x0x1x2x3
x25x2fx30x27 x31x32x33x34x35xax36x37x38x39x3ax3b
x4c x4d 12–1
1,x69x75x7x8x9x21x22x77,x73x74x23x33x4dx4? x19x33x4dx3ex3x68x4d.
(1)
+11
+3?+1
; (2)
2?2?2
+2?+1
;
(3)
0
@
12
+2+12
1+2 +1
1
A; (4)
0
@
1?2?
2 +1?2?2?1
1
A.
x50,(1)x33x4d,x4dx21x22x6c 14
+1+1
3?+1
.
(2)x33x4d,x4dx21x22x6c? 12
+12 +?
2?2?2
.
(3)x33x4d,x4dx21x22x6c
0
@
2+12
2 +2+12
1 0 1
1
A.
(4)xdx33x4d.
2,x3x69x75x7x8x9x21x22x4x58x5bx1e:
(1)
+1?
11
; (2)
11
1?2?2?+1
;
(3)
0
@
12?1
31?2 +2? 3?2?1
+1?2?2 +1
1
A; (4)
0
@
2?2?1 3?2
22 +3?2?2?3?
2 +2 +? 2?2 +2?
1
A;
(5)
0
@
+2 0 0
1?+2 0
0?1?+2
1
A; (6)
0
@
0 0?(1)
0?2?1 0
(1)2 0 0
1
A.
x50,(1) diag(1;1).
(2) diag(1;(1)(2)).
(3) diag(1;?;0).
(4) diag(1;?(?+1);?(?+1)2
12
·
).
(5) diag(1;1;(?+2)3).
(6) diag(1;?(1)(?+1);?(1)2(?+1)).
3,xax74x69x75x7x8x9x21x22x23xcx12x54:
(1) A =
0
@
3?2?4?+3
22 25?2?4?+3
22 (2)2
1
A; B =
0
@
2?3?+3 233
2?2?+1 47 25
2?3?+2 242
1
A.
(2) A =
0
@
22?2?1?+1
0?+1 1
(?+1)2?2 ++1
1
A; B =
0
@
1 2?2 +11
2?2 +?
1+1
1
A.
x50,(1)x12x54; (2)xdx12x54.
￠ 1 ￠
4,x2A(?)x6cx6ax6bx7x8x9x21x22,x30x31,A(?)x33x4dx4x43x1x44x45x46x47x23x40x3x26x4x2bx28c,A(c)x78x33x4d.
x4ex4f,())x2A(?)x33x4d,x3d
jA(?)j = a 6= 0 2C:
x18x40x63x64x4c 2C,jA(c)j = a,x3x48A(c)x33x4d.
(()x15x16f(?) = jA(?)j,x3dx40x63x64x4c 2C,f(c) 6= 0,x18f(?)x25Cx77x37x38,x3x48f(?) = a 6= 0 2C,
jA(?)j = a 6= 0 2C,xfx61A(?)x33x4d.
5,x69x75x11x12x23xcx5bx13,(x39x5bx13,x3dx1x48x30x31,x39xdx5bx13,x3dx2x18xfx3.)
x49x6bx7x8x9x21x22x12x54x4x43x1x44x45x46x47x23,x40x3x26x4k 2 K,A(k)x14B(k)x78x12x54.
x50,xdx5bx13,x39
A(?) =
1
; B(?) =
1
2
;
x3dA(?)x14B(?)xdx12x54,x68x40x63x64x4k 2 K,A(k)x14B(k)x12x54.
x4c x4d 12–2
1,x3x69x75x7x8x9x21x22x4x1c:
(1)
0
@
2?1?+1 21
+1?2 +2?+1?1
2 +2 +3?+22
1
A; (2)
0
@
+1?1?2
22?1?2
1?2
1
A.
x50,(1) 3; (2) 2.
2,x2A(?)x6cx6ax6bx7x8x9x21x22,x30x31,rankA(?) = maxfrankA(k)jk 2 Kg.
x50,x2rankA(?) = r,x3dA(?)x26x6ax6br x1x10x9Mr+1(?) = 0,x18x40x3x26x4k 2 K,Mr+1(k) = 0,x5f
xfx31rankA(k) 6 r,x79xfMr(?) 6= 0,x6dx25c 2 Kx3cMr(c) 6= 0,x5fxfx31r = maxfrankA(k) j k 2 Kg.
3,x3ex3x69x75x21x22x4xdxexfx10:
(1)
0
@
1?1 0
01?1
0 01
1
A; (2)
0
@
+2 (1)2+1
1?2 0
2?2?(1)2?2?1
1
A;
(3)
0
BB
B@
+fi fl 1 0
fl?+fi 0 1
0 0?+fi fl
0 0?fl?+fi
1
CC
CA; (4)
0
BB
B@
1 1 0 0
01 1 0
0 01 1
0 0 01
1
CC
CA;
(5)
0
BB
BB
B@
fi fl fl fl ￠￠￠ fl
0fi fl fl ￠￠￠ fl
0 0fi fl ￠￠￠ fl
:::::::::::::::::::::::::::::::::::::
0 0 0 0 ￠￠￠fi
1
CC
CC
CA; (6)
0
BB
BB
B@
0 0 ￠￠￠ 0 an
1? 0 ￠￠￠ 0 an?1
0?1? ￠￠￠ 0 an?2
::::::::::::::::::::::::::::::
0 0 0 ￠￠￠?1?+a1
1
CC
CC
CA.
x50,(1) 1;1;(1)3.
(2) 1;1;?(1).
(3)x39fl 6= 0,1;1;1;[(?+fi)2 +fl2]2;x39fl = 0,1;1;(?+fi)2;(?+fi)2.
(4) 1;1;1;(1)4.
(5)x39fl 6= 0,1;1;￠￠￠ ;1;(fi)n;x39fl = 0,fi;fi;￠￠￠ ;fi.
(6) 1;1;￠￠￠ ;1;?n +a1?n?1 +￠￠￠+an.
4,x2Dk(?) (k = 1;2;￠￠￠ ;r)x6cA(?)x4x33x75x9xfx10,x30x31:
D2k(?) j Dk?1(?)Dk+1(?); k = 2;3;￠￠￠ ;r?1:
￠ 2 ￠
x4ex4f,x2A(?)x4xdxexfx10x6c
d1(?);d2(?);￠￠￠ ;dn(?);
x3d
Dk?1(?) = d1(?)d2(?)￠￠￠dk?1(?);
Dk(?) = d1(?)d2(?)￠￠￠dk(?) = Dk?1(?)dk(?);
Dk+1(?) = d1(?)d2(?)￠￠￠dk+1(?);
x3x48
D2k(?) = D2k?1(?)d2k(?) j Dk?1(?)Dk(?)dk(?)dk+1(?);
D2k(?) j Dk?1(?)Dk+1(?):
5,x2A(?)x6cnx1x3fx22,x30x31,A(?)x14AT(?)x12x54.
x4ex4f,x6dx25x33x4dx21x22P(?);Q(?),x3c
P(?)A(?)Q(?) =
0
BB
B@
d1(?)
d2(?)
...
dn(?)
1
CC
CA;
x18
P(?)A(?)Q(?) =
0
BB
B@
d1(?)
d2(?)
...
dn(?)
1
CC
CA
T
= Q(?)TA(?)TP(?)T;
x4cx23A(?)x14AT(?)x12x54.
6,x2f1(x);￠￠￠ ;fn(x) 2 K[x],x62(f1(x);￠￠￠ ;fn(x)) = 1.
x30x31,x6dx25x7x8x9fij(x) 2 K[x] (i = 2;3;￠￠￠ ;n; j = 1;2;￠￠￠ ;n),x3c
flfl
flfl
flfl
flfl
fl
f1(x) f2(x) ￠￠￠ fn(x)
f21(x) f22(x) ￠￠￠ f2n(x)
...,..,..,..
fn1(x) fn2(x) ￠￠￠ fnn(x)
flfl
flfl
flfl
flfl
fl
= 1:
x4ex4f,x15x16x7x8x9x21x22
A(x) = (f1(x);f2(x);￠￠￠ ;fn(x));
x10x1fx20,A(x)x4xdxexfx10x6c1,x18x6dx25x33x4dx21x22P(x),x3c
A(x)P(x) = (1;0;￠￠￠ ;0),(*)
x2jP(x)j = c 6= 0,x3dx6dx25x33x4dx21x22Q(x),x3c
Q(x)P(x) =
0
BB
BB
B@
1
1
...
1
c
1
CC
CC
CA,(**)
x7
Q(x) = (fij(x));
￠ 3 ￠
x31
B(x) =
0
BB
B@
f1(x) f2(x) ￠￠￠ fn(x)
f21(x) f22(x) ￠￠￠ f2n(x)
...,..,..,..
fn1(x) fn2(x) ￠￠￠ fnn(x)
1
CC
CA;
x3dx10(*)x14(**)x20
B(x)P(x) =
0
BB
BB
B@
1
1
...
1
c
1
CC
CC
CA;
x4cx23jB(x)jjP(x)j = c,x79xfjP(x)j = c,x4ejB(x)j = 1,x6ax6bfij(x)x1x6cx3x3.
x4c x4d 12–3
1,xax74x69x75x21x22x23xcxdxe:
(1) A =
0
@
3 2?5
2 6?10
1 2?3
1
A; B =
0
@
6 20?34
6 32?51
4 20?32
1
A.
(2) A =
0
@
6 6?15
1 5?5
1 2?2
1
A; B =
0
@
37?20?4
34?17?4
119?70?11
1
A.
(3) A =
0
@
2?2 1
1?1 1
1?2 2
1
A; B =
0
@
1?3 3
2?6 13
1?4 8
1
A.
x50,(1)x23; (2)x23; (3) xc.
