111a108a27,a56
T = T1
parenleftbigg1 0
0 T2
parenrightbigg
,
a57
T =
a1?
a2
...
0 an
a72a18a220a170a171a29,a44ai > 0,a56C = BT = (β1 |B1T1),a57Ca29a49a89
a97
a15a149a150a111a108,a103a100β1
a96
B1a29a49a89a183
a185,a71B1Ta29a49a89a72B1a29a30a31a27
a172,
a168Ca29a89a111a108a27a72a149a150a183a185a111a108a27.
(b) a14Aa72na148
a105a151
a192a109a110,a57a65(a)a67,a73a74a45a170a30a18a91a211a19a72a183a29a18a220a170a171a109a110S,a64ASa29a89
a111a108a27a72a149a150a183a185a111a108a27,a70a71
U = AS
a72a40a109a110,a56T = S?1,a57Ta72a18a220a170a171a109a110,a103a100Sa29a45a170a30a18a91a211a19a183,a168Ta29a45a170a30a18a91a211a19a183,
a44
A = UT.
(c)a14a136a48a40a109a110U1a60a45a170a30a18a91a211a19a183a29a18a220a170a171a109a110T1,a64A = U1T1,a57
UT = U1T1,
a70a71
TT?11 = U?1U1.
a18a205a50a51a15a18a220a170a171a110,a52a51a72a183a185a110,a70a71U?1U1a72a45a170a110,a103a100a101a109a110a29a45a170a30a18a91a211a19a183,a168
U?1U1 = E,a99a15
U = U1,T = T1.
a75a35a31
a68
a33.
5,a33a34,a45a46a35a192a109a110A,a118a73a74a40a109a110U,a64U?1AUa72a18a220a170a171a109a110.
· 27 ·
a37a38:
a45Aa29a148a25na227a228a229a230,n = 1a95
a158a159
a39a40a28a160,a161
a209
a125
a158a159
a45a148a25a166
a99n
a29a109a110a28a160,a164
a165na148a109a110A.
a14λ1a72Aa29a46a35a17a22
a225,α
1 ∈ Cna72Aa29a28
a99
a17a22
a225λ
1a29a149a150a17a22a111a108,a8
α1
a29
a116a72a40a32a17Cn
a29a11a193a183a185a53α1,···,αn,a56
U1 = (α1,α2,···,αn),
a57U
1a72a40a109a110,a44
AU1 = U1
parenleftbiggλ
1?
0 A1
parenrightbigg
,
a57
U?11 AU1 =
parenleftbiggλ
1?
0 A1
parenrightbigg
.
a65
a228a229
a209
a14,a73a74n?1a148a40a109a110U2,a64
U?12 A1U2 =
λ2?
...
0 λn
.
a56
U = U1
parenleftbigg1 0
0 U2
parenrightbigg
,
a57U
a72a40a109a110,a44
U?1AU =
λ1?
λ2
...
0 λn
.
6,a33a34,a45a46a35a40a109a110A,a118a48a40a109a110U,a64U?1AUa72a45a170a110.
a37a38,a65
a18a62,a73a74a40a109a110Ua64
U?1AU = B
a72a18a220a170a171a109a110,a100a18a205a50a51a72a40a109a110,a168Ba72a40a109a110,a99a15Ba53a15a40a109a110a103a15a18a220a170a171a109a110,a118a72a45
a170a110.
7,a33a34,a42a43a44a17a109a110a29a17a22
a225
a19a15a23a25,a44a52a29a28
a99
a169a61
a17a22
a225
a29a17a22a111a108
a123
a41a183a185.
a37a38,(a)
a14λa72a42a43a44a17a109a110Ha29a35a36a17a22
a225,α∈ Cn
a72Ha29a28
a99
a17a22
a225λ
a29a17a22a111a108,a57
λαTα = αTAα = αTATα = AαTα = λαTα.
a65a99αTα> 0,
a26a49λ = λ,λ∈ R.
(b) a14α,βa117a5a72Ha29a28
a99
a169a61
a17a22
a225λ
1,λ2a29a17a22a111a108,
a57
λ2αTβ = αTAβ = αTATβ = AαTβ = λ1αTβ.
(a226a47,λ1 ∈R)a99a15(λ1?λ2)αTβ = 0,a65λ1 negationslash= λ2
a105
a68αTβ = 0,
a132α⊥β.
8,a33a34,a45a46a35a42a43a44a17a109a110H,a118a48a40a109a110U,a64U?1HUa72a45a170a171.
a37a38,a65a145
a625,a73a74a40a109a110U,a64T = U
1HU
a15a18a220a170a171a109a110,a103
TT = (U?1HU)T = UTHU
T
= UTHU = U?1HU = T.
a100a101Ta15a45a170a110.
9.
a14Aa72a192a109a110,a121a152ATA = AAT,a57a153Aa72a11a193a114a110,a33a34,a45a46a35a11a193a114a110,a118a48a40a109a110U,
a64U?1AUa72a45a170a171.
