x0x1x2x3
x0x1x2 x3x4x5x6
x0 x1 7–1
1,x7V x8nx9xaxbxcxd,·1;·2;￠￠￠ ;·nx8V xexfx10x11.
fi1 = ·1 +·2 +￠￠￠+·n,fi2 = ·2 +￠￠￠+·n,￠￠￠,fin = ·n
(1)x12x13,fi1;fi2;￠￠￠ ;finx8V xexfx10x11;
(2)x14x15x11·1;·2;￠￠￠ ;·nx16x11fi1;fi2;￠￠￠ ;finxex17x18x19x1a;
(3)x7fix1bx11·1;·2;￠￠￠ ;·nx1cxex1dx1ex8(a1;a2;￠￠￠ ;an),x14fix1bx11fi1;fi2;￠￠￠ ;finx1cxex1dx1e.
x4,(1),(2)x1fx8
(fi1;fi2;￠￠￠ ;fin) = ("1;"2;￠￠￠ ;"n)
0
BB
B@
1 0 ￠￠￠ 0
1 1 ￠￠￠ 0.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;
x7
T =
0
BB
B@
1 0 ￠￠￠ 0
1 1 ￠￠￠ 0.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;
x20Tx21x22,x23x24fi1;fi2;￠￠￠ ;finx8V xex11,x25x15x11"1;"2;￠￠￠ ;"nx16x11fi1;fi2;￠￠￠ ;finxex17x18x19x1ax8T.
(3)x7
fi = ("1;"2;￠￠￠ ;"n)
0
B@
a1
...
an
1
CA;
x20
fi = (fi1;fi2;￠￠￠ ;fin)A?1
0
B@
a1
...
an
1
CA;
x26x27fix1bx11fi1;fi2;￠￠￠ ;finx1cxex1dx1ex8(a1;a2?a1;a3?a2;￠￠￠ ;an?an?1).
2,x1bK4x28,x14x15x11"1;"2;"3;"4x16x11·1;·2;·3;·4xex17x18x19x1a,x29x14x2ax2b?x1bx26x2cx11x1cxex1dx1e.
(1) "1 = (1;0;0;0);"2 = (0;1;0;0);"3 = (0;0;1;0);"4 = (0;0;0;1);
·1 = (2;1;?1;1);·2 = (0;?1;1;0);·3 = (?1;?1;2;1);·4 = (2;1;1;3);
= (x1;x2;x3;x4)x1b·1;·2;·3;·4x1cxex1dx1e;
(2) "1 = (1;2;?1;0);"2 = (1;?1;1;1);"3 = (?1;2;1;1);"4=(?1;?1;0;1);
·1 = (2;1;0;1);·2 = (0;1;2;2);·3 = (?3;?1;?1;1),·4 = (1;3;1;2);
= (1;0;0;0)x1b"1;"2;"3;"4x1cxex1dx1e;
(3) "1 = (1;1;1;1);"2 = (1;1;?1;?1);"3=(1;?1;1;?1);"4=(1;?1;?1;1);
·1 = (1;1;0;1);·2 = (2;1;2;1);·3 = (1;1;1;0);·4 = (0;1;?1;?1);
￠ 1 ￠
= (1;0;0;?1)x1b·1;·2;·3;·4x1cxex1dx1e.
x4,(1) T =
0
BB
B@
2 0?1 2
1?1?1 1
1 1 2 1
1 0 1 3
1
CC
CA,
0
BB
B@
y1
y2
y3
y4
1
CC
CA = T?1
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 12
0
BB
B@
1?5?5 4
2 0 2?2
2?4?4 4
1 3 3?2
1
CC
CA
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 12
0
BB
B@
x1?5x2?5x3 +4x4
2x1 +2x3?2x4
2x1?4x2?4x3 +4x4
x1 +3x2 +3x3?2x4
1
CC
CA:
(2) T =
0
BB
B@
1 0 0 1
1 1?1 1
0 1 0 1
0 0 2 0
1
CC
CA,?x1bx26x2cx11x1cxex1dx1ex8 113
0
BB
B@
3
5
2
3
1
CC
CA.
(3) T = 14
0
BB
B@
3 6 3?1
1 0 1 3
1 2 1?1
1 0?1?1
1
CC
CA,?x1bx26x2cx11x1cxex1dx1ex8
0
BB
B@
2
4
5
3
1
CC
CA.
3,x2dx2ex2f(2),x14xfx2ax2b,x30x1bx11"1;"2;"3;"4x1cxex1dx1ex31x1bx11·1;·2;·3;·4x1cxex1dx1exe2x32.
x4,·1 +·2 +·3 +·4 = (0;4;2;6).
4,x7K[x]nx33x34x15K[x]x28x35x36x37x38nxex39x3ax3bx3cx3dxexaxbxcxd.
fi(x) = (x?a1)￠￠￠(x?ai?1)(x?ai+1)￠￠￠(x?an); i = 1;￠￠￠ ;n;
x3ex28ai 2 K (i = 1;2;￠￠￠ ;n)x8x3fx40x41x42xex36.
(1)x12x13,f1(x);f2(x);￠￠￠ ;fn(x)x3cx3dK[x]nxexfx10x11;
(2)x43a1;a2;￠￠￠ ;anx8x44x45nx35x46x47x48,x14x15x111;x;x2;￠￠￠ ;xn?1x16x11f1(x);f2(x);￠￠￠ ;fn(x)xex17
x18x19x1a.
x4,(1)x49x4ax12f1(x);f2(x);￠￠￠ ;fn(x)xaxbx4bx4cx4dx21,x7
k1f1(x)+k2f2(x)+￠￠￠+knfn(x) = 0;
x4ex4fx27x = aix50x51x2ex3b,x52
kifi(ai) = 0:
x1fx8fi(ai) 6= 0,x26x27ki = 0,i = 1;2;￠￠￠ ;n,x53f1(x);f2(x);￠￠￠ ;fn(x)xaxbx4bx4c.x54x1fdimK[x]n = n,x21
x55f1(x);f2(x);￠￠￠ ;fn(x)x8K[x]nxex11.
(2)x7x44x56nx35x46x47x48x311;"1;￠￠￠ ;"n?1,x20
fi(x) = x
n?1
x?"i =
xn?"ni
x?"i = x
n?1 +"ixn?2 +"2
ix
n?3 +￠￠￠+"n?1
i ;
x53x26x14x17x18x19x1ax8 0
BB
B@
1 "n?11 ￠￠￠ "n?1n?1
1 "n?21 ￠￠￠ "n?2n?1
...,..,..,..
1 1 ￠￠￠ 1
1
CC
CA:
5,x1bK[x]x28,x57
hxi0 = 1;hxi = x;hxik = x(x?1)(x?2)￠￠￠(x?k +1); k > 1
(1)x14K[x]5x28x15x111;hxi;hxi2;hxi3;hxi4x16x111;x;x2;x3;x4xex17x18x19x1a;
￠ 2 ￠
(2)x14K[x]5x28x39x3ax3bf(x) = 1+x+x2 +x3 +x4x1bx111;hxi;hxi2;hxi3;hxi4x1cxex1dx1e;
(3)x12x13:
nX
x=0
hxik = 1k +1hn+1ik+1;
(4)x15x58x59x5ax36x5bDn =
nX
k=0
k4xex5cx3ax5dx3b.
x4,(1) 1 = 1
x = hxi
x2 = 0+x+x(x?1) = 0+hxi+hxi2
x3 = x+3x(x?1)+x(x?1)(x?2) = hxi+3hxi2 +hxi3
x4 = hxi+7hxi2 +6hxi3 +hxi4
x53x26x14x17x18x19x1ax8
T =
0
BB
BB
B@
1 0 0 0 0
0 1 1 1 1
0 0 1 3 7
0 0 0 1 6
0 0 0 0 1
1
CC
CC
CA:
(2) (1;4;11;7;1).
(3)x5ex55hx+1ik+1?hxik+1 = (k +1)hxik,x26x27
nX
x=0
hxik = 1k +1
nX
x=0
[hx+1ik+1?hxik+1]
= 1k +1
"n+1X
x=1
hxik+1?
nX
x=0
hxik+1
#
= 1k +1 (hn+1ik+1?h0ik+1)
= 1k +1 hn+1ik+1:
(4)x1fx8x4 = hxi+7hxi2 +6hxi3 +hxi4,x26x27
Dn =
nX
x=0
x4 =
nX
x=0
hxi+7hxi2 +6hxi3 +hxi4￠
= 12 hn+1i2 + 73 hn+1i3 + 64 hn+1i4 + 15 hn+1i5
= 130 n(n+1)(2n+1)(3n2 +3n?1):
x0 x1 7–2
1,x2cx5fK3xex60x10x11:
"1 = (1;1;?1);
"2 = (1;0;?1);
"3 = (1;1;1);
·1 = (1;?1;2);
·2 = (2;?1;2);
·3 = (?2;1;1):
x7x41 x8K3xexaxbx61x62,x63:
x41"i = ·i i = 1;2;3:
(1)x14x15x11"1;"2;"3x16x11·1;·2;·3xex17x18x19x1a;
(2)x14x41 x1bx11"1;"2;"3x1cxex19x1a;
￠ 3 ￠
(3)x14x41 x1bx11·1;·2;·3x1cxex19x1a;
(4)x7fi = (2;?1;3),x4ex4fx14x41fix1bx11"1;"2;"3x64x11·1;·2;·3x1cxex1dx1e.
x4,x7K3x1ex65x11x8?1 = (1;0;0),?2 = (0;1;0),?3 = (0;0;1),x66
B =
0
@
1 1 1
1 0 1
1?1 1
1
A; C =
0
@
1 2?2
1?1 1
2 2 1
1
A;
x20x67
("1;"2;"3) = (?1;?2;?3)B; (·1;·2;·3) = (?1;?2;?3)C:
(1)x15x38(·1;·2;·3) = (?1;?2;?3)C = ("1;"2;"3)B?1C,x53x15x11"1;"2;"3x16x11·1;·2;·3xex17x18x19x1ax8
T = B?1C =
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(2)x15x38(x41("1);x41("2);x41("3)) = (·1;·2;·3) = ("1;"2;"3)B?1C,x53x41 x1bx11"1;"2;"3x1cxex19x1ax8
A = B?1C =
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(3)x7x41 x1bx11·1;·2;·3x1cxex19x1ax8A0,x20
A0 = T?1AT = (B?1C)?1(B?1C)(B?1C) = B?1C ==
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(4) fix11"1;"2;"3x64x11·1;·2;·3x1cxex1dx1ex4ex4fx8
B?1
0
@
2
1
3
1
A x64 C?1
0
@
2
1
3
1
A;
x1fx58x41fix1bx11"1;"2;"3x1cxex1dx1ex8
AB?1
0
@
2
1
3
1
A = 12
0
@
7
11
1
1
A;
x41 x1bx11·1;·2;·3x1cxex1dx1ex8
A0B?1
0
@
2
1
3
1
A = 12
0
@
7
6
5
1
A:
2,x7A? C,B? D,x12x13,
A 0
0 B
C 0
0 D
:
x2x3,x68x1bx21x22x19x1aT1;T2,x63x52
T?11 AT1 = C; T?12 BT2 = D;
x1fx58T =
T
1 0
0 T2
x21x22,x25
T?1
A 0
0 B
T =
T?1
1 AT1 0
0 T?12 BT2
=
C 0
0 D
:
x26x27
A 0
0 B
C 0
0 D
:
3,x7Ax21x22,x12x13,ABx64BAx41x69.