2,x30x31,x63x42x3fx22Ax14x6dx4x7cx4x21x22AT xdxe.
x4ex4f,x10x4c?E?AT = (?E?A)Tx12x54x4c?E?A(xbx6912–2.5),xfx61Ax14AT xdxe.
3,x2Ax14B x6cnx1x3fx22,x30x31,(AB)? = B?A?
x4ex4f,x15x16x12x9
[(?E +A)(?E +B)][(?E +A)(?E +B)]?
= j(?E +A)(?E +B)jE = j?E +AjE ￠j?E +BjE
= (?E +A)(?E +B)(?E +B)?(?E +A)?:
x3x48
(?E +A)(?E +B)f[(?E +A)(?E +B)](?E +B)?(?E +A)?g = 0:
x63x64x32x9x49x79x4x56x28,x20
[(?E +A)(?E +B)](?E +B)?(?E +A)? = 0;
x1
[(?E +A)(?E +B)]? = (?E +B)?(?E +A)?:
x67? = 0x72x26
(AB)? = B?A?:
4,x30x31,x39x3ax21x22Ax14B xdxe,x3dx6dx72x4x5x6x21x22A?x14B? x0xdxe.
￠ 4 ￠
x4ex4f,x3eAx14B xdxe,x3dx6dx25x33x4dx21x22P,x3cP?1AP = B,x4cx23x55x3dxbx694x4x11x12,
B? = (P?1AP)? = P?A?(P?1)? = P?A?(P?)?1:
x18A?x14B? xdxe.
5,x30x31,x21x22x4xdxex14x28x29x4x67x7x37x76.
x4ex4f,x2A;Bx23x28x29K1x77x4x21x22,x3d?E?Ax14?E?Bx4xdxexfx10x78x23x36x28x25K1x77x4x7x8x9.
x2x28x29K1 ‰ K2,x8x51x5fx74x7x8x9x0x33x48x6fx5bx36x28x25K2x77x4x7x8x9,x6ax6bxdxexfx10x5ax14x28x29x4x67x7x37
x76(x2bx7x9x6ax6bxax28xfx10),x6bx21x22A;B xdxex1bx62x43x1b?E?Ax14?E?Bx26xdx8x4xdxexfx10x5a.xfx61x21
x22x4xdxex14x28x29x4x67x7x37x76.
6,x2Ax6cnx1x3fx22,?0 x6cAx4x6ax6bx5x6x13,x30x31,x5x6x13?0x4x7fx28x6cx28> n?rank(?0E?A).
x4ex4f,x2?0 x6c A x4 r x6cx5x6x13,x2 d1(?);￠￠￠ ;dn(?) x6c A x4xdxexfx10,x3d0 j dn(?),x68
0 -dn?r(?),(xcx3d,x390 j dn?r(?),x3d0 j dn?r+1(?);￠￠￠ ;0 j dn(?),x4cx230x4x6c
x28> r +1)xfx61x6dx25x33x4dx21x22P(?);Q(?)x3c
P(?)(?E?A)Q(?) =
0
BB
BB
BB
BB
B@
d1(?)
...
dn?r(?)
dn?r+1(?)
...
dn(?)
1
CC
CC
CC
CC
CA;
x18
P(?0)(?0E?A)Q(?0) =
0
BB
BB
BB
BB
B@
d1(?0)
...
dn?r(?0)
dn?r+1(?0)
...
0
1
CC
CC
CC
CC
CA;
x10x4cd1(?0) 6= 0;￠￠￠ ;dn?r(?0) 6= 0,x3x48
rank(?0E?A) > rankP(?0)(?0E?A)Q(?0) > n?r:
xfx61
r > n?rank(?0E?A):
x4c x4d 12–4
1,x3x69x75x7x8x9x21x22x4x11x12xfx10:
(1)
0
@
2 +231?2 +23
2?2 +35?2?1?2 +34
2 +2 01
1
A; (2)
0
@
2?2?2 +1 2?2?2
2 +1?2 +1 2?2?2
2 +2?2 +1 3?2?5
1
A.
x50,(1)?;1;1;1;?+3.
(2)?+1;3.
2,x1fx20x7x8x9x21x22A(?)x4x11x12xfx10,x1crx14x1x28n,x3A(?)x4x58x5bx1e:
(1)?+1;?+1;(?+1)2;1;(1)2; r = 4,n = 5;
(2)2;(2)2;(2)3;?+2;(?+2)3; r = 4,n = 4;
￠ 5 ￠
(3)1;(1)2;(1)3;?+2;(?+2)2; r = 3,n = 5.
x50,(1) diag(1;?+1;(?+1)(1);(?+1)2(1)2;0).
(2) diag(1;2;(2)2(?+2);(2)3(?+2)3).
(3) diag(1;(1)2(?+2);(1)3(?+2)2;0;0).
3,x3x69x75x21x22x4x58x5bx1e:
(1)
0
BB
B@
0?(?+1)2 0 0
2(1) 0 0 0
0 0 0?2?1
0 0?(?+1)2 0
1
CC
CA;
(2)
0
BB
B@
2?4 0 0 0
0?2 +2? 0 0
0 0?3?2?2 0
0 0 0?3?4?
1
CC
CA;
(3)
0
BB
B@
2 +23?2 +2 0 0
2?2 +24 2?2 +3 0 0
0 0?+1?+2
0 0?2?1?2 +2
1
CC
CA;
(4)
0
BB
B@
22 0?3 +?21 0
2?4 0?3 +2?22 0
0?2 +2? 0?2 +62
0?2 +2 0?2 +57
1
CC
CA.
x50,(1) diag(1;?(?+1);?(?+1)2(1);?2(?+1)2(1)).
(2) diag(1;?(?2?4);?(?2?4);?2(?2?4)).
(3) diag(1;1;(1)(?+1);0).
(4) diag(1;1;?2?4;0).
4,x3x69x75x21x22x4xdxexfx10,x33x75x9xfx10x14x11x12xfx10:
(1)
0
@
4 2?5
6 4?9
5 3?7
1
A; (2)
0
@
2 1 3
6?3?9
4?2?6
1
A;
(3)
0
BB
B@
1 1 1 ￠￠￠ 1
1 1 1 ￠￠￠ 1
::::::::::::::::
1 1 1 ￠￠￠ 1
1
CC
CA; (4)
0
BB
B@
2?3 0 0
3?4 0 0
1 5 1?2
0 2 2?3
1
CC
CA.
x50,(1)xdxexfx10,1;1;?2(1),x33x75x9xfx10,1;1;?2(1),x11x12xfx10,?2;1.
(2)xdxexfx10,1;?;?(?+1),x33x75x9xfx10,1;?;?2(?+1),x11x12xfx10,?;?;?+1.
(3)xdxexfx10,1;?;￠￠￠ ;?| {z }
n?2x0;?(n),x33x75x9xfx10,1;?;?2;￠￠￠ ;?n?2;?n?1(n),x11x12xfx10,?;￠￠￠ ;?| {z }
n?1x0;
n.
(4)xdxexfx10,1;1;1;(?+1)4,x33x75x9xfx10,1;1;1;(?+1)4,x11x12xfx10,(?+1)4.
5,x2?0 x6c n x1x21x22Ax4x6ax6bx5x6x13,x30x31,x21x22Ax4x66x4cx5x6x13?0 x4x11x12xfx10x4x6bx28x12x4c
n?rank(?0E?A).
x4ex4f,x2d1(?);￠￠￠ ;dn(?)x6cAx4xdxexfx10,x39Ax4x66x4cx5x6x13?0x4x11x12xfx10x4x6bx28x6cr,x3d
0 j dn(?);￠￠￠ ;0 j dn?r+1(?);0 -dn?r(?);￠￠￠ ;0 -d1(?):
￠ 6 ￠
xfx61x6dx25x33x4dx21x22P(?);Q(?)x3c
P(?)(?E?A)Q(?) =
0
B@
d1(?)
...
dn(?)
1
CA;
P(?0)(?0E?A)Q(?0) =
0
BB
BB
BB
BB
@
d1(?0)
...
dn?r(?0)
0
...
0
1
CC
CC
CC
CC
A;
x4cx23
n?r = rankP(?0)(?0E?A)Q(?0) = rank(?0E?A);
x1
r = n?rank(?0E?A):
x4c x4d 12–5
1,x3x69x75x21x22x4x19x1ax1bx1cx1dx1e:
(1)
0
@
1?1 0
0?1 0
1 2 1
1
A; (2)
0
@
2 6?15
1 1?5
1 2?6
1
A;
(3)
0
@
13 16 14
6?7?6
6?8?7
1
A; (4)
0
@
9?6?2
18?12?3
18?9?6
1
A;
(5)
0
@
1?3 3
2?6 13
1?4 8
1
A; (6)
0
@
1?2?1
2 4 2
3?6?3
1
A;
(7)
0
@
1?1 1
3?3 3
2?2 2
1
A; (8)
0
@
5 2 6
2 0 3
2 1?2
1
A;
(9)
0
BB
B@
2 1 1?2
5?4 2 9
3 1 2?2
2?4 3 8
1
CC
CA; (10)
0
BB
B@
3?4 0 2
4?5?2 4
0 0 3?2
0 0 2?1
1
CC
CA;
(11)
0
BB
BB
B@
0 1 0 ￠￠￠ 0 0
0 0 1 ￠￠￠ 0 0
:::::::::::::::::::
0 0 0 ￠￠￠ 0 1
1 0 0 ￠￠￠ 0 0
1
CC
CC
CA; (12)
0
BB
B@
1 2 3 ￠￠￠ n
0 1 2 ￠￠￠ n?1
::::::::::::::::::::
0 0 0 ￠￠￠ 1
1
CC
CA.
x50,(1)
0
@
1 0 0
0 1 1
0 0 1
1
A; (2)
0
@
1 0 0
0?1 1
0 0?1
1
A;
(3)
0
@
1 0 0
0?1 1
0 0?1
1
A; (4)
0
@
3 0 0
0?3 1
0 0?3
1
A;
￠ 7 ￠
(5)
0
@
1 1 0
0 1 1
0 0 1
1
A; (6)
0
@
2 0 0
0 0 0
0 0 0
1
A;
(7)
0
@
0 1 0
0 0 0
0 0 0
1
A; (8)
0
@
1 0 0
0 1+2p6 0
0 0 1?2p6
1
A;
(9)
0
BB
B@
1 0 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA; (10)
0
BB
B@
1 1 0 0
0 1 0 0
0 0?1 1
0 0 0?1
1
CC
CA;
(11) diag(1;"1;"2;￠￠￠ ;"n?1),1;"1;"2;￠￠￠ ;"n?1x23xn?1x4nx6bx38;
(12)
0
BB
BB
B@
1 1 0 ￠￠￠ 0
0 1 1 ￠￠￠ 0.