· 28 ·
a37a38,a65a145
a625,a73a74a40a109a110U,a64T = U?1AUa15a18a220a170a171a109a110,a103
TTT = (U?1AU)T ·U?1AU = UTAU
T
·UTAU
= UTATAU = UTAATU
= U?1AU ·(U?1AU)T = TTT.
a100a101T
a106
a15a11a193a114a110,a65a109a110a29a54a230
a76a77
a33a34,a18a220a170a171a29a11a193a114a110a118a72a45a170a110,a100a101
a158a159
a28a160.
a12 a13 8–7
1,a74K3a59,a54a53(1,0,2),(1,2,1),(0,2,1)a29a45a55a53.
a55:
a14K3a29a138a40a53a72ε1 = (1,0,0),ε2 = (0,1,0),ε3 = (0,0,1),f1,f2,f3a72ε1,ε2,ε3a29a45a55a53,
g1,g2,g3a72α1 = (1,0,2),α2 = (1,2,1),α3 = (0,2,1)a29a45a55a53,a56(α1,α2,α3) = (ε1,ε2,ε3)A,a57
A =
1 1 0
0 2 2
2 1 1
.
a65a56
a627.1a67,
(g1,g2,g3) = (f1,f2,f3)A?T,
a83a244
A?T = 14
0 4?4
1 1 1
2?2 2
.
a100a101a45a46a47a29α = (x,y,z) ∈K3,a48
g1(x,y,z) = 14 (?f2 + 2f3)(x,y,z) =? 12 y+ 12 z,
g2(x,y,z) = 14 (4f1 +f2?2f3)(x,y,z) = x+ 14 y? 12 z,
g3(x,y,z) = 14 (?4f1 +f2 + 2f3)(x,y,z) =?x+ 14 y+ 12 z.
2,a14η1,η2,η3a15a30a31a32a17V a29a35a36a53,f1,f2,f3a15a52a29a45a55a53,
α1 = η1 + 2η2 + 3η2,α2 = η1 +η2?η3,α3 = η1 +η2.
a63a33α1,α2,α3a15a52a29a35a36a53a46
a54
a58a45a55a53(a227f1,f2,f3a178a128).
a55:
a14
(α1,α2,α3) = (η1,η2,η3)A,
a57
A =
1 1 1
2 1 1
3?1 0
.
a77a67A
a105a151
,a44
A?1 =
1 1 0
3 3?1
5?4 1
.
a100a101α1,α2,α3a15V a29a35a36a53,a14fprime1,fprime2,fprime3a15α1,α2,α3a29a45a55a53,a56
(fprime1,fprime2,fprime3) = (f1,f2,f3)S,
· 29 ·
a57a65a56
a627.1a68
S = A?T =
1?3 5
1 3?4
0?1 1
,
(fprime1,fprime2,fprime3) = (f1,f2,f3)
1?3 5
1 3?4
0?1 1
.
3,a14Va15a25a50Ka18a29a35a36a30a31a32a17,f1,···,fsa15V a29sa36
a102a157
a30a31a24a25,a33a34,a73a74a111a108α∈V,
a64
fi(α) negationslash= 0,i = 1,···,s.
a37a38:
a14
Wi = {α∈V |f(α) = 0},i = 1,2,···,s.
a57W
ia15V a29a98a32a17,a103a100a72fi negationslash= 0,Wi negationslash= V,a56
W = W1 ∪W2 ∪···∪Ws,
a57W
a169
a15Va29a30a31a98a32a17(a122a220a19a145a623–4.4),a100a101W negationslash= V,a103W?V,a118a48α∈V,αnegationslash∈W,a99a15a45a26
a48a29i = 1,···,sa48αnegationslash∈Wi,a132fi(α) negationslash= 0.
4,a14α1,···,αsa15a30a31a32a17V a59a29sa36
a102a157
a111a108,a33a34,a73a74V a18a29a30a31a24a25f,a64
f(αi) negationslash= 0,i = 1,···,s.
a37a38:
a164a165a45a55a32a17V?,a57α1,···,αs
a105a38
a210V?a18a29sa36a30a31a24a25,a168
a65
a18a62,a73a74f ∈V?,a64
α?i (f) = f(αi) negationslash= 0,i = 1,···,s.
5,a14V a15a25a50Ka18a29a35a36a30a31a32a17,f1,···,fsa15V a29sa36a30a31a24a25,a57
a172
W = {α∈V |fi(α) = 0,i = 1,···,s}.
a33a34,(1) Wa15V a29a35a36a30a31a98a32a17(a153a72a30a31a24a25f1,···,fsa29
a157a143
a98a32a17);
(2) V a29a46a47a30a31a98a32a17
a97
a15a245a58a30a31a24a25a29
a157a143
a98a32a17.
a37a38,(1) a65f
i(0) = 0,i = 1,···,s,
a680 ∈W,W
a102
a32,a14α,β ∈W,k ∈K,a57a45i = 1,···,s,
fi(α+β) = fi(α) +fi(β) = 0,
fi(kα) = kfi(α) = 0,
a26a49α+β ∈W,kα∈W,Wa15V a29a30a31a98a32a17.