￠ 4 ￠
x2x3,x15x38A?1(AB)A = BA,x53AB? BA.
4,x7Ax21x22,x25A? B,x12x13,Bx6ax21x22,x25A?1? B?1.
x2x3,x15x38T;Ax6bx21x22,x26x27Bx21x22,x25
B?1 = (T?1AT)?1 = TAT?1;
x53A?1? B?1.
5,x7A? B,x12x13,AT? BT.
x2x3,x68x1bx21x22x19x1aT,x63x52T?1AT = B,x53BT = (T?1AT)T = TTATT?T.
6,x7A? B,f(x) 2 K[x],x12x13,f(A)? f(B).
x2x3,x68x1bx21x22x19x1aT,x63x52T?1AT = B,x53
T?1(f(A))T = f(T?1AT) = f(B):
7,x12x13,0
BB
B@
1
2
...
n
1
CC
CA?
0
BB
B@
i1
i2
...
in
1
CC
CA;
x3ex28(i1;i2;￠￠￠ ;in)x31(1;2;￠￠￠ ;n)xexfx10x6cx5b.
x2x3,x7V x31nx9xaxbxcxd,"1;￠￠￠ ;"nx31V xex11,x41 x8V xexaxbx61x62,x5fx6dx8
x41"i =?i"i;
x20x41 x1bx11"1;￠￠￠ ;"nx1cxex19x1ax8
A =
0
BB
B@
1
2
...
n
1
CC
CA:
x15x38(i1;i2;￠￠￠ ;in)x31(1;2;￠￠￠ ;n)xexfx10x6cx5b,x1fx58"i1;￠￠￠ ;"inx6ex8V xex11,x24
x41"ij =?ij"ij; j = 1;￠￠￠ ;n:
x53x41 x1bx11"i1;￠￠￠ ;"inx1cxex19x1ax8
B =
0
BB
B@
i1
i2
...
in
1
CC
CA;
x23x24A? B.
8,x7A;B 2 Mn(R),x12x13,x6fx70x68x1bx21x22x19x1aU 2 Mn(C),x63A = U?1BU,x20x71x68x1bx21x22x19x1a
T 2 Mn(R),x63A = T?1BT.
x72,x73x74x75x76x77x78x79x7ax7bx7c:
x7dK,Fx75x7ex7fx0x1,x2x3Fx75Kx7ax4x1(x5K F),A;Bx75x0x1Kx6x7ax7ex7fx7x8,x9
xaA,BxbFx6xcxd,xexfx10x11xbKx6xcxd.
x2x3,x7U = T1 +iT2,T1;T2 2 Mn(R),x20(T1 +iT2)A = B(T1 +iT2),x54x1fA;B 2 Mn(R),x21x12x5a
T1A = BT1; T2A = BT2:
￠ 5 ￠
x13x14f(?) = jT1 +?T2j,x15x38f(i) = jUj 6= 0,x15x13f(?)x40x31x16x39x3ax3b,x71x67?0 2Rx63f(?0) 6= 0,x23x24
T = T1 +?0T2x21x22,x15T1A = BT1,T2A = BT2x21x52TA = BT,x53A = T?1BT.
9,x7x;y;z 2 K,x66
A =
0
@
x y z
y z x
z x y
1
A; B =
0
@
z x y
x y z
y z x
1
A; C =
0
@
y z x
z x y
x y z
1
A:
x12x13,A;B;Cx17x58x41x69.
x2x3,x43x18x62x19x1a
P =
0
@
0 1 0
0 0 1
1 0 0
1
A; Q =
0
@
0 0 1
1 0 0
0 1 0
1
A;
x20P;Qx6bx21x22,x25
P?1AP = C; Q?1AQ = B;
x26x27A? B,A? C,x15x41x69x4cx19xex1ax1bxb,x21x52B? C.
10,x12x13,0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA
n
0
BB
B@
n 0 ￠￠￠ 0
0 0 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ 0
1
CC
CA:
x2x3,x7V x31nx9xaxbxcxd,"1;￠￠￠ ;"nx31V xex11,x41 x8V xexaxbx61x62,x5fx6dx8
x41"i = "1 +"2 +￠￠￠+"n; i = 1;2;￠￠￠ ;n:
x20x41 x1bx11"1;￠￠￠ ;"nx1cxex19x1ax8
A =
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA
n
:
x54x5ex55
fi1 = "1 +"2 +￠￠￠+"n; fi2 = "1?"2;￠￠￠ ;fin = "1?"n
x6ex8V xex11,x25x41 x1bx11fi1;￠￠￠ ;finx1cxex19x1ax8
B =
0
BB
B@
n 0 ￠￠￠ 0
0 0 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ 0
1
CC
CA:
x23x24A? B.
x0 x1 7–3
1,x14x1cx36x1dx2exaxbxcxdV xexaxbx61x62x41 xex1ex1fx20x64x1ex1fx2ax2b,x7x41 x1bV xexfx10x11x1cxex19x1ax31:
(1) A =
2 5
4 3
; (2) A =
0 a
a 0
(a 6= 0);
(3) A =
0
BB
B@
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
1
CC
CA; (4) A =
0
@
5 6?3
1 0 1
1 2?1
1
A;
￠ 6 ￠
(5) A =
0
@
0 0 1
0 1 0
1 0 0
1
A; (6) A =
0
@
0?2?1
2 0?1
1?3 1
1
A;
(7) A =
0
@
4 3 0
3?2 0
2?6 2
1
A:
x4,x36x21x33x34x1ex1fx20,x22x23x1bx24xex2ax2bx25x31x41x26xexfx10x1ex1fx2ax2b.
(1) 7,(1;1);?2,(5;?4).
(2) ai,(1;i);?ai,(i;1).
(3) 2,k(1;1;0;0)+l(1;0;1;0)+m(1;0;0;1);?2,(?1;1;1;1).
(4) 2,(?2;1;0); 1+p3,(?3;1;?2+p3); 1?p3,(?3;1;?2?p3).
(5) 1,k(1;0;1)+l(0;1;0);?1,(?1;0;1).
(6)?3,(1;1;1); 2,(1;1;?4).
(7) 2,(0;0;1); 1,(?1;1;8).
2,x7?1,?2x31xaxbx61x62x41xex60x10x40x42xex1ex1fx20,"1,"2x4ex4fx31x41xex27x38x1ex1fx20?1,?2xex1ex1fx2ax2b.
x12x13,"1 +"2x40x31x41 xex1ex1fx2ax2b.
x2x3,(x28x12)x6fx70"1 +"2x31x41 xex27x38x29x10x1ex1fx20?0xex1ex1fx2ax2b,x20
x41("1 +"2) =?0("1 +"2):
x54x41("1 +"2) = x41"1 + x41"2 =?1"1 +?2"2,x26x27
(?10)"1 +(?20)"2 = 0:
x15?1 6=?2x21x52"1;"2xaxbx4bx4c,x1fx58
10 = 0;?20 = 0;
x52x16?1 =?0 =?2,x2ax2b.
3,x12x13,x6fx70xaxbx61x62x41 x27x2cx10x2dx16x2ax2bx2ex8x30xex1ex1fx2ax2b,x20x41 x8x1ex2bx2fx30x61x62.
x2x3,x7x31x29x10x2dx16x2ax2bfix67x41fi = kfi,x31x32xfx10x2dx16x2ax2bfl,x67x41fl = mfl,x6fx70k 6= m,x20x48x33
x34x2f2xex35x36,fi+flx40x31x41xex1ex1fx2ax2b,x6fx70fi+fl = 0,x20x67x41fl =?x41fi =?kfi = kfl,x64k 6= mx2ax2b.
x1fx58fi+flx31x2dx16x2ax2b,x64x2fx7x2ax2b.
4,x12x13,ABx64BAx67x41x42xex1ex1fx20.
x2x3,x48x33x34x2f4–8xex374x2f,x38jAj6= 0,AC = CAx39,x67fl
flfl
fl
A B
C D
flfl
flfl = jAD?CBj:
x1fx58 fl
flfl
fl
E B
A E
flfl
flfl = j?E?ABj:
x54x1f
0 E
E 0
E B
A E
0 E
E 0
=
E A
B?E
;
x60x3ax43x3bx5bx3b,x4dx52 fl
flfl
fl
E A
B?E
flfl
flfl = j?E?BAj =
flfl
flfl?E B
A E
flfl
flfl = j?E?ABj:
x1fx58ABx64BAx67x41x42xex1ex1fx39x3ax3b,x23x24x67x41x42xex1ex1fx20.
￠ 7 ￠
5,x12x13,x3cx3dx3ex52xcxdxex3fx40x61x62xex1ex1fx20(x6fx67xex41)x49x42x31§1.
x2x3,x7fix31x27x38x3fx40x61x62x41 xex1ex1fx20?0xex1ex1fx2ax2b,x20
0 6= (fi;fi) = (x41fi;x41fi) =?20(fi;fi);
x1fx58?20 = 1,?0 = §1.
6,x12x13,x43x16x19x1axex1ex1fx20x44x8x16.
x2x3,x7fix31x27x38x43x16x19x1aAxex1ex1fx20?0xex1ex1fx2ax2b,x20Afi =?0fi,x15x38
0 = Akfi =?k0fi;
x21x52?k0 = 0,?0 = 0.