..,..,..,..,..
0 0 ￠￠￠ 1 1
0 0 ￠￠￠ 0 1
1
CC
CC
CA.
2,x2x21x22
A =
0
@
2 0 0
a 2 0
b c 2
1
A:
(1)x21x22Ax33x54x26xbxcx4x19x1ax1bx1cx1dx1e?
(2)x3ex44x59Ax33x40x41x42x4x46x47.
x50,(1) Ax43x26x6ax6bx5x6x13?0 = 2,x3x48Ax4x19x1ax1bx6dx4x6dx28= Ax4x11x12xfx10x4x6bx28= rank(?0E?
A) (xdx45xbx6912–4.5) x6b
rank(?0E?A) =
8>
<
>:
2 x1bac 6= 0,
1 x1ba;cx77x6ax6bx12x4c0,x4x6ax6bxdx12x4c0,xea;cx78x230,x68b 6= 0x52,
0 x1ba = b = c = 0x52.
xfx61x1bac 6= 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 1 0
0 2 1
0 0 2
1
A;x1ba;cx77x6ax6bx12x4c0,x4x6ax6bxdx12x4c0,xea;cx78x230,
x68b 6= 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 0 0
0 2 1
0 0 2
1
A;x1ba = b = c = 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 0 0
0 2 0
0 0 2
1
A.
(2) Ax33x40x41x42 () a = b = c = 0.
3,x2x21x22Ax4x5x6x7x8x9
′A(?) =?5 +?4?5?32 +84:
x3ex3x18Ax3x26x33x54x4x19x1ax1bx1cx1dx1e.
x50,′A(?) = (1)3(?+2)2,xfx61Ax4x33x54x4x11x12xfx10x6c:
(a)1;1;1;?+2;?+2;
(b) (1)2;1;?+2;?+2;
(c) (1)3;?+2;?+2;
(d)1;1;1;(?+2)2;
(e) (1)2;1;(?+2)2;
￠ 8 ￠
(f) (1)3;(?+2)2.
x18Ax4x33x54x4x19x1ax1bx1cx1dx1ex6c:
0
BB
BB
B@
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 1 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 1 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA:
4,x2x21x22Ax4x1cx6c1,x30x31,Ax4x19x1ax1bx1cx1dx1exex33x54x6c
0
BB
BB
B@
fl
0
0
...
0
1
CC
CC
CA; x39fl = TrA 6= 0,
xe 0
BB
BB
B@
0 1
0 0
0
...
0
1
CC
CC
CA; x39TrA = 0.
x4ex4f,x10x4cAx4x1cx12x4c1,xfx61JAx4x1cx0x12x4c1,x18Ax4x19x1ax1bx6dx77x43x26x6ax6bx4x1cx6c1,x68x2ex4x1c
x78x12x4c0,x6bx1cx6c0x4x19x1ax1bx6dx72x23x6ax1x6fx21x22(0),x1cx6c1x4x19x1ax1bx6dx33x54x23x6ax1x22(fl)xe2x1x19x1ax1b
x6d
0 1
0 0
,x3x48Ax4x19x1ax1bx1cx1dx1exex33x54x6c
0
BB
BB
B@
fl
0
0
...
0
1
CC
CC
CA; xe
0
BB
BB
B@
0 1
0 0
0
...
0
1
CC
CC
CA:
x79xfTrJA = TrA,x1x4ex3x71x11x12.
5,x55x3dx32x69x4x11x12xfx10x69x75x21x22x4x33x75x9:
(1)
0
BB
BB
BB
@
a1 x x ￠￠￠ x
x a2 x ￠￠￠ x
x x a3 ￠￠￠ x
...,..,..,..,..
x x x ￠￠￠ an
1
CC
CC
CC
A
,ai6=x;x6=0; (2)
0
BB
BB
BB
@
x0 a1 a2 ￠￠￠ an
a0 x1 a2 ￠￠￠ an
a0 a1 x2 ￠￠￠ an
...,..,..,..,..
a0 a1 a2 ￠￠￠ xn
1
CC
CC
CC
A
,xi6=ai.
x50,(1) jAj=
flfl
flfl
flfl
fl
0
B@
a1?x
...
an?x
1
CA+x
0
B@
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CA
flfl
flfl
flfl
fl
￠ 9 ￠
=
nQ
i=1
(ai?x)
flfl
flfl
flfl
flfl
flfl
fl
E +x
0
BB
BB
B@
1
a1?x ￠￠￠
1
a1?x1
a2?x ￠￠￠
1
a2?x
...,..,..
1
an?x ￠￠￠
1
an?x
1
CC
CC
CA
flfl
flfl
flfl
flfl
flfl
fl
=
nQ
i=1
(ai?x)
flfl
flfl
flfl
flfl
flfl
E +x
0
BB
BB
@
nP
i=1
1
ai?x 0
0
...
0 0
1
CC
CC
A
flfl
flfl
flfl
flfl
flfl
=
nQ
i=1
(ai?x)
1+
nP
i=1
x
ai?x
:
(2) x8xcx4x3fx40x33x4e
jAj =
nY
i=0
(xi?ai)
"
1+
nX
i=0
ai
xi?ai
#
:
6,x2?0 x6cnx1x21x22Ax4x6ax6bx5x6x13,x67
n0 = rankE = n;nk = rank(?0E?A)k;
ak = nk?1?nk;bk = ak?ak+1; k = 1;2;￠￠￠
x39x69x1dx3x11:
n0 n1 n2 n3 n4 ￠￠￠
a1
a2
a3
a4 ￠￠￠
b1
b2
b3 ￠￠￠
x30x31,(1)x21x22Ax4x66x4cx5x6x13?0x4x19x1ax1bx6dx4x6dx28x12x4ca1;
(2)x21x22Ax4x66x4cx5x6x13?0x4kx1x19x1ax1bx6dx4x6dx28x12x4cbk;
x4ex4f,(1) x10xbx6912–4.5x13x1x33x4e.
(2) x10x4cnix23x21x22x4xdxexdxex16,x18x3x26x4ai;bi x0x78x23x21x22x4xdxexdxex16,x2Ax4x66x4cx5x6x13?0
x4kx1x19x1ax1bx6dx4x6dx28x6cmk,x6bx68x2exdx66x4cx5x6x13?0x4xbx19x1ax1bx6dx4x1x28x3fx1ex6cm,x3d
n0 =
X
k>1
mkk +m;
n1 =
X
k>1
mk(k?1)+m;
n2 =
X
k>2
mk(k?2)+m;
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
nr =
X
k>r
mk(k?r)+m;
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
￠ 10 ￠
x6ax6b
a1 =
0
@X
k>1
mkk +m
1
A?
0
@X
k>1
mk(k?1)+m
1
A = X
k>1
mk;
a2 =
0
@X
k>1
mk(k?1)+m
1
A?
0
@X
k>2
mk(k?2)+m
1
A
=
0
@X
k>2
mk(k?1)+m
1
A?
0
@X
k>2
mk(k?2)+m
1
A = X
k>2
mk
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
ar =
0
@ X
k>r?1
mk(k?r)+m
1
A?
0
@X
k>r
mk(k?r)+m
1
A
=
0
@X
k>r
mk(k?r)+m
1
A?
0
@X
k>r
mk(k?r)+m
1
A = X
k>r
mk
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
x3x48
br = ar?ar+1 =
X
k>r
mk?
X
k>r+1
mk = mr:
7,x55x3dx32x69x4x11x12xfx10x69x75x21x22x4x19x1ax1bx1cx1dx1e:
(1)
0
BB
B@
1?3 0 3
2?6 0 13
0?3 1 3
1?4 0 8
1
CC
CA; (2)
0
BB
B@
3?1 0 0
1 1 0 0
3 0 5?3
4?1 3?1
1
CC
CA.
x50,(1) x70x20,?0 = 1x23x21x22x4x6ax6bx5x6x13,x33x4ex69x1d:
4 2 1 0 0 0 ￠￠￠
2 1 1 0 0 ￠￠￠
1 0 1 0 ￠￠￠
x3x48
b1 = 1; b2 =; b3 = 1:
x1x61x21x22x261x1x143x1x4x19x1ax1bx6dxb1x6b,x6ax21x22x4x1x28x33x20x6dx12x26x24x4x5x6x13,xfx61x68x19x1ax1bx1cx1dx1e
x6c 0
BB
B@
1 0 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA:
(2)x61x21x22x43x261x6bx5x6x13?0 = 2,x33x4ex69x1d:
4 2 0 0
2 2 0
0 2
￠ 11 ￠
x18x61x21x22x262x6b2x1x19x1ax1bx6d,xfx61x68x19x1ax1bx1cx1dx1ex6c0
BB
B@
2 1 0 0
0 2 0 0
0 0 2 1
0 0 0 2
1
CC
CA:
8,x2x21x22Ax4x5x6x13(x25x2bx28x1dx27x2c)x55x231,x30x31,Akx14Axdxe,x68x77,k x6cx63x6ax6ex6fx35x28(x58x4
xex34x4).
x4ex4f,x77x2Ax6cx19x1ax1bx6d:
J =
0
BB
BB
BB
@
1 1 0
1,..