(2)a14Wa72Va29a35a36a30a31a98a32a17,a14α1,···,αra15Wa29a53,
a8
a52
a29
a116a72Va29a53α1,···,αn,a45a46a47
a29
α = x1α1 +···+xrαr +xr+1αr+1 +···+xnαn,
a125a126
f1(α) = xr+1,f2(α) = xr+2,···fn?r(α) = xr+n,
a57a77a67f
1,···,fn?r a97
a15V a29a30a31a24a25,a39a40a45a46a47a29α ∈ W a48fi(α) = 0,i = 1,···,n?r,a103a239
α =
nsummationtext
i=1
xiαia78a79
fi(α) = 0,i = 1,···,n?r,
a57
a48xr+1 = ··· = xn = 0,a70a71
α = x1α1 +···+xrαr ∈W.
· 30 ·
a100a101Wa15f1,···,fn?ra29
a157a143
a98a32a17.
6,a14fa72na51a30a31a32a17V a18a29
a102a157
a30a31a24a25,a33a34,a73a74V a29a53η1,···,ηn,a64
α =
nsummationdisplay
i=1
xiηi,
a97
a48f(α) = x1.
a37a38,a65a99f
a102a157
,a168a73a74γ ∈V a64
a68
f(γ) = cnegationslash= 0 ∈K.
a56α = γc,a57αnegationslash= 0,a44f(α) = 1.
a8
α
a29
a116a72V a29a53α1 = α,α2,···,αn,a56
η1 = α1,η2 = α2?f(α2)α1,···,ηi = αi?f(αi)α1,···,ηn = αn?f(αn)α1,
a57η
1,···,ηna106
a15V a29a53,a44
f(η1) = 1,f(ηi) = 0,i = 2,···,n.
a70a71a45a46a47a29α =
nsummationtext
i=1
xiηi,a48
f(α) =
nsummationdisplay
i=1
xif(ηi) = x1.
7,a14a27a72a25a50Ka18na51a30a31a32a17V a29a30a31a59a232,η1,···,ηna72V a29a53,f1,···,fn a72η1,···,ηna29
a45a55a53.
(1)a33a34,a45V a29a46a35a30a31a24a25f,fa27
a174
a15V a29a30a31a24a25;
(2)a125a126V?a41a138a60a29a42a43a27?a72:
a27
, f mapsto?→f
a27
a33a34,a27
a15V?a29a30a31a59a232;
(3)a121a27a74a53η1,···,ηna88a29a109a110a15A,a63a54a27?a74a53f1,···,fna88a29a109a110.
a37a38,(1)
a39a40fa27a15V a41Ka29a42a43,a45a46a47a29α,β ∈V,k ∈K,a48
(fa27)(α+β) = f(a27(α+β)) = f(a27α+a27β) = f(a27α) +f(a27β) = (fa27)(α) + (fa27)(β),
(fa27)(kα) = f(a27(kα)) = f(ka27α) = kf(a27α) = k(fa27)(α),
a26a49fa27a15V a18a29a30a31a24a25.
(2) a65(1)a67,a27
a15V?a29a35a36a59a232,a45a46a47a29f,g ∈V?,k∈K,α∈V,a48
(a27
(f +g))(α) = (f +g)(
a27α) = f(a27α) +g(a27α) = (fa27)(α) + (ga27)(α) = (a27
f)(α) + (
a27
g)(α),
a65α
a29a46a47a31
a105
a68
a27
(f +g) =
a27
f +
a27
g.
a103
a65
(a27
(kf))(α) = (kf)(
a27α) = k(fa27)(α) = k(a27
f)(α),
a65α
a29a46a47a31
a105
a68
a27
(kf) = k
a27
f.
a100a101a27
a15V?a29a30a31a59a232.
(3) a65a66a67,
(a27η1,···,a27ηn) = (η1,···,ηn)A,
a14
(a27
f1,···,
a27
fn) = (f1,···,fn)S,
· 31 ·
a57
a27
fj =
nsummationdisplay
k=1
skjfk,j = 1,···,n.
a70a71
(a27
fj)(ηi) =
nsummationdisplay
k=1
skjfk(ηi) = sij,i,j = 1,···,n.
a136a35a114a186,
(a27
fj)(ηi) = fj(
a27ηi) = fj
parenleftBigg nsummationdisplay
l=1
aljηl
parenrightBigg
=
nsummationdisplay
l=1
aljfj(ηl) = aji,i,j = 1,···,n.
a99
a15
aji = sij,i,j = 1,···,n.
a65
a101
a68
S = AT.
· 32 ·