7,x7A = (a1;a2;￠￠￠ ;an) 2Rn (aix40x44x8x16),x14x19x1aATAxex1ex1fx20x64x1ex1fx2ax2b.
x4,x7 ai 6= 0,x1ex1fx20 0 x31x26xex1ex1fx2ax2bx31 fij = (0;￠￠￠ ;0;ai;￠￠￠ ;?aj;0;￠￠￠ ;0)
j i
(j=1;￠￠￠ ;i?1;i+1;￠￠￠ ;n)xexaxbx3cx44.x1ex1fx20
nP
j=1
a2jxex1ex1fx2ax2bx31(a1;￠￠￠ ;an).
8,x7A = (aij) 2 Mn(K),x45x3b
flfl
flfl
flfl
flfl
fl
ai1i1 ai1i2 ￠￠￠ ai1ik
ai2i1 ai2i2 ￠￠￠ ai2ik
...,..,..,..
aiki1 aiki2 ￠￠￠ aikik
flfl
flfl
flfl
flfl
fl; (1 6 i1 < i2 < ￠￠￠ < ik 6 n);
x46x8Axexfx10kx47x48x45x3b,x66x1ex1fx39x3ax3b
′A(?) = j?E?Aj =?n?a1?n?1 +￠￠￠+(?1)n?1an?1?+(?1)nan:
x12x13,akx49x38Axex44x56kx47x48x45x3bx4ax4b.
x2x3,x4cj?E?Ajxex2cx5b 0
BB
BB
BB
BB
B@
a1j
a2j
...
ajj
...
anj
1
CC
CC
CC
CC
CA
x4dx4ex3dx60x5b,0
BB
BB
BB
BB
BB
@
0.
..
0
0.
..
0
1
CC
CC
CC
CC
CC
A
x64
0
BB
BB
BB
BB
B@
a1j
a2j
...
ajj
...
anj
1
CC
CC
CC
CC
CA
x20x3bx5bx3bj?E?Ajx21x4ex4fx82nx10nx47x3bx5bx3bx4ax4b,x3ex28x2cx10x3bx5bx3bxex5bx4dx31x2ex50x60x51x52x3bx4axf.
x7Akx8x53xfx54x67kx10?xex45x3bx5bx3b,x3e?x55x38j1;￠￠￠ ;jkx5b,x56Akx57x58kx5bx59x5a,x52
Ak =?k ￠(?1)n?kDn?k;
x3ex28Dn?kx8x1bAx28x5bx5cx37j1;￠￠￠ ;jkx5bx5dx37j1;￠￠￠ ;jkx3bx24x52x16xen?kx47x48x45x3b,x38x58kx10?x43x5en
x47x3bx5bx3bx28x26x67x21x42xekx10x47x18,x20Dn?kx25x43x5ex26x67Cknx10x48x45x3b,x23x24′A(?)x28?n?kxex19x36x49x38
￠ 8 ￠
(?1)kx2fx27Axex26x67kx47x48x45x3bx4ax4b.x1fx58akx8Axex26x67kx47x48x45x3bx4ax4b.
9,x7A 2 Mn(K),x12x13,x68x1bKx2exexfx10x35x36x40x5fx17n2xex39x3ax3bf(x),x63f(A) = 0.
x2x3,x1fx8Mn(K)x31Kx2en2x9xaxbxcxd,x53E;A;A2;￠￠￠ ;An2?1;An2xaxbx41x4c.x38x31x68x1bx40x44x8
x16xeai 2 K,i = 1;￠￠￠ ;n2x63x52
a0E +a1A+￠￠￠+an2?1An2?1 +an2An2 = 0:
x66
f(x) = an2xn2 +an2?1xn2?1 +￠￠￠+a1x+a0;
x20f(A) = 0.
10,x7A 2 Mn(C),x12x13,x68x1bx21x22x19x1aT 2 GL(n;C),x63T?1ATx8x2ex60x61x19x1a.
x2x3,x31nx62x36x63x64x65x66,x38n = 1x39x35x36x67x68x3dx69,x6ax7x35x36x31n?1x47x19x1ax3dx69.
x7?1x31Axexfx10x1ex1fx20,x41x26xex1ex1fx2ax2bx31fi1 2 Cn,x4cfi1x6bx6cx3dCnxex11fi1;fi2;￠￠￠ ;fin,x66
T1 = (fi1;fi2;￠￠￠ ;fin),x20T1x21x22,x25
AT1 = T1
1?
0 A1
; x4d T?11 AT1 =
1?
0 A1
;
x3ex28A1 2 Mn?1(C),x15x64x65x6dx7,x68x1bx21x22x19x1aT2 2 Mn?1(C),x63x52
T?12 A1T2 =
0
B@
2?
...
0?n
1
CA;
x66T = T1
1 0
0 T2
,x20Tx21x22,x25
T?1AT =
0
B@
1?
...
0?n
1
CA:
11,x7A 2 Mn(C),f(x)x8xfx1cx19x36x39x3ax3b,x12x13,x6fx70Axex44x56x1ex1fx20x8?1,?2,￠￠￠,?n,x20f(A)
xex44x56x1ex1fx20x8f(?1),f(?2),￠￠￠,f(?n).
x2x3,x15x34x2f10,x68x1bx21x22x19x1aT,x63
T?1AT =
0
BB
B@
1?
2
...
0?n
1
CC
CA;
x58x3e?1;￠￠￠ ;?nx31Axex44x56x1ex1fx20,x23x24
T?1f(A)T = f(T?1AT) = f
0
BB
B@
0
BB
B@
1?
2
...
0?n
1
CC
CA
1
CC
CA =
0
BB
B@
f(?1)?
f(?2)
...
0 f(?n)
1
CC
CA:
x26x27f(?1);f(?2);￠￠￠ ;f(?n)x8f(A)xex44x56x1ex1fx20.
x0 x1 7–4
1,x34x2f7–3x37xfx2fx28xex19x1a,x6ex6fx31x21x27x31x61x70xe? x1bx21x31x61x70xex71x72x1c,x14x5ax41x26xex17x18x19x1a
x4bx31x61x19x1a.
￠ 9 ￠
x4,(1) T =
1 5
1?4
,T?1AT = diag(7;?2).
(2) T =
1 i
i 1
,T?1AT = diag(ai;?ai).
(3) T =
0
BB
B@
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
1
CC
CA,T?1AT = diag(?2;2;2;2).
(4) T =
0
@
2?3?3
1 1 1
0?2+p3?2?p3
1
A,T?1AT = diag(2;1+p3;1?p3).
(5) T =
0
@
1 1 0
0 0 1
1 1 0
1
A,T?1AT = diag(?1;1;1).
(6),(7)x40x21x31x61x70.
2,x1bK[x]nx28,x14x73x4ex61x62x44:
x44(f(x)) = f0(x)
xex1ex1fx39x3ax3b,x29x12x13,x44x1bx53x74xfx10x11x1cxex19x1ax4dx40x21x42x31x31x61x19x1a.
x4,x43K[x]nxex111;x;x2;￠￠￠ ;xn?1,x20x44x1bx58x10x11x1cxex19x1ax31
D =
0
BB
B@
0 1 0 ￠￠￠ 0
0 0 2 ￠￠￠ 0.
..,..,..,..,..
0 0 0 ￠￠￠ 0
1
CC
CA:
x23x24x44xex1ex1fx39x3ax3bx8′D(?) =?n,x6fx70x44x21x31x61x70,x20x68x1bx21x22x19x1aTx63x52T?1AT = 0,x4dD = 0,
x24x44x40x31x16x61x62,x2ax2b.
3,x7A 2 Mn(K),x12x13,x6fx70rankA+rank(A?E) = n,x20Ax21x31x61x70.
x2x3,x15x34x2f4–8.13x55,rankA+rank(A?E) = nxex6cx4ex71x4ax75x76x31A2 = A,x4dx31Axex53xfx5bx2a
x2bfix67Afi = fi,x54A(A?E) = 0,x53x31A?Exex53xfx5bx2ax2bflx67Afl = 0.
x7Axex5bx2ax2bx3cxex77x78x4bx4cx3cx8fi1;￠￠￠ ;fir,A? E xex77x78x4bx4cx5bx2ax2bx3cx8fl1;￠￠￠ ;fln?r (x1fx8
rankA+rank(A?E) = n),x1cx12fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?rxaxbx4bx4c.
x7x67
rX
i=1
kifii +
n?rX
j=1
mjflj = 0:
x20
rX
i=1
kiAfii +
n?rX
j=1
mjAflj =
rX
i=1
kiAfii =
rX
i=1
kifii = 0:
x38x31k1 = ￠￠￠ = kr = 0,x79x24m1 = ￠￠￠ = mn?r = 0,x26x27fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?rxaxbx4bx4c.
x66T = (fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r),x20Tx21x22,x25
AT = A(fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r) = (fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r)
E
r 0
0 0
;
x4d
AT = T
E
r 0
0 0
;
￠ 10 ￠
x23x24
T?1AT =
E
r 0
0 0
:
4,x7A 2 Mn(K),x12x13,x6fx70rank(A+E)+rank(A?E) = n,x20Ax21x31x61x70.
x2x3,x64x2ex2fx7ax69,x7b.
5,x7x41 x8x36x1dKx2enx9xaxbxcxdV xexaxbx61x62,x7cx7dx412 = x41,x14x41 xex1ex1fx20,x29x12x13x41 x21x31x61
x70.
x2x3,x7
V1 = fx41fi j fi 2 Vg; V2 = ffi?x41fi j fi 2 Vg:
x20x31x53x7exefi 2 V,x67fi = x41fi+(fi?x41fi),x1fx58
V = V1 +V2:
x7ffi 2 V1 \V2,x20
fi = x41fl =?x41 ; fl; 2 V:
x38x31
x41fi = x412fl = x41fl = fi; (*)
x41fi = x41?x412 = x41?x41 = 0,(**)
x26x27fi = x41fi = 0,x4dV1 \V2 = 0,x58x0x25x12x13x1
V = V1 'V2:
x31x38V1x28xex2ax2bfi,x67(*)x3bx3dx69,x15x13V1x31x27x38x1ex1fx201xex1ex1fx45xcxd,x7ax69x2x15(**)x3bx21x52V2x31x27
x38x1ex1fx200xex1ex1fx45xcxd,x48x33x12x364.5,x41 x21x31x61x70,x41 xex1ex1fx20x80,1.