...,..
..,1
0 1
1
CC
CC
CC
A
:
x19k > 0,x3d
Jk =
0
BB
BB
BB
@
1 k?
1,..
...,..
..,k
0 1
1
CC
CC
CC
A
:
x4cx23Jkx4x19x1ax1bx6dx4x6dx28= r?rank(E?Jk) = r?(r?1) = 1,x3x48Jkx4x19x1ax1bx1cx1dx1ex0x23J,x6ax6b
Jkx14J xdxe.
x79xf
J?1 =
0
BB
BB
BB
@
1?1?
1,..
...,..
..,?1
0 1
1
CC
CC
CC
A
:
x8x27x33x30J?1x14J xdxe,x4cx23J?kx14Jk xdxe,x6ax6bx0x14J xdxe.
x40x4cx6ax13x4x24x1e,x2Ax4x19x1ax1bx1cx1dx1ex6c
JA =
0
BB
B@
J1
J2
...
Js
1
CC
CA:
x3d
Ak? JkA?
0
BB
B@
Jk1
Jk2
...
Jks
1
CC
CA? JA? A:
x4c x4d 12–6
1,x3x69x75x21x22x4xbxcx7x8x9:
(1)
0
@
1 2?3
1 1 2
1?1 4
1
A; (2)
0
@
2?5 2
1 5?3
1 0?1
1
A;
￠ 12 ￠
(3)
0
BB
B@
3?1 3?1
1 3?1 3
3 1?3 1
1?3 1?3
1
CC
CA; (4)
0
BB
B@
1 2 0 0
2?3 0 0
0 0 0?1
0 0 1?2
1
CC
CA;
(5)
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;?(6)
0
BB
BB
B@
1 2 3 ￠￠￠ n?1 n
n 1 2 ￠￠￠ n?2 n?1
n?1 n 1 ￠￠￠ n?3 n?2.
..,..,..,..,..,..
2 3 4 ￠￠￠ n 1
1
CC
CC
CA.
x50,(1) (2)3.
(2)?3?6?2?4?.
(3)?2.
(4) (?+1)2.
(5)?(n).
(6)x2
P =
0
BB
BB
B@
0 1 0 ￠￠￠ 0
0 0 1 ￠￠￠ 0.
..,..,..,..,..
0 0 0 ￠￠￠ 1
1 0 0 ￠￠￠ 0
1
CC
CC
CA; f(?) = 1+2?+￠￠￠+n?
n?1:
x3d
A = E +2P +3P2 +￠￠￠+nPn?1 = f(P):
x10x4cPx4x5x6x13x6c1;";"2;￠￠￠ ;"n?1,x68x77"x6cnx56x4fx68xcxdx38,x3x48Ax4x5x6x7x8x9x6c
′A(?) = (f(1))(f("))￠￠￠(f("n?1)) =
n(n+1)2
g(?):
xfx6c
f(?) = n?
n+1?(n+1)?n +1
(1)2 ;
x3x48
f("k) = n"?n(1?"k)2 =? n1?"k, (*)
g(?) =
n?1Y
k=1
+ n1?"k
= 1n?1Q
k=1
(1?"k)
n?1Y
k=1
(?+n"k)
= 1n
n?1Y
k=1
+n
"
k
=?
n?1
n ￠
+n
·n
1
+n
1
=?
n?1
n2
+n
n
1
= 1n2 [(?+n)nn]:
x3x48
′A(?) = 1n2
n(n+1)2
[(?+n)nn]:
x79x10(?)x20,′A(?)x37x6cx38,x18Ax4xbxcx7x8x9x72x23x68x5x6x7x8x9,x6ax6bAx4xbxcx7x8x9x6c
1
n2
n(n+1)2
[(?+n)nn]:
￠ 13 ￠
2,x2Ax6cnx1x3fx22,m(?)x23x6dx4xbxcx7x8x9,g(?)x6cx63x6ax7x8x9,d(?) = (m(?);g(?)).
x30x31,(1) rankd(A) = rankg(A);
(2) g(A)x33x4dx4x43x1x44x45x46x47x23g(?)x14m(?)x72x3b;
(3)x39g(A)x33x4d,x3dg?1(A)x6ax59x23Ax4x7x8x9.
x4ex4f,(1) x6dx25x7x8x9u(?);v(?),x3c
m(?)u(?)+g(?)v(?) = d(?):
x18
m(A)u(A)+g(A)v(A) = d(A):
x10m(A) = 0x33x4ed(A) = g(A)v(A),x3x48
rankd(A) 6 rankg(A):
x79xfd(?) j g(?),x6dx25h(?)x3cd(?)h(?) = g(?),x1d(A)h(A) = g(A),x4cx23
rankg(A) 6 rankd(A):
x2bx2cx4e
rankg(A) = rankd(A):
(2) ())x2?1;￠￠￠ ;?n x6cAx4x55x35x5x6x13,x3dg(A)x4x55x35x5x6x13x6cg(?1);￠￠￠ ;g(?n),x39g(A)x33x4d,
x3dg(A)x4x36x6bx5x6x13g(?i) 6= 0,x10x4cm(?)x4x38x78x23Ax4x5x6x13,xfx61g(?)x14m(?)x37x14x15x38,x6ax6b
(g(?);m(?)) = 1.
(()x39(g(?);m(?)) = 1,x3dx10(1)x3x30,d(?) = 1,xfx61d(A) = E,x18x40x4c(1)x77x4v(?),x26
g(A)v(A) = E;
g(A)x33x4d.
(3)x39g(A)x33x4d,x25(2)x4x43x1x75x4x30x31x77,x1fx4eg(A)v(A) = E,x3x48g(A)?1 = v(A)x6cAx4x7x8
x9.
3,x30x31,x21x22A(x25x2bx28x29x32)x33x40x41x42x4x43x1x44x45x46x47x23x68xbxcx7x8x9x37x6cx38.
x4ex4f,()) Ax33x40x41x42,x6ax6bx61x40x41x1ex72x23Ax4x19x1ax1bx1cx1dx1e,xfx61Ax4x19x1ax1bx6dx55x23x6ax1x4,A
x4x11x12xfx10x55x23x6ax56x4,x6bAx4xbxcx7x8x9x31x6cx11x12xfx10x4x2bxcx14x37x9,x6ax59x23xdx8x6ax56xfx10x4x7cx29,
x6ax6bx37x6cx38.
(()x39Ax4xbxcx7x8x9x37x6cx38,x3dx61xbxcx7x8x9x23xdx8x6ax56xfx10x4x7cx29,x4cx23Ax4x11x12xfx10x78x23
x6ax56x4,x1x19x1ax1bx1cx1dx1ex77x4x19x1ax1bx6dx78x23x6ax1x4,x23x6ax6bx40x41x21x22,xfx31Ax33x40x41x42.
4,x2A =
0
@
1 1?1
1 0 0
0 1?1
1
A,x3A100.
x50,Ax4x5x6x7x8x9x6c?(?2?2),x67
100 =?(?2?2)g(?)+a?2 +b?+c;
x1x24x48? = 0;p2;?p2x7fx2ex32x9,x4
c = 0; 250 = 2a+p2b; 250 = 2a?p2b:
x2fx4eb = 0,a = 249,x3x48
A100 = 249A2 = 249
0
@
2 0 0
1 1?1
1?1 1
1
A:
￠ 14 ￠
5,x30x31,x39x3ax40x63x64k 2Nx78x26Tr(Ak) = 0,x3d′A(?) =?n.
x4ex4f,x10x4cx21x22x4x16x72x23x21x22x4x55x35x5x6x13x3fx1e,x2Ax4x5x6x13x6c?1;￠￠￠ ;?n,x3d
Tr(Ak) =?k1 +￠￠￠+?kn = sk:
x10Tr(Ak) = 0x33x4esk = 0,x6ax17x18x14x9x33x4e?1;￠￠￠ ;?nx4x3x26x11x12x40x67x7x8x9 1 = ￠￠￠ = n = 0,x4cx23
′A(?) = (1)￠￠￠(n) =?n:
6,x2Ax4x5x6x7x8x9′(?) = h(?)g(?),x62(h(?);g(?)) = 1,
x30x31,rankh(A) = degg(?),rankg(A) = degh(?).
x4ex4f,x2Ax4x5x6x13x6c?1;￠￠￠ ;?n,x3dh(A)x4x5x6x13x6ch(?1);￠￠￠ ;h(?n),g(A)x4x5x6x13x6cg(?1);￠￠￠ ;
g(?n),x10x4c′(?) = h(?)g(?)x62(h(?);g(?)) = 1,xfx61fh(?i)gx770x4x6bx28x12x4cdegh(?),fg(?i)gx770x4
x6bx28x12x4cdegg(?),x62degh(?)+degg(?) = n.
x10xbx6912–3.6x20,
degh(?) > n?rank(h(A)) (1)
degg(?) > n?rank(g(A)) (2)
xfx61
n = degh(?)+degg(?) > 2n?(rankh(A)+rankg(A));
rankh(A)+rankg(A) > n:
x79xf
h(A)g(A) = ′(A) = 0;
rankh(A)+rankg(A) 6 n:
x4cx23
rankh(A)+rankg(A) = n:
x6ax6b(1),(2)x9x55x78x4fx12x2b,x3cx4e
rankh(A) = n?degh(?) = degg(?);
rankg(A) = n?degg(?) = degh(?):
￠ 15 ￠
x25x2fx30x27 x31x32x33x34x35xax36x37x38x39x3ax3b
x4c x4d 12–1
1,x69x75x7x8x9x21x22x77,x73x74x23x33x4dx4? x19x33x4dx3ex3x68x4d.
(1)
+11
+3?+1
; (2)
2?2?2
+2?+1
;
(3)
0
@
12
+2+12
1+2 +1
1
A; (4)
0
@
1?2?
2 +1?2?2?1
1
A.
x50,(1)x33x4d,x4dx21x22x6c 14
+1+1
3?+1
.
(2)x33x4d,x4dx21x22x6c? 12
+12 +?
2?2?2
.
(3)x33x4d,x4dx21x22x6c
0
@
2+12
2 +2+12
1 0 1
1
A.
(4)xdx33x4d.
2,x3x69x75x7x8x9x21x22x4x58x5bx1e:
(1)
+1?
11
; (2)
11
1?2?2?+1
;
(3)
0
@
12?1
31?2 +2? 3?2?1
+1?2?2 +1
1
A; (4)
0
@
2?2?1 3?2
22 +3?2?2?3?
2 +2 +? 2?2 +2?
1
A;
(5)
0
@
+2 0 0
1?+2 0
0?1?+2
1
A; (6)
0
@
0 0?(1)
0?2?1 0
(1)2 0 0
1
A.
x50,(1) diag(1;1).
(2) diag(1;(1)(2)).
(3) diag(1;?;0).
(4) diag(1;?(?+1);?(?+1)2
12
·
).
(5) diag(1;1;(?+2)3).
(6) diag(1;?(1)(?+1);?(1)2(?+1)).
3,xax74x69x75x7x8x9x21x22x23xcx12x54:
(1) A =
0
@
3?2?4?+3
22 25?2?4?+3
22 (2)2
1
A; B =
0
@
2?3?+3 233
2?2?+1 47 25
2?3?+2 242
1
A.
(2) A =
0
@
22?2?1?+1
0?+1 1
(?+1)2?2 ++1
1
A; B =
0
@
1 2?2 +11
2?2 +?
1+1
1
A.
x50,(1)x12x54; (2)xdx12x54.
￠ 1 ￠
4,x2A(?)x6cx6ax6bx7x8x9x21x22,x30x31,A(?)x33x4dx4x43x1x44x45x46x47x23x40x3x26x4x2bx28c,A(c)x78x33x4d.
x4ex4f,())x2A(?)x33x4d,x3d
jA(?)j = a 6= 0 2C:
x18x40x63x64x4c 2C,jA(c)j = a,x3x48A(c)x33x4d.
(()x15x16f(?) = jA(?)j,x3dx40x63x64x4c 2C,f(c) 6= 0,x18f(?)x25Cx77x37x38,x3x48f(?) = a 6= 0 2C,
jA(?)j = a 6= 0 2C,xfx61A(?)x33x4d.
5,x69x75x11x12x23xcx5bx13,(x39x5bx13,x3dx1x48x30x31,x39xdx5bx13,x3dx2x18xfx3.)
x49x6bx7x8x9x21x22x12x54x4x43x1x44x45x46x47x23,x40x3x26x4k 2 K,A(k)x14B(k)x78x12x54.
x50,xdx5bx13,x39
A(?) =
1
; B(?) =
1
2
;
x3dA(?)x14B(?)xdx12x54,x68x40x63x64x4k 2 K,A(k)x14B(k)x12x54.
x4c x4d 12–2
1,x3x69x75x7x8x9x21x22x4x1c:
(1)
0
@
2?1?+1 21
+1?2 +2?+1?1
2 +2 +3?+22
1
A; (2)
0
@
+1?1?2
22?1?2
1?2
1
A.
x50,(1) 3; (2) 2.
2,x2A(?)x6cx6ax6bx7x8x9x21x22,x30x31,rankA(?) = maxfrankA(k)jk 2 Kg.
x50,x2rankA(?) = r,x3dA(?)x26x6ax6br x1x10x9Mr+1(?) = 0,x18x40x3x26x4k 2 K,Mr+1(k) = 0,x5f
xfx31rankA(k) 6 r,x79xfMr(?) 6= 0,x6dx25c 2 Kx3cMr(c) 6= 0,x5fxfx31r = maxfrankA(k) j k 2 Kg.
3,x3ex3x69x75x21x22x4xdxexfx10:
(1)
0
@
1?1 0
01?1
0 01
1
A; (2)
0
@
+2 (1)2+1
1?2 0
2?2?(1)2?2?1
1
A;
(3)
0
BB
B@
+fi fl 1 0
fl?+fi 0 1
0 0?+fi fl
0 0?fl?+fi
1
CC
CA; (4)
0
BB
B@
1 1 0 0
01 1 0
0 01 1
0 0 01
1
CC
CA;
(5)
0
BB
BB
B@
fi fl fl fl ￠￠￠ fl
0fi fl fl ￠￠￠ fl
0 0fi fl ￠￠￠ fl
:::::::::::::::::::::::::::::::::::::
0 0 0 0 ￠￠￠fi
1
CC
CC
CA; (6)
0
BB
BB
B@
0 0 ￠￠￠ 0 an
1? 0 ￠￠￠ 0 an?1
0?1? ￠￠￠ 0 an?2
::::::::::::::::::::::::::::::
0 0 0 ￠￠￠?1?+a1
1
CC
CC
CA.
x50,(1) 1;1;(1)3.
(2) 1;1;?(1).
(3)x39fl 6= 0,1;1;1;[(?+fi)2 +fl2]2;x39fl = 0,1;1;(?+fi)2;(?+fi)2.
(4) 1;1;1;(1)4.
(5)x39fl 6= 0,1;1;￠￠￠ ;1;(fi)n;x39fl = 0,fi;fi;￠￠￠ ;fi.
(6) 1;1;￠￠￠ ;1;?n +a1?n?1 +￠￠￠+an.
4,x2Dk(?) (k = 1;2;￠￠￠ ;r)x6cA(?)x4x33x75x9xfx10,x30x31:
D2k(?) j Dk?1(?)Dk+1(?); k = 2;3;￠￠￠ ;r?1:
￠ 2 ￠
x4ex4f,x2A(?)x4xdxexfx10x6c
d1(?);d2(?);￠￠￠ ;dn(?);
x3d
Dk?1(?) = d1(?)d2(?)￠￠￠dk?1(?);
Dk(?) = d1(?)d2(?)￠￠￠dk(?) = Dk?1(?)dk(?);
Dk+1(?) = d1(?)d2(?)￠￠￠dk+1(?);
x3x48
D2k(?) = D2k?1(?)d2k(?) j Dk?1(?)Dk(?)dk(?)dk+1(?);
D2k(?) j Dk?1(?)Dk+1(?):
5,x2A(?)x6cnx1x3fx22,x30x31,A(?)x14AT(?)x12x54.
x4ex4f,x6dx25x33x4dx21x22P(?);Q(?),x3c
P(?)A(?)Q(?) =
0
BB
B@
d1(?)
d2(?)
...
dn(?)
1
CC
CA;
x18
P(?)A(?)Q(?) =
0
BB
B@
d1(?)
d2(?)
...
dn(?)
1
CC
CA
T
= Q(?)TA(?)TP(?)T;
x4cx23A(?)x14AT(?)x12x54.
6,x2f1(x);￠￠￠ ;fn(x) 2 K[x],x62(f1(x);￠￠￠ ;fn(x)) = 1.
x30x31,x6dx25x7x8x9fij(x) 2 K[x] (i = 2;3;￠￠￠ ;n; j = 1;2;￠￠￠ ;n),x3c
flfl
flfl
flfl
flfl
fl
f1(x) f2(x) ￠￠￠ fn(x)
f21(x) f22(x) ￠￠￠ f2n(x)
...,..,..,..
fn1(x) fn2(x) ￠￠￠ fnn(x)
flfl
flfl
flfl
flfl
fl
= 1:
x4ex4f,x15x16x7x8x9x21x22
A(x) = (f1(x);f2(x);￠￠￠ ;fn(x));
x10x1fx20,A(x)x4xdxexfx10x6c1,x18x6dx25x33x4dx21x22P(x),x3c
A(x)P(x) = (1;0;￠￠￠ ;0),(*)
x2jP(x)j = c 6= 0,x3dx6dx25x33x4dx21x22Q(x),x3c
Q(x)P(x) =
0
BB
BB
B@
1
1
...
1
c
1
CC
CC
CA,(**)
x7
Q(x) = (fij(x));
￠ 3 ￠
x31
B(x) =
0
BB
B@
f1(x) f2(x) ￠￠￠ fn(x)
f21(x) f22(x) ￠￠￠ f2n(x)
...,..,..,..
fn1(x) fn2(x) ￠￠￠ fnn(x)
1
CC
CA;
x3dx10(*)x14(**)x20
B(x)P(x) =
0
BB
BB
B@
1
1
...
1
c
1
CC
CC
CA;
x4cx23jB(x)jjP(x)j = c,x79xfjP(x)j = c,x4ejB(x)j = 1,x6ax6bfij(x)x1x6cx3x3.
x4c x4d 12–3
1,xax74x69x75x21x22x23xcxdxe:
(1) A =
0
@
3 2?5
2 6?10
1 2?3
1
A; B =
0
@
6 20?34
6 32?51
4 20?32
1
A.
(2) A =
0
@
6 6?15
1 5?5
1 2?2
1
A; B =
0
@
37?20?4
34?17?4
119?70?11
1
A.
(3) A =
0
@
2?2 1
1?1 1
1?2 2
1
A; B =
0
@
1?3 3
2?6 13
1?4 8
1
A.
x50,(1)x23; (2)x23; (3) xc.
2,x30x31,x63x42x3fx22Ax14x6dx4x7cx4x21x22AT xdxe.
x4ex4f,x10x4c?E?AT = (?E?A)Tx12x54x4c?E?A(xbx6912–2.5),xfx61Ax14AT xdxe.