(x3,x6ax21x4x62x34x2f3xex5x66x6x27x12x13)
6,x7
A =
1 1
1 0
:
(1)x7x56x19x1aAx31x61x70,x8x14An;
(2)x4x62x2ex50x35x70,x14x9xaxbxcx36x5bxex5cx3ax5dx3b.
x4,(1) Axex1ex1fx20x8 1§
p5
2,x41x26xex1ex1fx2ax2bx8?
1+p5
21
!;
1?p5
21
!
:
x66
T =
1+p5
2
1?p5
21 1
!;
x20
T?1AT =
1+p5
2 0
0 1?
p5
2
!
:
￠ 11 ￠
x23x24
An = T
0
B@
1+p5
2
n
0
0
1?p5
2
n
1
CAT?1
= 1p5
1+p5
2
1?p5
21 1
!0
B@
1+p5
2
n
0
0
1?p5
2
n
1
CA
1? 1?p5
2
1 1+
p5
2
!
=
p5
5
fin+1?fln+1 fin?fln
fin?fln fin?1?fln?1
;
x3ex28fi = 1+
p5
2,fl =
1?p5
2,
(2)x66
fi0 =
1
0
;fi1 =
1
1
;￠￠￠ ;fin =
a
n+1
an
;
x3ex28aix31x37ix10x9xaxbxcx36,x20
fin = Anfi0:
x26x27
an+1
an
= Anfi0 =
p5
5
fin+1?fln+1 fin?fln
fin?fln fin?1?fln?1
1
0
;
xdx52
an =
p5
5 (fi
n?fln) =
p5
5
"?
1+p5
2
!n
1?p5
2
!n#
:
7,x7A = (aij)x8nx47x2ex60x61x19x1a,x12x13:
(1)x7faii 6= ajj (i 6= j),x20Ax21x31x61x70;
(2)x7fa11 = a22 = ￠￠￠ = ann,x25xexfx67xfx10aij 6= 0 (i 6= j),x20Ax40x21x31x61x70.
x2x3,(1)x15x38Ax31x2ex60x61x19x1a,x53a11;a22;￠￠￠ ;annx8Axenx10x1ex1fx20,x7fx38i 6= jx39aii 6= ajj,x20
Ax67nx10x40x42xex1ex1fx20,x23x24Ax21x31x61x70.
(2) (x28x12)x10x55Axex1ex1fx20x8?0 = a11 = ￠￠￠ = ann,x6fAx21x31x61x70,x20x68x1bx21x22x19x1aT,x63x52
T?1AT = diag(?0;￠￠￠ ;?0):
x38x31
A = T
0
B@
0
...
0
1
CAT?1 =
0
B@
0
...
0
1
CA:
x4dAx8x11x2bx1a,x64x6dx7x2ax2b.
8,x7x67x4ex12x31x61x19x1a
A =
0
BB
B@
A1 0 ￠￠￠ 0
0 A2 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ As
1
CC
CA:
x12x13,Ax21x31x61x70xex6cx4ex71x4ax75x76x31x2cx10Aix6bx21x31x61x70.
x2x3,x6cx4exbx31x13x68xe,x1cx12x71x4axb.
￠ 12 ￠
x7Ax21x31x61x70,x20x68x1bx21x22x19x1aT,x63x52
T?1AT =
0
BB
B@
1
2
...
n
1
CC
CA:
x38x31
AT = T
0
BB
B@
1
2
...
n
1
CC
CA:
x66
T =
0
BB
B@
T1
T2
...
Ts
1
CC
CA;
x3ex28Tixex3bx36x49x38Aixex47x36ri,x20
AiTi = Ti
0
BB
B@
1
2
...
n
1
CC
CA:
x58x15x13Tixex2dx16x5bx2ax2bx4dx31Aixex1ex1fx2ax2b,x54x1fTx21x22,x53Txex3bx2ax2bxaxbx4bx4c,x1fx24Tixex3bx2ax2bx6a
xaxbx4bx4c.x38x31rankTi = Aixex47x36,Tixex5bx14x6ax49x38Aixex47x36,x1fx58Aix67rix10xaxbx4bx4cxex1ex1fx2ax2b,
x15x13Aix6bx21x31x61x70.
9,x7A;B 2 Mn(K),x25AB = BA,x12x13,x6fx70A,Bx4dx21x31x61x70,x20x68x1bx21x22x19x1aT 2 Mn(K),x63
T?1ATx64T?1BTx42x8x31x61x19x1a.
x2x3,x7Axex40x42x1ex1fx20x8?1;?2;￠￠￠ ;?s,x3ex28?ixex15x36x8ri,x15x38Ax21x31x61x70,x68x1bx21x22x19x1aT,
x63
T?1AT = A1 =
0
B@
1Er1
...
sErs
1
CA:
x66B1 = T?1BT,x20A1B1 = B1A1.
x66B1 = (Bij),x3ex4ex12x5x3bx64A1x41x42,x20x15A1B1 = B1A1x52
iBij = Bij?j
x38x31x38i 6= jx39x67Bij = 0,x4d
B1 =
0
BB
B@
B11
B22
...
Bss
1
CC
CA:
x54x1fBx21x31x61x70,B1x6ax21x31x61x70,x23x24x15x2ex2fx55x2cx10Biix4dx21x31x61x70,x4dx68x1bx21x22x19x1aSi 2 Mri(K)x63
S?1i BiiSi =
0
B@
i1
...
iri
1
CA:
￠ 13 ￠
x66
T = T1
0
B@
S1
...
Ss
1
CA;
x20Tx21x22,x25
T?1AT =
0
B@
S?11
...
S?1s
1
CA
0
B@
1Er1
...
sErs
1
CA
0
B@
S1
...
Ss
1
CA =
0
B@
1Er1
...
sErs
1
CA;
T?1BT =
0
B@
S?11
...
S?1s
1
CA
0
B@
B11
...
Bss
1
CA
0
B@
S1
...
Ss
1
CA
=
0
B@
S?11 B11S1
...
S?1s BssSs
1
CA =
0
BB
BB
BB
BB
BB
BB
@
11
...
1r1
...
s1
...
srs
1
CC
CC
CC
CC
CC
CC
A
x42x8x31x61x52.
x0 x1 7–5
1,x7x41 x31xaxbxcxdV xexaxbx61x62,x10x55x41 x1bx11·1;·2;￠￠￠ ;·nx1cxex19x1ax31
A =
0
BB
B@
0 1
...,..
..,1
0
1
CC
CA:
x14x41 xex26x67x40x61x45xcxd.
x4,x7Wx31x41 xexfx10x2dx16x40x61x45xcxd,x16x7x12x13,Wx71x17x54x29x10x11x2ax2b·i.
x7
0 6= fi = ak·k +￠￠￠+an·n 2 W; ak 6= 0:
x20x41k?1fi = ak·1 2 W,·1 2 W.
x8x7·kx31x17x54x38Wx28xex1cx1ex18x78xex11x2ax2b,x20
·k?1 = x41·k;·k?2 = x412·k;￠￠￠ ;·1 = x41k?1·k 2 W:
x1cx19x12
W = L(·1;￠￠￠ ;·k):
x62x28x12x66,x6fx67fi 2 W,x1afi 62 L(·1;￠￠￠ ;·k),x20
fi = a1·1 +￠￠￠+am·m; x3ex28am 6= 0;m > k:
x38x31
am?k·1 +￠￠￠+am?1·k +am·k+1 = x41m?k?1fi 2 W:
￠ 14 ￠
x54x1fam?k·1 +￠￠￠+am?1·k 2 W,am 6= 0,x21x52·k+1 2 W,x2ax2b,x1fx58x41xex26x67x40x61x45xcxdx8x16x45xcxd
x27x1bL(·1;￠￠￠ ;·k),k = 1;￠￠￠ ;n.
2,x7x41 x8x3cx3dx3ex52xcxdxex3fx40x61x62,x12x13,x41 xex40x61x45xcxdxex3fx40x1cx6ax31x41 xex40x61x45xcxd.
x2x3,x7Wx31x41 xex40x61x45xcxd,x20x41(W) W,x15x38x3fx40x61x62x71x21x22,x1fx58x41(W) = W.
x31x38x53x7exefl 2 W?,fi 2 W,x71x68x1b 2 Wx63fi = x41, x38x31
(x41fl;fi) = (x41fl;x41 ) = (fl; ) = 0:
x58x15x13fl 2 W?,x53W?x31x41 xex40x61x45xcxd.
3,x7x41 x31xaxbxcxdV xexaxbx61x62,Wx8x41 xex40x61x45xcxd,x12x13,x41(W)x1dx31x41 xex40x61x45xcxd.
x2x3,x15x38Wx31x41 xex40x61x45xcxd,x41(W) W,x1fx58x41(x41(W)) x41(W),x15x13x41(W)x6ax31x41 xex40
x61x45xcxd.
4,x7V x31x1cx36x1dx2exenx9xaxbxcxd,x41,x42x31V xexaxbx61x62,x25x41x42 = x42x41,x12x13:
(1)x6fx70?x31x41 xexfx10x1ex1fx20,xbx1e,V?x31x42xex40x61x45xcxd;
(2) x41,x42xexfx67xfx10x5dx1fxex1ex1fx2ax2b.
x2x3,(1)x31x53x7exefi 2 V?,
x41(x42fi) = x42(x41fi) = x42(?fi) =?x42fi:
x1fx58x42fi 2 V?,x15x13V?x6ax31x42xex40x61x45xcxd.
(2)x15x2ex55,V?x31x42xex40x61x45xcxd,x23x24x42jV?x31V?x2exexaxbx61x62,x38x31x42jV?x67x1ex1fx20?x27x1bx41
x26xex1ex1fx2ax2bfl 2 V?,x63
x42fl = x42jV?(fl) =?fl:
x54x1ffl 2 V?,x41fl =?fl,x26x27flx31x41 x64x42xex5dx1fx1ex1fx2ax2b.