3,x2Ax14B x6cnx1x3fx22,x30x31,(AB)? = B?A?
x4ex4f,x15x16x12x9
[(?E +A)(?E +B)][(?E +A)(?E +B)]?
= j(?E +A)(?E +B)jE = j?E +AjE ￠j?E +BjE
= (?E +A)(?E +B)(?E +B)?(?E +A)?:
x3x48
(?E +A)(?E +B)f[(?E +A)(?E +B)](?E +B)?(?E +A)?g = 0:
x63x64x32x9x49x79x4x56x28,x20
[(?E +A)(?E +B)](?E +B)?(?E +A)? = 0;
x1
[(?E +A)(?E +B)]? = (?E +B)?(?E +A)?:
x67? = 0x72x26
(AB)? = B?A?:
4,x30x31,x39x3ax21x22Ax14B xdxe,x3dx6dx72x4x5x6x21x22A?x14B? x0xdxe.
￠ 4 ￠
x4ex4f,x3eAx14B xdxe,x3dx6dx25x33x4dx21x22P,x3cP?1AP = B,x4cx23x55x3dxbx694x4x11x12,
B? = (P?1AP)? = P?A?(P?1)? = P?A?(P?)?1:
x18A?x14B? xdxe.
5,x30x31,x21x22x4xdxex14x28x29x4x67x7x37x76.
x4ex4f,x2A;Bx23x28x29K1x77x4x21x22,x3d?E?Ax14?E?Bx4xdxexfx10x78x23x36x28x25K1x77x4x7x8x9.
x2x28x29K1 ‰ K2,x8x51x5fx74x7x8x9x0x33x48x6fx5bx36x28x25K2x77x4x7x8x9,x6ax6bxdxexfx10x5ax14x28x29x4x67x7x37
x76(x2bx7x9x6ax6bxax28xfx10),x6bx21x22A;B xdxex1bx62x43x1b?E?Ax14?E?Bx26xdx8x4xdxexfx10x5a.xfx61x21
x22x4xdxex14x28x29x4x67x7x37x76.
6,x2Ax6cnx1x3fx22,?0 x6cAx4x6ax6bx5x6x13,x30x31,x5x6x13?0x4x7fx28x6cx28> n?rank(?0E?A).
x4ex4f,x2?0 x6c A x4 r x6cx5x6x13,x2 d1(?);￠￠￠ ;dn(?) x6c A x4xdxexfx10,x3d0 j dn(?),x68
0 -dn?r(?),(xcx3d,x390 j dn?r(?),x3d0 j dn?r+1(?);￠￠￠ ;0 j dn(?),x4cx230x4x6c
x28> r +1)xfx61x6dx25x33x4dx21x22P(?);Q(?)x3c
P(?)(?E?A)Q(?) =
0
BB
BB
BB
BB
B@
d1(?)
...
dn?r(?)
dn?r+1(?)
...
dn(?)
1
CC
CC
CC
CC
CA;
x18
P(?0)(?0E?A)Q(?0) =
0
BB
BB
BB
BB
B@
d1(?0)
...
dn?r(?0)
dn?r+1(?0)
...
0
1
CC
CC
CC
CC
CA;
x10x4cd1(?0) 6= 0;￠￠￠ ;dn?r(?0) 6= 0,x3x48
rank(?0E?A) > rankP(?0)(?0E?A)Q(?0) > n?r:
xfx61
r > n?rank(?0E?A):
x4c x4d 12–4
1,x3x69x75x7x8x9x21x22x4x11x12xfx10:
(1)
0
@
2 +231?2 +23
2?2 +35?2?1?2 +34
2 +2 01
1
A; (2)
0
@
2?2?2 +1 2?2?2
2 +1?2 +1 2?2?2
2 +2?2 +1 3?2?5
1
A.
x50,(1)?;1;1;1;?+3.
(2)?+1;3.
2,x1fx20x7x8x9x21x22A(?)x4x11x12xfx10,x1crx14x1x28n,x3A(?)x4x58x5bx1e:
(1)?+1;?+1;(?+1)2;1;(1)2; r = 4,n = 5;
(2)2;(2)2;(2)3;?+2;(?+2)3; r = 4,n = 4;
￠ 5 ￠
(3)1;(1)2;(1)3;?+2;(?+2)2; r = 3,n = 5.
x50,(1) diag(1;?+1;(?+1)(1);(?+1)2(1)2;0).
(2) diag(1;2;(2)2(?+2);(2)3(?+2)3).
(3) diag(1;(1)2(?+2);(1)3(?+2)2;0;0).
3,x3x69x75x21x22x4x58x5bx1e:
(1)
0
BB
B@
0?(?+1)2 0 0
2(1) 0 0 0
0 0 0?2?1
0 0?(?+1)2 0
1
CC
CA;
(2)
0
BB
B@
2?4 0 0 0
0?2 +2? 0 0
0 0?3?2?2 0
0 0 0?3?4?
1
CC
CA;
(3)
0
BB
B@
2 +23?2 +2 0 0
2?2 +24 2?2 +3 0 0
0 0?+1?+2
0 0?2?1?2 +2
1
CC
CA;
(4)
0
BB
B@
22 0?3 +?21 0
2?4 0?3 +2?22 0
0?2 +2? 0?2 +62
0?2 +2 0?2 +57
1
CC
CA.
x50,(1) diag(1;?(?+1);?(?+1)2(1);?2(?+1)2(1)).
(2) diag(1;?(?2?4);?(?2?4);?2(?2?4)).
(3) diag(1;1;(1)(?+1);0).
(4) diag(1;1;?2?4;0).
4,x3x69x75x21x22x4xdxexfx10,x33x75x9xfx10x14x11x12xfx10:
(1)
0
@
4 2?5
6 4?9
5 3?7
1
A; (2)
0
@
2 1 3
6?3?9
4?2?6
1
A;
(3)
0
BB
B@
1 1 1 ￠￠￠ 1
1 1 1 ￠￠￠ 1
::::::::::::::::
1 1 1 ￠￠￠ 1
1
CC
CA; (4)
0
BB
B@
2?3 0 0
3?4 0 0
1 5 1?2
0 2 2?3
1
CC
CA.
x50,(1)xdxexfx10,1;1;?2(1),x33x75x9xfx10,1;1;?2(1),x11x12xfx10,?2;1.
(2)xdxexfx10,1;?;?(?+1),x33x75x9xfx10,1;?;?2(?+1),x11x12xfx10,?;?;?+1.
(3)xdxexfx10,1;?;￠￠￠ ;?| {z }
n?2x0;?(n),x33x75x9xfx10,1;?;?2;￠￠￠ ;?n?2;?n?1(n),x11x12xfx10,?;￠￠￠ ;?| {z }
n?1x0;
n.
(4)xdxexfx10,1;1;1;(?+1)4,x33x75x9xfx10,1;1;1;(?+1)4,x11x12xfx10,(?+1)4.
5,x2?0 x6c n x1x21x22Ax4x6ax6bx5x6x13,x30x31,x21x22Ax4x66x4cx5x6x13?0 x4x11x12xfx10x4x6bx28x12x4c
n?rank(?0E?A).
x4ex4f,x2d1(?);￠￠￠ ;dn(?)x6cAx4xdxexfx10,x39Ax4x66x4cx5x6x13?0x4x11x12xfx10x4x6bx28x6cr,x3d
0 j dn(?);￠￠￠ ;0 j dn?r+1(?);0 -dn?r(?);￠￠￠ ;0 -d1(?):
￠ 6 ￠
xfx61x6dx25x33x4dx21x22P(?);Q(?)x3c
P(?)(?E?A)Q(?) =
0
B@
d1(?)
...
dn(?)
1
CA;
P(?0)(?0E?A)Q(?0) =
0
BB
BB
BB
BB
@
d1(?0)
...
dn?r(?0)
0
...
0
1
CC
CC
CC
CC
A;
x4cx23
n?r = rankP(?0)(?0E?A)Q(?0) = rank(?0E?A);
x1
r = n?rank(?0E?A):
x4c x4d 12–5
1,x3x69x75x21x22x4x19x1ax1bx1cx1dx1e:
(1)
0
@
1?1 0
0?1 0
1 2 1
1
A; (2)
0
@
2 6?15
1 1?5
1 2?6
1
A;
(3)
0
@
13 16 14
6?7?6
6?8?7
1
A; (4)
0
@
9?6?2
18?12?3
18?9?6
1
A;
(5)
0
@
1?3 3
2?6 13
1?4 8
1
A; (6)
0
@
1?2?1
2 4 2
3?6?3
1
A;
(7)
0
@
1?1 1
3?3 3
2?2 2
1
A; (8)
0
@
5 2 6
2 0 3
2 1?2
1
A;
(9)
0
BB
B@
2 1 1?2
5?4 2 9
3 1 2?2
2?4 3 8
1
CC
CA; (10)
0
BB
B@
3?4 0 2
4?5?2 4
0 0 3?2
0 0 2?1
1
CC
CA;
(11)
0
BB
BB
B@
0 1 0 ￠￠￠ 0 0
0 0 1 ￠￠￠ 0 0
:::::::::::::::::::
0 0 0 ￠￠￠ 0 1
1 0 0 ￠￠￠ 0 0
1
CC
CC
CA; (12)
0
BB
B@
1 2 3 ￠￠￠ n
0 1 2 ￠￠￠ n?1
::::::::::::::::::::
0 0 0 ￠￠￠ 1
1
CC
CA.
x50,(1)
0
@
1 0 0
0 1 1
0 0 1
1
A; (2)
0
@
1 0 0
0?1 1
0 0?1
1
A;
(3)
0
@
1 0 0
0?1 1
0 0?1
1
A; (4)
0
@
3 0 0
0?3 1
0 0?3
1
A;
￠ 7 ￠
(5)
0
@
1 1 0
0 1 1
0 0 1
1
A; (6)
0
@
2 0 0
0 0 0
0 0 0
1
A;
(7)
0
@
0 1 0
0 0 0
0 0 0
1
A; (8)
0
@
1 0 0
0 1+2p6 0
0 0 1?2p6
1
A;
(9)
0
BB
B@
1 0 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA; (10)
0
BB
B@
1 1 0 0
0 1 0 0
0 0?1 1
0 0 0?1
1
CC
CA;
(11) diag(1;"1;"2;￠￠￠ ;"n?1),1;"1;"2;￠￠￠ ;"n?1x23xn?1x4nx6bx38;
(12)
0
BB
BB
B@
1 1 0 ￠￠￠ 0
0 1 1 ￠￠￠ 0.
..,..,..,..,..
0 0 ￠￠￠ 1 1
0 0 ￠￠￠ 0 1
1
CC
CC
CA.
2,x2x21x22
A =
0
@
2 0 0
a 2 0
b c 2
1
A:
(1)x21x22Ax33x54x26xbxcx4x19x1ax1bx1cx1dx1e?
(2)x3ex44x59Ax33x40x41x42x4x46x47.
x50,(1) Ax43x26x6ax6bx5x6x13?0 = 2,x3x48Ax4x19x1ax1bx6dx4x6dx28= Ax4x11x12xfx10x4x6bx28= rank(?0E?
A) (xdx45xbx6912–4.5) x6b
rank(?0E?A) =
8>
<
>:
2 x1bac 6= 0,
1 x1ba;cx77x6ax6bx12x4c0,x4x6ax6bxdx12x4c0,xea;cx78x230,x68b 6= 0x52,
0 x1ba = b = c = 0x52.
xfx61x1bac 6= 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 1 0
0 2 1
0 0 2
1
A;x1ba;cx77x6ax6bx12x4c0,x4x6ax6bxdx12x4c0,xea;cx78x230,
x68b 6= 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 0 0
0 2 1
0 0 2
1
A;x1ba = b = c = 0x52,Ax4x19x1ax1bx1cx1dx1ex23
0
@
2 0 0
0 2 0
0 0 2
1
A.
(2) Ax33x40x41x42 () a = b = c = 0.
3,x2x21x22Ax4x5x6x7x8x9
′A(?) =?5 +?4?5?32 +84:
x3ex3x18Ax3x26x33x54x4x19x1ax1bx1cx1dx1e.
x50,′A(?) = (1)3(?+2)2,xfx61Ax4x33x54x4x11x12xfx10x6c:
(a)1;1;1;?+2;?+2;
(b) (1)2;1;?+2;?+2;
(c) (1)3;?+2;?+2;
(d)1;1;1;(?+2)2;
(e) (1)2;1;(?+2)2;
￠ 8 ￠
(f) (1)3;(?+2)2.
x18Ax4x33x54x4x19x1ax1bx1cx1dx1ex6c:
0
BB
BB
B@
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 1 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 0
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 0 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA;
0
BB
BB
B@
1 1 0 0 0
0 1 1 0 0
0 0 1 0 0
0 0 0?2 1
0 0 0 0?2
1
CC
CC
CA:
4,x2x21x22Ax4x1cx6c1,x30x31,Ax4x19x1ax1bx1cx1dx1exex33x54x6c
0
BB
BB
B@
fl
0
0
...
0
1
CC
CC
CA; x39fl = TrA 6= 0,
xe 0
BB
BB
B@
0 1
0 0
0
...
0
1
CC
CC
CA; x39TrA = 0.
x4ex4f,x10x4cAx4x1cx12x4c1,xfx61JAx4x1cx0x12x4c1,x18Ax4x19x1ax1bx6dx77x43x26x6ax6bx4x1cx6c1,x68x2ex4x1c
x78x12x4c0,x6bx1cx6c0x4x19x1ax1bx6dx72x23x6ax1x6fx21x22(0),x1cx6c1x4x19x1ax1bx6dx33x54x23x6ax1x22(fl)xe2x1x19x1ax1b
x6d
0 1
0 0
,x3x48Ax4x19x1ax1bx1cx1dx1exex33x54x6c
0
BB
BB
B@
fl
0
0
...
0
1
CC
CC
CA; xe
0
BB
BB
B@
0 1
0 0
0
...
0
1
CC
CC
CA:
x79xfTrJA = TrA,x1x4ex3x71x11x12.
5,x55x3dx32x69x4x11x12xfx10x69x75x21x22x4x33x75x9:
(1)
0
BB
BB
BB
@
a1 x x ￠￠￠ x
x a2 x ￠￠￠ x
x x a3 ￠￠￠ x
...,..,..,..,..
x x x ￠￠￠ an
1
CC
CC
CC
A
,ai6=x;x6=0; (2)
0
BB
BB
BB
@
x0 a1 a2 ￠￠￠ an
a0 x1 a2 ￠￠￠ an
a0 a1 x2 ￠￠￠ an
...,..,..,..,..
a0 a1 a2 ￠￠￠ xn
1
CC
CC
CC
A
,xi6=ai.
x50,(1) jAj=
flfl
flfl
flfl
fl
0
B@
a1?x
...
an?x
1
CA+x
0
B@
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CA
flfl
flfl
flfl
fl
￠ 9 ￠
=
nQ
i=1
(ai?x)
flfl
flfl
flfl
flfl
flfl
fl
E +x
0
BB
BB
B@
1
a1?x ￠￠￠
1
a1?x1
a2?x ￠￠￠
1
a2?x
...,..,..
1
an?x ￠￠￠
1
an?x
1
CC
CC
CA
flfl
flfl
flfl
flfl
flfl
fl
=
nQ
i=1
(ai?x)
flfl
flfl
flfl
flfl
flfl
E +x
0
BB
BB
@
nP
i=1
1
ai?x 0
0
...
0 0
1
CC
CC
A
flfl
flfl
flfl
flfl
flfl
=
nQ
i=1
(ai?x)
1+
nP
i=1
x
ai?x
:
(2) x8xcx4x3fx40x33x4e
jAj =
nY
i=0
(xi?ai)
"
1+
nX
i=0
ai
xi?ai
#
:
6,x2?0 x6cnx1x21x22Ax4x6ax6bx5x6x13,x67
n0 = rankE = n;nk = rank(?0E?A)k;
ak = nk?1?nk;bk = ak?ak+1; k = 1;2;￠￠￠
x39x69x1dx3x11:
n0 n1 n2 n3 n4 ￠￠￠
a1
a2
a3
a4 ￠￠￠
b1
b2
b3 ￠￠￠
x30x31,(1)x21x22Ax4x66x4cx5x6x13?0x4x19x1ax1bx6dx4x6dx28x12x4ca1;
(2)x21x22Ax4x66x4cx5x6x13?0x4kx1x19x1ax1bx6dx4x6dx28x12x4cbk;
x4ex4f,(1) x10xbx6912–4.5x13x1x33x4e.
(2) x10x4cnix23x21x22x4xdxexdxex16,x18x3x26x4ai;bi x0x78x23x21x22x4xdxexdxex16,x2Ax4x66x4cx5x6x13?0
x4kx1x19x1ax1bx6dx4x6dx28x6cmk,x6bx68x2exdx66x4cx5x6x13?0x4xbx19x1ax1bx6dx4x1x28x3fx1ex6cm,x3d
n0 =
X
k>1
mkk +m;
n1 =
X
k>1
mk(k?1)+m;
n2 =
X
k>2
mk(k?2)+m;
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
nr =
X
k>r
mk(k?r)+m;
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
￠ 10 ￠
x6ax6b
a1 =
0
@X
k>1
mkk +m
1
A?
0
@X
k>1
mk(k?1)+m
1
A = X
k>1
mk;
a2 =
0
@X
k>1
mk(k?1)+m
1
A?
0
@X
k>2
mk(k?2)+m
1
A
=
0
@X
k>2
mk(k?1)+m
1
A?
0
@X
k>2
mk(k?2)+m
1
A = X
k>2
mk
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
ar =
0
@ X
k>r?1
mk(k?r)+m
1
A?
0
@X
k>r
mk(k?r)+m
1
A
=
0
@X
k>r
mk(k?r)+m
1
A?
0
@X
k>r
mk(k?r)+m
1
A = X
k>r
mk
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠
x3x48
br = ar?ar+1 =
X
k>r
mk?
X
k>r+1
mk = mr:
7,x55x3dx32x69x4x11x12xfx10x69x75x21x22x4x19x1ax1bx1cx1dx1e:
(1)
0
BB
B@
1?3 0 3
2?6 0 13
0?3 1 3
1?4 0 8
1
CC
CA; (2)
0
BB
B@
3?1 0 0
1 1 0 0
3 0 5?3
4?1 3?1
1
CC
CA.
x50,(1) x70x20,?0 = 1x23x21x22x4x6ax6bx5x6x13,x33x4ex69x1d:
4 2 1 0 0 0 ￠￠￠
2 1 1 0 0 ￠￠￠
1 0 1 0 ￠￠￠
x3x48
b1 = 1; b2 =; b3 = 1:
x1x61x21x22x261x1x143x1x4x19x1ax1bx6dxb1x6b,x6ax21x22x4x1x28x33x20x6dx12x26x24x4x5x6x13,xfx61x68x19x1ax1bx1cx1dx1e
x6c 0
BB
B@
1 0 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA:
(2)x61x21x22x43x261x6bx5x6x13?0 = 2,x33x4ex69x1d:
4 2 0 0
2 2 0
0 2
￠ 11 ￠
x18x61x21x22x262x6b2x1x19x1ax1bx6d,xfx61x68x19x1ax1bx1cx1dx1ex6c0
BB
B@
2 1 0 0
0 2 0 0
0 0 2 1
0 0 0 2
1
CC
CA:
8,x2x21x22Ax4x5x6x13(x25x2bx28x1dx27x2c)x55x231,x30x31,Akx14Axdxe,x68x77,k x6cx63x6ax6ex6fx35x28(x58x4
xex34x4).
x4ex4f,x77x2Ax6cx19x1ax1bx6d:
J =
0
BB
BB
BB
@
1 1 0
1,..