￠ 15 ￠
x0x1x2 x3x4x5x6
x0 x1 7–1
1,x7V x8nx9xaxbxcxd,·1;·2;￠￠￠ ;·nx8V xexfx10x11.
fi1 = ·1 +·2 +￠￠￠+·n,fi2 = ·2 +￠￠￠+·n,￠￠￠,fin = ·n
(1)x12x13,fi1;fi2;￠￠￠ ;finx8V xexfx10x11;
(2)x14x15x11·1;·2;￠￠￠ ;·nx16x11fi1;fi2;￠￠￠ ;finxex17x18x19x1a;
(3)x7fix1bx11·1;·2;￠￠￠ ;·nx1cxex1dx1ex8(a1;a2;￠￠￠ ;an),x14fix1bx11fi1;fi2;￠￠￠ ;finx1cxex1dx1e.
x4,(1),(2)x1fx8
(fi1;fi2;￠￠￠ ;fin) = ("1;"2;￠￠￠ ;"n)
0
BB
B@
1 0 ￠￠￠ 0
1 1 ￠￠￠ 0.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;
x7
T =
0
BB
B@
1 0 ￠￠￠ 0
1 1 ￠￠￠ 0.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;
x20Tx21x22,x23x24fi1;fi2;￠￠￠ ;finx8V xex11,x25x15x11"1;"2;￠￠￠ ;"nx16x11fi1;fi2;￠￠￠ ;finxex17x18x19x1ax8T.
(3)x7
fi = ("1;"2;￠￠￠ ;"n)
0
B@
a1
...
an
1
CA;
x20
fi = (fi1;fi2;￠￠￠ ;fin)A?1
0
B@
a1
...
an
1
CA;
x26x27fix1bx11fi1;fi2;￠￠￠ ;finx1cxex1dx1ex8(a1;a2?a1;a3?a2;￠￠￠ ;an?an?1).
2,x1bK4x28,x14x15x11"1;"2;"3;"4x16x11·1;·2;·3;·4xex17x18x19x1a,x29x14x2ax2b?x1bx26x2cx11x1cxex1dx1e.
(1) "1 = (1;0;0;0);"2 = (0;1;0;0);"3 = (0;0;1;0);"4 = (0;0;0;1);
·1 = (2;1;?1;1);·2 = (0;?1;1;0);·3 = (?1;?1;2;1);·4 = (2;1;1;3);
= (x1;x2;x3;x4)x1b·1;·2;·3;·4x1cxex1dx1e;
(2) "1 = (1;2;?1;0);"2 = (1;?1;1;1);"3 = (?1;2;1;1);"4=(?1;?1;0;1);
·1 = (2;1;0;1);·2 = (0;1;2;2);·3 = (?3;?1;?1;1),·4 = (1;3;1;2);
= (1;0;0;0)x1b"1;"2;"3;"4x1cxex1dx1e;
(3) "1 = (1;1;1;1);"2 = (1;1;?1;?1);"3=(1;?1;1;?1);"4=(1;?1;?1;1);
·1 = (1;1;0;1);·2 = (2;1;2;1);·3 = (1;1;1;0);·4 = (0;1;?1;?1);
￠ 1 ￠
= (1;0;0;?1)x1b·1;·2;·3;·4x1cxex1dx1e.
x4,(1) T =
0
BB
B@
2 0?1 2
1?1?1 1
1 1 2 1
1 0 1 3
1
CC
CA,
0
BB
B@
y1
y2
y3
y4
1
CC
CA = T?1
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 12
0
BB
B@
1?5?5 4
2 0 2?2
2?4?4 4
1 3 3?2
1
CC
CA
0
BB
B@
x1
x2
x3
x4
1
CC
CA = 12
0
BB
B@
x1?5x2?5x3 +4x4
2x1 +2x3?2x4
2x1?4x2?4x3 +4x4
x1 +3x2 +3x3?2x4
1
CC
CA:
(2) T =
0
BB
B@
1 0 0 1
1 1?1 1
0 1 0 1
0 0 2 0
1
CC
CA,?x1bx26x2cx11x1cxex1dx1ex8 113
0
BB
B@
3
5
2
3
1
CC
CA.
(3) T = 14
0
BB
B@
3 6 3?1
1 0 1 3
1 2 1?1
1 0?1?1
1
CC
CA,?x1bx26x2cx11x1cxex1dx1ex8
0
BB
B@
2
4
5
3
1
CC
CA.
3,x2dx2ex2f(2),x14xfx2ax2b,x30x1bx11"1;"2;"3;"4x1cxex1dx1ex31x1bx11·1;·2;·3;·4x1cxex1dx1exe2x32.
x4,·1 +·2 +·3 +·4 = (0;4;2;6).
4,x7K[x]nx33x34x15K[x]x28x35x36x37x38nxex39x3ax3bx3cx3dxexaxbxcxd.
fi(x) = (x?a1)￠￠￠(x?ai?1)(x?ai+1)￠￠￠(x?an); i = 1;￠￠￠ ;n;
x3ex28ai 2 K (i = 1;2;￠￠￠ ;n)x8x3fx40x41x42xex36.
(1)x12x13,f1(x);f2(x);￠￠￠ ;fn(x)x3cx3dK[x]nxexfx10x11;
(2)x43a1;a2;￠￠￠ ;anx8x44x45nx35x46x47x48,x14x15x111;x;x2;￠￠￠ ;xn?1x16x11f1(x);f2(x);￠￠￠ ;fn(x)xex17
x18x19x1a.
x4,(1)x49x4ax12f1(x);f2(x);￠￠￠ ;fn(x)xaxbx4bx4cx4dx21,x7
k1f1(x)+k2f2(x)+￠￠￠+knfn(x) = 0;
x4ex4fx27x = aix50x51x2ex3b,x52
kifi(ai) = 0:
x1fx8fi(ai) 6= 0,x26x27ki = 0,i = 1;2;￠￠￠ ;n,x53f1(x);f2(x);￠￠￠ ;fn(x)xaxbx4bx4c.x54x1fdimK[x]n = n,x21
x55f1(x);f2(x);￠￠￠ ;fn(x)x8K[x]nxex11.
(2)x7x44x56nx35x46x47x48x311;"1;￠￠￠ ;"n?1,x20
fi(x) = x
n?1
x?"i =
xn?"ni
x?"i = x
n?1 +"ixn?2 +"2
ix
n?3 +￠￠￠+"n?1
i ;
x53x26x14x17x18x19x1ax8 0
BB
B@
1 "n?11 ￠￠￠ "n?1n?1
1 "n?21 ￠￠￠ "n?2n?1
...,..,..,..
1 1 ￠￠￠ 1
1
CC
CA:
5,x1bK[x]x28,x57
hxi0 = 1;hxi = x;hxik = x(x?1)(x?2)￠￠￠(x?k +1); k > 1
(1)x14K[x]5x28x15x111;hxi;hxi2;hxi3;hxi4x16x111;x;x2;x3;x4xex17x18x19x1a;
￠ 2 ￠
(2)x14K[x]5x28x39x3ax3bf(x) = 1+x+x2 +x3 +x4x1bx111;hxi;hxi2;hxi3;hxi4x1cxex1dx1e;
(3)x12x13:
nX
x=0
hxik = 1k +1hn+1ik+1;
(4)x15x58x59x5ax36x5bDn =
nX
k=0
k4xex5cx3ax5dx3b.
x4,(1) 1 = 1
x = hxi
x2 = 0+x+x(x?1) = 0+hxi+hxi2
x3 = x+3x(x?1)+x(x?1)(x?2) = hxi+3hxi2 +hxi3
x4 = hxi+7hxi2 +6hxi3 +hxi4
x53x26x14x17x18x19x1ax8
T =
0
BB
BB
B@
1 0 0 0 0
0 1 1 1 1
0 0 1 3 7
0 0 0 1 6
0 0 0 0 1
1
CC
CC
CA:
(2) (1;4;11;7;1).
(3)x5ex55hx+1ik+1?hxik+1 = (k +1)hxik,x26x27
nX
x=0
hxik = 1k +1
nX
x=0
[hx+1ik+1?hxik+1]
= 1k +1
"n+1X
x=1
hxik+1?
nX
x=0
hxik+1
#
= 1k +1 (hn+1ik+1?h0ik+1)
= 1k +1 hn+1ik+1:
(4)x1fx8x4 = hxi+7hxi2 +6hxi3 +hxi4,x26x27
Dn =
nX
x=0
x4 =
nX
x=0
hxi+7hxi2 +6hxi3 +hxi4￠
= 12 hn+1i2 + 73 hn+1i3 + 64 hn+1i4 + 15 hn+1i5
= 130 n(n+1)(2n+1)(3n2 +3n?1):
x0 x1 7–2
1,x2cx5fK3xex60x10x11:
"1 = (1;1;?1);
"2 = (1;0;?1);
"3 = (1;1;1);
·1 = (1;?1;2);
·2 = (2;?1;2);
·3 = (?2;1;1):
x7x41 x8K3xexaxbx61x62,x63:
x41"i = ·i i = 1;2;3:
(1)x14x15x11"1;"2;"3x16x11·1;·2;·3xex17x18x19x1a;
(2)x14x41 x1bx11"1;"2;"3x1cxex19x1a;
￠ 3 ￠
(3)x14x41 x1bx11·1;·2;·3x1cxex19x1a;
(4)x7fi = (2;?1;3),x4ex4fx14x41fix1bx11"1;"2;"3x64x11·1;·2;·3x1cxex1dx1e.
x4,x7K3x1ex65x11x8?1 = (1;0;0),?2 = (0;1;0),?3 = (0;0;1),x66
B =
0
@
1 1 1
1 0 1
1?1 1
1
A; C =
0
@
1 2?2
1?1 1
2 2 1
1
A;
x20x67
("1;"2;"3) = (?1;?2;?3)B; (·1;·2;·3) = (?1;?2;?3)C:
(1)x15x38(·1;·2;·3) = (?1;?2;?3)C = ("1;"2;"3)B?1C,x53x15x11"1;"2;"3x16x11·1;·2;·3xex17x18x19x1ax8
T = B?1C =
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(2)x15x38(x41("1);x41("2);x41("3)) = (·1;·2;·3) = ("1;"2;"3)B?1C,x53x41 x1bx11"1;"2;"3x1cxex19x1ax8
A = B?1C =
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(3)x7x41 x1bx11·1;·2;·3x1cxex19x1ax8A0,x20
A0 = T?1AT = (B?1C)?1(B?1C)(B?1C) = B?1C ==
0
@
52?3 32
2 3?3
3
2 2?
1
2
1
A:
(4) fix11"1;"2;"3x64x11·1;·2;·3x1cxex1dx1ex4ex4fx8
B?1
0
@
2
1
3
1
A x64 C?1
0
@
2
1
3
1
A;
x1fx58x41fix1bx11"1;"2;"3x1cxex1dx1ex8
AB?1
0
@
2
1
3
1
A = 12
0
@
7
11
1
1
A;
x41 x1bx11·1;·2;·3x1cxex1dx1ex8
A0B?1
0
@
2
1
3
1
A = 12
0
@
7
6
5
1
A:
2,x7A? C,B? D,x12x13,
A 0
0 B
C 0
0 D
:
x2x3,x68x1bx21x22x19x1aT1;T2,x63x52
T?11 AT1 = C; T?12 BT2 = D;
x1fx58T =
T
1 0
0 T2
x21x22,x25
T?1
A 0
0 B
T =
T?1
1 AT1 0
0 T?12 BT2
=
C 0
0 D
:
x26x27
A 0
0 B
C 0
0 D
:
3,x7Ax21x22,x12x13,ABx64BAx41x69.
￠ 4 ￠
x2x3,x15x38A?1(AB)A = BA,x53AB? BA.
4,x7Ax21x22,x25A? B,x12x13,Bx6ax21x22,x25A?1? B?1.
x2x3,x15x38T;Ax6bx21x22,x26x27Bx21x22,x25
B?1 = (T?1AT)?1 = TAT?1;
x53A?1? B?1.
5,x7A? B,x12x13,AT? BT.
x2x3,x68x1bx21x22x19x1aT,x63x52T?1AT = B,x53BT = (T?1AT)T = TTATT?T.
6,x7A? B,f(x) 2 K[x],x12x13,f(A)? f(B).
x2x3,x68x1bx21x22x19x1aT,x63x52T?1AT = B,x53
T?1(f(A))T = f(T?1AT) = f(B):
7,x12x13,0
BB
B@
1
2
...
n
1
CC
CA?
0
BB
B@
i1
i2
...
in
1
CC
CA;
x3ex28(i1;i2;￠￠￠ ;in)x31(1;2;￠￠￠ ;n)xexfx10x6cx5b.
x2x3,x7V x31nx9xaxbxcxd,"1;￠￠￠ ;"nx31V xex11,x41 x8V xexaxbx61x62,x5fx6dx8
x41"i =?i"i;
x20x41 x1bx11"1;￠￠￠ ;"nx1cxex19x1ax8
A =
0
BB
B@
1
2
...
n
1
CC
CA:
x15x38(i1;i2;￠￠￠ ;in)x31(1;2;￠￠￠ ;n)xexfx10x6cx5b,x1fx58"i1;￠￠￠ ;"inx6ex8V xex11,x24
x41"ij =?ij"ij; j = 1;￠￠￠ ;n:
x53x41 x1bx11"i1;￠￠￠ ;"inx1cxex19x1ax8
B =
0
BB
B@
i1
i2
...
in
1
CC
CA;
x23x24A? B.
8,x7A;B 2 Mn(R),x12x13,x6fx70x68x1bx21x22x19x1aU 2 Mn(C),x63A = U?1BU,x20x71x68x1bx21x22x19x1a
T 2 Mn(R),x63A = T?1BT.
x72,x73x74x75x76x77x78x79x7ax7bx7c:
x7dK,Fx75x7ex7fx0x1,x2x3Fx75Kx7ax4x1(x5K F),A;Bx75x0x1Kx6x7ax7ex7fx7x8,x9
xaA,BxbFx6xcxd,xexfx10x11xbKx6xcxd.
x2x3,x7U = T1 +iT2,T1;T2 2 Mn(R),x20(T1 +iT2)A = B(T1 +iT2),x54x1fA;B 2 Mn(R),x21x12x5a
T1A = BT1; T2A = BT2:
￠ 5 ￠
x13x14f(?) = jT1 +?T2j,x15x38f(i) = jUj 6= 0,x15x13f(?)x40x31x16x39x3ax3b,x71x67?0 2Rx63f(?0) 6= 0,x23x24
T = T1 +?0T2x21x22,x15T1A = BT1,T2A = BT2x21x52TA = BT,x53A = T?1BT.
9,x7x;y;z 2 K,x66
A =
0
@
x y z
y z x
z x y
1
A; B =
0
@
z x y
x y z
y z x
1
A; C =
0
@
y z x
z x y
x y z
1
A:
x12x13,A;B;Cx17x58x41x69.
x2x3,x43x18x62x19x1a
P =
0
@
0 1 0
0 0 1
1 0 0
1
A; Q =
0
@
0 0 1
1 0 0
0 1 0
1
A;
x20P;Qx6bx21x22,x25
P?1AP = C; Q?1AQ = B;
x26x27A? B,A? C,x15x41x69x4cx19xex1ax1bxb,x21x52B? C.
10,x12x13,0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA
n
0
BB
B@
n 0 ￠￠￠ 0
0 0 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ 0
1
CC
CA:
x2x3,x7V x31nx9xaxbxcxd,"1;￠￠￠ ;"nx31V xex11,x41 x8V xexaxbx61x62,x5fx6dx8
x41"i = "1 +"2 +￠￠￠+"n; i = 1;2;￠￠￠ ;n:
x20x41 x1bx11"1;￠￠￠ ;"nx1cxex19x1ax8
A =
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA
n
:
x54x5ex55
fi1 = "1 +"2 +￠￠￠+"n; fi2 = "1?"2;￠￠￠ ;fin = "1?"n
x6ex8V xex11,x25x41 x1bx11fi1;￠￠￠ ;finx1cxex19x1ax8
B =
0
BB
B@
n 0 ￠￠￠ 0
0 0 ￠￠￠ 0.
..,..,..,..
0 0 ￠￠￠ 0
1
CC
CA:
x23x24A? B.
x0 x1 7–3
1,x14x1cx36x1dx2exaxbxcxdV xexaxbx61x62x41 xex1ex1fx20x64x1ex1fx2ax2b,x7x41 x1bV xexfx10x11x1cxex19x1ax31:
(1) A =
2 5
4 3
; (2) A =
0 a
a 0
(a 6= 0);
(3) A =
0
BB
B@
1 1 1 1
1 1?1?1
1?1 1?1
1?1?1 1
1
CC
CA; (4) A =
0
@
5 6?3
1 0 1
1 2?1
1
A;
￠ 6 ￠
(5) A =
0
@
0 0 1
0 1 0
1 0 0
1
A; (6) A =
0
@
0?2?1
2 0?1
1?3 1
1
A;
(7) A =
0
@
4 3 0
3?2 0
2?6 2
1
A:
x4,x36x21x33x34x1ex1fx20,x22x23x1bx24xex2ax2bx25x31x41x26xexfx10x1ex1fx2ax2b.
(1) 7,(1;1);?2,(5;?4).
(2) ai,(1;i);?ai,(i;1).
(3) 2,k(1;1;0;0)+l(1;0;1;0)+m(1;0;0;1);?2,(?1;1;1;1).
(4) 2,(?2;1;0); 1+p3,(?3;1;?2+p3); 1?p3,(?3;1;?2?p3).
(5) 1,k(1;0;1)+l(0;1;0);?1,(?1;0;1).
(6)?3,(1;1;1); 2,(1;1;?4).
(7) 2,(0;0;1); 1,(?1;1;8).
2,x7?1,?2x31xaxbx61x62x41xex60x10x40x42xex1ex1fx20,"1,"2x4ex4fx31x41xex27x38x1ex1fx20?1,?2xex1ex1fx2ax2b.
x12x13,"1 +"2x40x31x41 xex1ex1fx2ax2b.
x2x3,(x28x12)x6fx70"1 +"2x31x41 xex27x38x29x10x1ex1fx20?0xex1ex1fx2ax2b,x20
x41("1 +"2) =?0("1 +"2):
x54x41("1 +"2) = x41"1 + x41"2 =?1"1 +?2"2,x26x27
(?10)"1 +(?20)"2 = 0:
x15?1 6=?2x21x52"1;"2xaxbx4bx4c,x1fx58
10 = 0;?20 = 0;
x52x16?1 =?0 =?2,x2ax2b.
3,x12x13,x6fx70xaxbx61x62x41 x27x2cx10x2dx16x2ax2bx2ex8x30xex1ex1fx2ax2b,x20x41 x8x1ex2bx2fx30x61x62.
x2x3,x7x31x29x10x2dx16x2ax2bfix67x41fi = kfi,x31x32xfx10x2dx16x2ax2bfl,x67x41fl = mfl,x6fx70k 6= m,x20x48x33
x34x2f2xex35x36,fi+flx40x31x41xex1ex1fx2ax2b,x6fx70fi+fl = 0,x20x67x41fl =?x41fi =?kfi = kfl,x64k 6= mx2ax2b.
x1fx58fi+flx31x2dx16x2ax2b,x64x2fx7x2ax2b.
4,x12x13,ABx64BAx67x41x42xex1ex1fx20.
x2x3,x48x33x34x2f4–8xex374x2f,x38jAj6= 0,AC = CAx39,x67fl
flfl
fl
A B
C D
flfl
flfl = jAD?CBj:
x1fx58 fl
flfl
fl
E B
A E
flfl
flfl = j?E?ABj:
x54x1f
0 E
E 0
E B
A E
0 E
E 0
=
E A
B?E
;
x60x3ax43x3bx5bx3b,x4dx52 fl
flfl
fl
E A
B?E
flfl
flfl = j?E?BAj =
flfl
flfl?E B
A E
flfl
flfl = j?E?ABj:
x1fx58ABx64BAx67x41x42xex1ex1fx39x3ax3b,x23x24x67x41x42xex1ex1fx20.
￠ 7 ￠
5,x12x13,x3cx3dx3ex52xcxdxex3fx40x61x62xex1ex1fx20(x6fx67xex41)x49x42x31§1.
x2x3,x7fix31x27x38x3fx40x61x62x41 xex1ex1fx20?0xex1ex1fx2ax2b,x20
0 6= (fi;fi) = (x41fi;x41fi) =?20(fi;fi);
x1fx58?20 = 1,?0 = §1.