...,..
..,1
0 1
1
CC
CC
CC
A
:
x19k > 0,x3d
Jk =
0
BB
BB
BB
@
1 k?
1,..
...,..
..,k
0 1
1
CC
CC
CC
A
:
x4cx23Jkx4x19x1ax1bx6dx4x6dx28= r?rank(E?Jk) = r?(r?1) = 1,x3x48Jkx4x19x1ax1bx1cx1dx1ex0x23J,x6ax6b
Jkx14J xdxe.
x79xf
J?1 =
0
BB
BB
BB
@
1?1?
1,..
...,..
..,?1
0 1
1
CC
CC
CC
A
:
x8x27x33x30J?1x14J xdxe,x4cx23J?kx14Jk xdxe,x6ax6bx0x14J xdxe.
x40x4cx6ax13x4x24x1e,x2Ax4x19x1ax1bx1cx1dx1ex6c
JA =
0
BB
B@
J1
J2
...
Js
1
CC
CA:
x3d
Ak? JkA?
0
BB
B@
Jk1
Jk2
...
Jks
1
CC
CA? JA? A:
x4c x4d 12–6
1,x3x69x75x21x22x4xbxcx7x8x9:
(1)
0
@
1 2?3
1 1 2
1?1 4
1
A; (2)
0
@
2?5 2
1 5?3
1 0?1
1
A;
￠ 12 ￠
(3)
0
BB
B@
3?1 3?1
1 3?1 3
3 1?3 1
1?3 1?3
1
CC
CA; (4)
0
BB
B@
1 2 0 0
2?3 0 0
0 0 0?1
0 0 1?2
1
CC
CA;
(5)
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1.
..,..,..,..
1 1 ￠￠￠ 1
1
CC
CA;?(6)
0
BB
BB
B@
1 2 3 ￠￠￠ n?1 n
n 1 2 ￠￠￠ n?2 n?1
n?1 n 1 ￠￠￠ n?3 n?2.
..,..,..,..,..,..
2 3 4 ￠￠￠ n 1
1
CC
CC
CA.
x50,(1) (2)3.
(2)?3?6?2?4?.
(3)?2.
(4) (?+1)2.
(5)?(n).
(6)x2
P =
0
BB
BB
B@
0 1 0 ￠￠￠ 0
0 0 1 ￠￠￠ 0.
..,..,..,..,..
0 0 0 ￠￠￠ 1
1 0 0 ￠￠￠ 0
1
CC
CC
CA; f(?) = 1+2?+￠￠￠+n?
n?1:
x3d
A = E +2P +3P2 +￠￠￠+nPn?1 = f(P):
x10x4cPx4x5x6x13x6c1;";"2;￠￠￠ ;"n?1,x68x77"x6cnx56x4fx68xcxdx38,x3x48Ax4x5x6x7x8x9x6c
′A(?) = (f(1))(f("))￠￠￠(f("n?1)) =
n(n+1)2
g(?):
xfx6c
f(?) = n?
n+1?(n+1)?n +1
(1)2 ;
x3x48
f("k) = n"?n(1?"k)2 =? n1?"k, (*)
g(?) =
n?1Y
k=1
+ n1?"k
= 1n?1Q
k=1
(1?"k)
n?1Y
k=1
(?+n"k)
= 1n
n?1Y
k=1
+n
"
k
=?
n?1
n ￠
+n
·n
1
+n
1
=?
n?1
n2
+n
n
1
= 1n2 [(?+n)nn]:
x3x48
′A(?) = 1n2
n(n+1)2
[(?+n)nn]:
x79x10(?)x20,′A(?)x37x6cx38,x18Ax4xbxcx7x8x9x72x23x68x5x6x7x8x9,x6ax6bAx4xbxcx7x8x9x6c
1
n2
n(n+1)2
[(?+n)nn]:
￠ 13 ￠
2,x2Ax6cnx1x3fx22,m(?)x23x6dx4xbxcx7x8x9,g(?)x6cx63x6ax7x8x9,d(?) = (m(?);g(?)).
x30x31,(1) rankd(A) = rankg(A);
(2) g(A)x33x4dx4x43x1x44x45x46x47x23g(?)x14m(?)x72x3b;
(3)x39g(A)x33x4d,x3dg?1(A)x6ax59x23Ax4x7x8x9.
x4ex4f,(1) x6dx25x7x8x9u(?);v(?),x3c
m(?)u(?)+g(?)v(?) = d(?):
x18
m(A)u(A)+g(A)v(A) = d(A):
x10m(A) = 0x33x4ed(A) = g(A)v(A),x3x48
rankd(A) 6 rankg(A):
x79xfd(?) j g(?),x6dx25h(?)x3cd(?)h(?) = g(?),x1d(A)h(A) = g(A),x4cx23
rankg(A) 6 rankd(A):
x2bx2cx4e
rankg(A) = rankd(A):
(2) ())x2?1;￠￠￠ ;?n x6cAx4x55x35x5x6x13,x3dg(A)x4x55x35x5x6x13x6cg(?1);￠￠￠ ;g(?n),x39g(A)x33x4d,
x3dg(A)x4x36x6bx5x6x13g(?i) 6= 0,x10x4cm(?)x4x38x78x23Ax4x5x6x13,xfx61g(?)x14m(?)x37x14x15x38,x6ax6b
(g(?);m(?)) = 1.
(()x39(g(?);m(?)) = 1,x3dx10(1)x3x30,d(?) = 1,xfx61d(A) = E,x18x40x4c(1)x77x4v(?),x26
g(A)v(A) = E;
g(A)x33x4d.
(3)x39g(A)x33x4d,x25(2)x4x43x1x75x4x30x31x77,x1fx4eg(A)v(A) = E,x3x48g(A)?1 = v(A)x6cAx4x7x8
x9.
3,x30x31,x21x22A(x25x2bx28x29x32)x33x40x41x42x4x43x1x44x45x46x47x23x68xbxcx7x8x9x37x6cx38.
x4ex4f,()) Ax33x40x41x42,x6ax6bx61x40x41x1ex72x23Ax4x19x1ax1bx1cx1dx1e,xfx61Ax4x19x1ax1bx6dx55x23x6ax1x4,A
x4x11x12xfx10x55x23x6ax56x4,x6bAx4xbxcx7x8x9x31x6cx11x12xfx10x4x2bxcx14x37x9,x6ax59x23xdx8x6ax56xfx10x4x7cx29,
x6ax6bx37x6cx38.
(()x39Ax4xbxcx7x8x9x37x6cx38,x3dx61xbxcx7x8x9x23xdx8x6ax56xfx10x4x7cx29,x4cx23Ax4x11x12xfx10x78x23
x6ax56x4,x1x19x1ax1bx1cx1dx1ex77x4x19x1ax1bx6dx78x23x6ax1x4,x23x6ax6bx40x41x21x22,xfx31Ax33x40x41x42.
4,x2A =
0
@
1 1?1
1 0 0
0 1?1
1
A,x3A100.
x50,Ax4x5x6x7x8x9x6c?(?2?2),x67
100 =?(?2?2)g(?)+a?2 +b?+c;
x1x24x48? = 0;p2;?p2x7fx2ex32x9,x4
c = 0; 250 = 2a+p2b; 250 = 2a?p2b:
x2fx4eb = 0,a = 249,x3x48
A100 = 249A2 = 249
0
@
2 0 0
1 1?1
1?1 1
1
A:
￠ 14 ￠
5,x30x31,x39x3ax40x63x64k 2Nx78x26Tr(Ak) = 0,x3d′A(?) =?n.
x4ex4f,x10x4cx21x22x4x16x72x23x21x22x4x55x35x5x6x13x3fx1e,x2Ax4x5x6x13x6c?1;￠￠￠ ;?n,x3d
Tr(Ak) =?k1 +￠￠￠+?kn = sk:
x10Tr(Ak) = 0x33x4esk = 0,x6ax17x18x14x9x33x4e?1;￠￠￠ ;?nx4x3x26x11x12x40x67x7x8x9 1 = ￠￠￠ = n = 0,x4cx23
′A(?) = (1)￠￠￠(n) =?n:
6,x2Ax4x5x6x7x8x9′(?) = h(?)g(?),x62(h(?);g(?)) = 1,
x30x31,rankh(A) = degg(?),rankg(A) = degh(?).
x4ex4f,x2Ax4x5x6x13x6c?1;￠￠￠ ;?n,x3dh(A)x4x5x6x13x6ch(?1);￠￠￠ ;h(?n),g(A)x4x5x6x13x6cg(?1);￠￠￠ ;
g(?n),x10x4c′(?) = h(?)g(?)x62(h(?);g(?)) = 1,xfx61fh(?i)gx770x4x6bx28x12x4cdegh(?),fg(?i)gx770x4
x6bx28x12x4cdegg(?),x62degh(?)+degg(?) = n.
x10xbx6912–3.6x20,
degh(?) > n?rank(h(A)) (1)
degg(?) > n?rank(g(A)) (2)
xfx61
n = degh(?)+degg(?) > 2n?(rankh(A)+rankg(A));
rankh(A)+rankg(A) > n:
x79xf
h(A)g(A) = ′(A) = 0;
rankh(A)+rankg(A) 6 n:
x4cx23
rankh(A)+rankg(A) = n:
x6ax6b(1),(2)x9x55x78x4fx12x2b,x3cx4e
rankh(A) = n?degh(?) = degg(?);
rankg(A) = n?degg(?) = degh(?):
￠ 15 ￠