6,x12x13,x43x16x19x1axex1ex1fx20x44x8x16.
x2x3,x7fix31x27x38x43x16x19x1aAxex1ex1fx20?0xex1ex1fx2ax2b,x20Afi =?0fi,x15x38
0 = Akfi =?k0fi;
x21x52?k0 = 0,?0 = 0.
7,x7A = (a1;a2;￠￠￠ ;an) 2Rn (aix40x44x8x16),x14x19x1aATAxex1ex1fx20x64x1ex1fx2ax2b.
x4,x7 ai 6= 0,x1ex1fx20 0 x31x26xex1ex1fx2ax2bx31 fij = (0;￠￠￠ ;0;ai;￠￠￠ ;?aj;0;￠￠￠ ;0)
j i
(j=1;￠￠￠ ;i?1;i+1;￠￠￠ ;n)xexaxbx3cx44.x1ex1fx20
nP
j=1
a2jxex1ex1fx2ax2bx31(a1;￠￠￠ ;an).
8,x7A = (aij) 2 Mn(K),x45x3b
flfl
flfl
flfl
flfl
fl
ai1i1 ai1i2 ￠￠￠ ai1ik
ai2i1 ai2i2 ￠￠￠ ai2ik
...,..,..,..
aiki1 aiki2 ￠￠￠ aikik
flfl
flfl
flfl
flfl
fl; (1 6 i1 < i2 < ￠￠￠ < ik 6 n);
x46x8Axexfx10kx47x48x45x3b,x66x1ex1fx39x3ax3b
′A(?) = j?E?Aj =?n?a1?n?1 +￠￠￠+(?1)n?1an?1?+(?1)nan:
x12x13,akx49x38Axex44x56kx47x48x45x3bx4ax4b.
x2x3,x4cj?E?Ajxex2cx5b 0
BB
BB
BB
BB
B@
a1j
a2j
...
ajj
...
anj
1
CC
CC
CC
CC
CA
x4dx4ex3dx60x5b,0
BB
BB
BB
BB
BB
@
0.
..
0
0.
..
0
1
CC
CC
CC
CC
CC
A
x64
0
BB
BB
BB
BB
B@
a1j
a2j
...
ajj
...
anj
1
CC
CC
CC
CC
CA
x20x3bx5bx3bj?E?Ajx21x4ex4fx82nx10nx47x3bx5bx3bx4ax4b,x3ex28x2cx10x3bx5bx3bxex5bx4dx31x2ex50x60x51x52x3bx4axf.
x7Akx8x53xfx54x67kx10?xex45x3bx5bx3b,x3e?x55x38j1;￠￠￠ ;jkx5b,x56Akx57x58kx5bx59x5a,x52
Ak =?k ￠(?1)n?kDn?k;
x3ex28Dn?kx8x1bAx28x5bx5cx37j1;￠￠￠ ;jkx5bx5dx37j1;￠￠￠ ;jkx3bx24x52x16xen?kx47x48x45x3b,x38x58kx10?x43x5en
x47x3bx5bx3bx28x26x67x21x42xekx10x47x18,x20Dn?kx25x43x5ex26x67Cknx10x48x45x3b,x23x24′A(?)x28?n?kxex19x36x49x38
￠ 8 ￠
(?1)kx2fx27Axex26x67kx47x48x45x3bx4ax4b.x1fx58akx8Axex26x67kx47x48x45x3bx4ax4b.
9,x7A 2 Mn(K),x12x13,x68x1bKx2exexfx10x35x36x40x5fx17n2xex39x3ax3bf(x),x63f(A) = 0.
x2x3,x1fx8Mn(K)x31Kx2en2x9xaxbxcxd,x53E;A;A2;￠￠￠ ;An2?1;An2xaxbx41x4c.x38x31x68x1bx40x44x8
x16xeai 2 K,i = 1;￠￠￠ ;n2x63x52
a0E +a1A+￠￠￠+an2?1An2?1 +an2An2 = 0:
x66
f(x) = an2xn2 +an2?1xn2?1 +￠￠￠+a1x+a0;
x20f(A) = 0.
10,x7A 2 Mn(C),x12x13,x68x1bx21x22x19x1aT 2 GL(n;C),x63T?1ATx8x2ex60x61x19x1a.
x2x3,x31nx62x36x63x64x65x66,x38n = 1x39x35x36x67x68x3dx69,x6ax7x35x36x31n?1x47x19x1ax3dx69.
x7?1x31Axexfx10x1ex1fx20,x41x26xex1ex1fx2ax2bx31fi1 2 Cn,x4cfi1x6bx6cx3dCnxex11fi1;fi2;￠￠￠ ;fin,x66
T1 = (fi1;fi2;￠￠￠ ;fin),x20T1x21x22,x25
AT1 = T1
1?
0 A1
; x4d T?11 AT1 =
1?
0 A1
;
x3ex28A1 2 Mn?1(C),x15x64x65x6dx7,x68x1bx21x22x19x1aT2 2 Mn?1(C),x63x52
T?12 A1T2 =
0
B@
2?
...
0?n
1
CA;
x66T = T1
1 0
0 T2
,x20Tx21x22,x25
T?1AT =
0
B@
1?
...
0?n
1
CA:
11,x7A 2 Mn(C),f(x)x8xfx1cx19x36x39x3ax3b,x12x13,x6fx70Axex44x56x1ex1fx20x8?1,?2,￠￠￠,?n,x20f(A)
xex44x56x1ex1fx20x8f(?1),f(?2),￠￠￠,f(?n).
x2x3,x15x34x2f10,x68x1bx21x22x19x1aT,x63
T?1AT =
0
BB
B@
1?
2
...
0?n
1
CC
CA;
x58x3e?1;￠￠￠ ;?nx31Axex44x56x1ex1fx20,x23x24
T?1f(A)T = f(T?1AT) = f
0
BB
B@
0
BB
B@
1?
2
...
0?n
1
CC
CA
1
CC
CA =
0
BB
B@
f(?1)?
f(?2)
...
0 f(?n)
1
CC
CA:
x26x27f(?1);f(?2);￠￠￠ ;f(?n)x8f(A)xex44x56x1ex1fx20.
x0 x1 7–4
1,x34x2f7–3x37xfx2fx28xex19x1a,x6ex6fx31x21x27x31x61x70xe? x1bx21x31x61x70xex71x72x1c,x14x5ax41x26xex17x18x19x1a
x4bx31x61x19x1a.
￠ 9 ￠
x4,(1) T =
1 5
1?4
,T?1AT = diag(7;?2).
(2) T =
1 i
i 1
,T?1AT = diag(ai;?ai).
(3) T =
0
BB
B@
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
1
CC
CA,T?1AT = diag(?2;2;2;2).
(4) T =
0
@
2?3?3
1 1 1
0?2+p3?2?p3
1
A,T?1AT = diag(2;1+p3;1?p3).
(5) T =
0
@
1 1 0
0 0 1
1 1 0
1
A,T?1AT = diag(?1;1;1).
(6),(7)x40x21x31x61x70.
2,x1bK[x]nx28,x14x73x4ex61x62x44:
x44(f(x)) = f0(x)
xex1ex1fx39x3ax3b,x29x12x13,x44x1bx53x74xfx10x11x1cxex19x1ax4dx40x21x42x31x31x61x19x1a.
x4,x43K[x]nxex111;x;x2;￠￠￠ ;xn?1,x20x44x1bx58x10x11x1cxex19x1ax31
D =
0
BB
B@
0 1 0 ￠￠￠ 0
0 0 2 ￠￠￠ 0.
..,..,..,..,..
0 0 0 ￠￠￠ 0
1
CC
CA:
x23x24x44xex1ex1fx39x3ax3bx8′D(?) =?n,x6fx70x44x21x31x61x70,x20x68x1bx21x22x19x1aTx63x52T?1AT = 0,x4dD = 0,
x24x44x40x31x16x61x62,x2ax2b.
3,x7A 2 Mn(K),x12x13,x6fx70rankA+rank(A?E) = n,x20Ax21x31x61x70.
x2x3,x15x34x2f4–8.13x55,rankA+rank(A?E) = nxex6cx4ex71x4ax75x76x31A2 = A,x4dx31Axex53xfx5bx2a
x2bfix67Afi = fi,x54A(A?E) = 0,x53x31A?Exex53xfx5bx2ax2bflx67Afl = 0.
x7Axex5bx2ax2bx3cxex77x78x4bx4cx3cx8fi1;￠￠￠ ;fir,A? E xex77x78x4bx4cx5bx2ax2bx3cx8fl1;￠￠￠ ;fln?r (x1fx8
rankA+rank(A?E) = n),x1cx12fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?rxaxbx4bx4c.
x7x67
rX
i=1
kifii +
n?rX
j=1
mjflj = 0:
x20
rX
i=1
kiAfii +
n?rX
j=1
mjAflj =
rX
i=1
kiAfii =
rX
i=1
kifii = 0:
x38x31k1 = ￠￠￠ = kr = 0,x79x24m1 = ￠￠￠ = mn?r = 0,x26x27fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?rxaxbx4bx4c.
x66T = (fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r),x20Tx21x22,x25
AT = A(fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r) = (fi1;￠￠￠ ;fir;fl1;￠￠￠ ;fln?r)
E
r 0
0 0
;
x4d
AT = T
E
r 0
0 0
;
￠ 10 ￠
x23x24
T?1AT =
E
r 0
0 0
:
4,x7A 2 Mn(K),x12x13,x6fx70rank(A+E)+rank(A?E) = n,x20Ax21x31x61x70.
x2x3,x64x2ex2fx7ax69,x7b.
5,x7x41 x8x36x1dKx2enx9xaxbxcxdV xexaxbx61x62,x7cx7dx412 = x41,x14x41 xex1ex1fx20,x29x12x13x41 x21x31x61
x70.
x2x3,x7
V1 = fx41fi j fi 2 Vg; V2 = ffi?x41fi j fi 2 Vg:
x20x31x53x7exefi 2 V,x67fi = x41fi+(fi?x41fi),x1fx58
V = V1 +V2:
x7ffi 2 V1 \V2,x20
fi = x41fl =?x41 ; fl; 2 V:
x38x31
x41fi = x412fl = x41fl = fi; (*)
x41fi = x41?x412 = x41?x41 = 0,(**)
x26x27fi = x41fi = 0,x4dV1 \V2 = 0,x58x0x25x12x13x1
V = V1 'V2:
x31x38V1x28xex2ax2bfi,x67(*)x3bx3dx69,x15x13V1x31x27x38x1ex1fx201xex1ex1fx45xcxd,x7ax69x2x15(**)x3bx21x52V2x31x27
x38x1ex1fx200xex1ex1fx45xcxd,x48x33x12x364.5,x41 x21x31x61x70,x41 xex1ex1fx20x80,1.
(x3,x6ax21x4x62x34x2f3xex5x66x6x27x12x13)
6,x7
A =
1 1
1 0
:
(1)x7x56x19x1aAx31x61x70,x8x14An;
(2)x4x62x2ex50x35x70,x14x9xaxbxcx36x5bxex5cx3ax5dx3b.
x4,(1) Axex1ex1fx20x8 1§
p5
2,x41x26xex1ex1fx2ax2bx8?
1+p5
21
!;
1?p5
21
!
:
x66
T =
1+p5
2
1?p5
21 1
!;
x20
T?1AT =
1+p5
2 0
0 1?
p5
2
!
:
￠ 11 ￠
x23x24
An = T
0
B@
1+p5
2
n
0
0
1?p5
2
n
1
CAT?1
= 1p5
1+p5
2
1?p5
21 1
!0
B@
1+p5
2
n
0
0
1?p5
2
n
1
CA
1? 1?p5
2
1 1+
p5
2
!
=
p5
5
fin+1?fln+1 fin?fln
fin?fln fin?1?fln?1
;
x3ex28fi = 1+
p5
2,fl =
1?p5
2,
(2)x66
fi0 =
1
0
;fi1 =
1
1
;￠￠￠ ;fin =
a
n+1
an
;
x3ex28aix31x37ix10x9xaxbxcx36,x20
fin = Anfi0:
x26x27
an+1
an
= Anfi0 =
p5
5
fin+1?fln+1 fin?fln
fin?fln fin?1?fln?1
1
0
;
xdx52
an =
p5
5 (fi
n?fln) =
p5
5
"?
1+p5
2
!n
1?p5
2
!n#
:
7,x7A = (aij)x8nx47x2ex60x61x19x1a,x12x13:
(1)x7faii 6= ajj (i 6= j),x20Ax21x31x61x70;
(2)x7fa11 = a22 = ￠￠￠ = ann,x25xexfx67xfx10aij 6= 0 (i 6= j),x20Ax40x21x31x61x70.
x2x3,(1)x15x38Ax31x2ex60x61x19x1a,x53a11;a22;￠￠￠ ;annx8Axenx10x1ex1fx20,x7fx38i 6= jx39aii 6= ajj,x20
Ax67nx10x40x42xex1ex1fx20,x23x24Ax21x31x61x70.
(2) (x28x12)x10x55Axex1ex1fx20x8?0 = a11 = ￠￠￠ = ann,x6fAx21x31x61x70,x20x68x1bx21x22x19x1aT,x63x52
T?1AT = diag(?0;￠￠￠ ;?0):
x38x31
A = T
0
B@
0
...
0
1
CAT?1 =
0
B@
0
...
0
1
CA:
x4dAx8x11x2bx1a,x64x6dx7x2ax2b.
8,x7x67x4ex12x31x61x19x1a
A =
0
BB
B@
A1 0 ￠￠￠ 0
0 A2 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ As
1
CC
CA:
x12x13,Ax21x31x61x70xex6cx4ex71x4ax75x76x31x2cx10Aix6bx21x31x61x70.
x2x3,x6cx4exbx31x13x68xe,x1cx12x71x4axb.
￠ 12 ￠
x7Ax21x31x61x70,x20x68x1bx21x22x19x1aT,x63x52
T?1AT =
0
BB
B@
1
2
...
n
1
CC
CA:
x38x31
AT = T
0
BB
B@
1
2
...
n
1
CC
CA:
x66
T =
0
BB
B@
T1
T2
...
Ts
1
CC
CA;
x3ex28Tixex3bx36x49x38Aixex47x36ri,x20
AiTi = Ti
0
BB
B@
1
2
...
n
1
CC
CA:
x58x15x13Tixex2dx16x5bx2ax2bx4dx31Aixex1ex1fx2ax2b,x54x1fTx21x22,x53Txex3bx2ax2bxaxbx4bx4c,x1fx24Tixex3bx2ax2bx6a
xaxbx4bx4c.x38x31rankTi = Aixex47x36,Tixex5bx14x6ax49x38Aixex47x36,x1fx58Aix67rix10xaxbx4bx4cxex1ex1fx2ax2b,
x15x13Aix6bx21x31x61x70.
9,x7A;B 2 Mn(K),x25AB = BA,x12x13,x6fx70A,Bx4dx21x31x61x70,x20x68x1bx21x22x19x1aT 2 Mn(K),x63
T?1ATx64T?1BTx42x8x31x61x19x1a.
x2x3,x7Axex40x42x1ex1fx20x8?1;?2;￠￠￠ ;?s,x3ex28?ixex15x36x8ri,x15x38Ax21x31x61x70,x68x1bx21x22x19x1aT,
x63
T?1AT = A1 =
0
B@
1Er1
...
sErs
1
CA:
x66B1 = T?1BT,x20A1B1 = B1A1.
x66B1 = (Bij),x3ex4ex12x5x3bx64A1x41x42,x20x15A1B1 = B1A1x52
iBij = Bij?j
x38x31x38i 6= jx39x67Bij = 0,x4d
B1 =
0
BB
B@
B11
B22
...
Bss
1
CC
CA:
x54x1fBx21x31x61x70,B1x6ax21x31x61x70,x23x24x15x2ex2fx55x2cx10Biix4dx21x31x61x70,x4dx68x1bx21x22x19x1aSi 2 Mri(K)x63
S?1i BiiSi =
0
B@
i1
...
iri
1
CA:
￠ 13 ￠
x66
T = T1
0
B@
S1
...
Ss
1
CA;
x20Tx21x22,x25
T?1AT =
0
B@
S?11
...
S?1s
1
CA
0
B@
1Er1
...
sErs
1
CA
0
B@
S1
...
Ss
1
CA =
0
B@
1Er1
...
sErs
1
CA;
T?1BT =
0
B@
S?11
...
S?1s
1
CA
0
B@
B11
...
Bss
1
CA
0
B@
S1
...
Ss
1
CA
=
0
B@
S?11 B11S1
...
S?1s BssSs
1
CA =
0
BB
BB
BB
BB
BB
BB
@
11
...
1r1
...
s1
...
srs
1
CC
CC
CC
CC
CC
CC
A
x42x8x31x61x52.
x0 x1 7–5
1,x7x41 x31xaxbxcxdV xexaxbx61x62,x10x55x41 x1bx11·1;·2;￠￠￠ ;·nx1cxex19x1ax31
A =
0
BB
B@
0 1
...,..
..,1
0
1
CC
CA:
x14x41 xex26x67x40x61x45xcxd.
x4,x7Wx31x41 xexfx10x2dx16x40x61x45xcxd,x16x7x12x13,Wx71x17x54x29x10x11x2ax2b·i.
x7
0 6= fi = ak·k +￠￠￠+an·n 2 W; ak 6= 0:
x20x41k?1fi = ak·1 2 W,·1 2 W.
x8x7·kx31x17x54x38Wx28xex1cx1ex18x78xex11x2ax2b,x20
·k?1 = x41·k;·k?2 = x412·k;￠￠￠ ;·1 = x41k?1·k 2 W:
x1cx19x12
W = L(·1;￠￠￠ ;·k):
x62x28x12x66,x6fx67fi 2 W,x1afi 62 L(·1;￠￠￠ ;·k),x20
fi = a1·1 +￠￠￠+am·m; x3ex28am 6= 0;m > k:
x38x31
am?k·1 +￠￠￠+am?1·k +am·k+1 = x41m?k?1fi 2 W:
￠ 14 ￠
x54x1fam?k·1 +￠￠￠+am?1·k 2 W,am 6= 0,x21x52·k+1 2 W,x2ax2b,x1fx58x41xex26x67x40x61x45xcxdx8x16x45xcxd
x27x1bL(·1;￠￠￠ ;·k),k = 1;￠￠￠ ;n.
2,x7x41 x8x3cx3dx3ex52xcxdxex3fx40x61x62,x12x13,x41 xex40x61x45xcxdxex3fx40x1cx6ax31x41 xex40x61x45xcxd.
x2x3,x7Wx31x41 xex40x61x45xcxd,x20x41(W) W,x15x38x3fx40x61x62x71x21x22,x1fx58x41(W) = W.
x31x38x53x7exefl 2 W?,fi 2 W,x71x68x1b 2 Wx63fi = x41, x38x31
(x41fl;fi) = (x41fl;x41 ) = (fl; ) = 0:
x58x15x13fl 2 W?,x53W?x31x41 xex40x61x45xcxd.
3,x7x41 x31xaxbxcxdV xexaxbx61x62,Wx8x41 xex40x61x45xcxd,x12x13,x41(W)x1dx31x41 xex40x61x45xcxd.
x2x3,x15x38Wx31x41 xex40x61x45xcxd,x41(W) W,x1fx58x41(x41(W)) x41(W),x15x13x41(W)x6ax31x41 xex40
x61x45xcxd.
4,x7V x31x1cx36x1dx2exenx9xaxbxcxd,x41,x42x31V xexaxbx61x62,x25x41x42 = x42x41,x12x13:
(1)x6fx70?x31x41 xexfx10x1ex1fx20,xbx1e,V?x31x42xex40x61x45xcxd;
(2) x41,x42xexfx67xfx10x5dx1fxex1ex1fx2ax2b.
x2x3,(1)x31x53x7exefi 2 V?,
x41(x42fi) = x42(x41fi) = x42(?fi) =?x42fi:
x1fx58x42fi 2 V?,x15x13V?x6ax31x42xex40x61x45xcxd.
(2)x15x2ex55,V?x31x42xex40x61x45xcxd,x23x24x42jV?x31V?x2exexaxbx61x62,x38x31x42jV?x67x1ex1fx20?x27x1bx41
x26xex1ex1fx2ax2bfl 2 V?,x63
x42fl = x42jV?(fl) =?fl:
x54x1ffl 2 V?,x41fl =?fl,x26x27flx31x41 x64x42xex5dx1fx1ex1fx2ax2b.
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