x0x1x2x3
x0x15x2 x16x17x18x19x11x16x17x18x1ax1b
x1 x2 4–1
1,x18x1fx20flx3ex4ex1fx20x42fi1;fi2;￠￠￠ ;fisx74x26x1bx1c,x71x55x63x4efi1;fi2;￠￠￠ ;fis?1x74x26x1bx1c,x53x54:x1fx20
x42fi1;fi2;￠￠￠ ;fisx42x1fx20x42fi1;fi2;￠￠￠ ;fis?1;flx56x15.
x3x4,x4ex31x18,x31x6ba1;￠￠￠ ;as 2 Kx27x50
fl = a1fi1 +a2fi2 +￠￠￠+asfis:
x8x1fas = 0,x4aflx3ex24x49fi1;fi2;￠￠￠ ;fis?1x74x26x1bx1c,x42x31x18x34x35,x21x4fas 6= 0,x3cx13
fis = 1a
s
fl? a1a
s
fi1?￠￠￠? as?1a
s
fis?1;
x4fisx3ex24x4efi1;fi2;￠￠￠ ;fis?1;flx74x26x1bx1c,x43x25x1fx20x42fi1;fi2;￠￠￠ ;fisx3ex49fi1;fi2;￠￠￠ ;fis?1;flx74x26x1bx1c.
x16x48x40xf,x3dx3ex31x18,x1fx20x42fi1;fi2;￠￠￠ ;fis?1;flx3ex24x49x1fx20x42fi1;fi2;￠￠￠ ;fisx74x26x1bx1c,x21x4fx77x37x66x1fx20
x42x56x15.
2,(x17x18x19x1a) x18x1fx20x42fi1;fi2;￠￠￠ ;fis x74x26x2cx2a,x3fx3ex4ex1fx20x42fl1;fl2;￠￠￠ ;flt x74x26x1bx1c,x53x54:
x31x6bfl1;fl2;￠￠￠ ;flt x15x48x66x59x4afli1;fli2;￠￠￠ ;flit,x27x1fx20x42fi1;fi2;￠￠￠ ;fir;flir+1;flir+2;￠￠￠ ;flit x42x1fx20x42
fl1;fl2;￠￠￠ ;fltx56x15(r = 1;￠￠￠ ;s).
x3x4,x21x22fi1;fi2;￠￠￠ ;fisx74x26x2cx2a,x3fx3ex4ex1fx20x42fl1;fl2;￠￠￠ ;fltx74x26x1bx1c,x21s 6 t.
x1dxfx1ax2x50x44x53x54x1bx4ax1dx5.
(i)x18s = 1.
x21x22fi1x3ex4efl1;￠￠￠ ;fltx74x26x1bx1c,x21x31x6bai 2 Kx27x50fi1 =
tP
i=1
aifli,x25fi1x74x26x2cx2a,x4fi1 6= 0,x23
x24a1;￠￠￠ ;atx55x33x22x6f,x40x47al 6= 0 (1 6 l 6 t),x4a
fll = 1a
l
fi1?
tX
i=1i6=l
ai
al fli;
x21x4fx1fx20x42fi1;fl1;￠￠￠ ;fll?1;fll+1;￠￠￠ ;fltx42x1fx20x42fl1;fl2;￠￠￠ ;fltx56x15.
x49fli1 = fll,fli2 = fl1,:::,flil = fll?1,flil+1 = fll+1,:::,flit = flt,x4x50x22x23.
(ii)x31x1dx22x23x2s?1x2ax2b,xbx3sx66x74x26x2cx2ax15x1fx20fi1;fi2;￠￠￠ ;fis.
x21fi1;fi2;￠￠￠ ;fis?1x74x26x2cx2a,x4ex2x50x31x18,x31x6bfl1;￠￠￠ ;fltx15x48x66x59x4aflj1;￠￠￠ ;fljt,x27
ffi1;￠￠￠ ;fir;fljr+1;￠￠￠ ;fljtg?= ffl1;￠￠￠ ;fltg (r = 1;￠￠￠ ;s?1):
x51fisx3ex4efl1;￠￠￠ ;fltx74x26x1bx1c,x23x24fisx3ex24x4efi1;￠￠￠ ;fis?1;fljs;￠￠￠ ;fljt x74x26x1bx1c,x21x31x6bki;lk 2 K,
i = 1;￠￠￠ ;s?1,k = s;￠￠￠ ;t,x27x50
fis =
s?1X
i=1
kifii +
tX
k=s
lkfljk:
￠ 1 ￠
x4ex3cfi1;￠￠￠ ;fis x74x26x2cx2a,x21ls;￠￠￠ ;lt x55x33x22x6f,x18x3dx48x66x55x22x6fx15x13lh,x4ah > s,x43x25fljh x3ex24
x4efi1;￠￠￠ ;fis;fljh+1;￠￠￠ ;fljt x74x26x1bx1c,x49flis = fljh,fli1 = flj1,:::,flis?1 = fljs?1,flis+1 = fljs+1,:::,
flit = fljt,x4a
ffi1;￠￠￠ ;fis;flis+1;￠￠￠ ;flitg?= ffl1;￠￠￠ ;fltg:
x4ex2x50x44x4bx5x3exbx22x23x2ax2b.
3,x18x1fx20x42fi1;fi2;￠￠￠ ;fisx15x9x22r,fii1;fii2;￠￠￠ ;fiirx13x38x15x48x66x7cx11x42.x53x54:x8x1ffi1;fi2;￠￠￠ ;fis
x3ex4efii1;fii2;￠￠￠ ;fiirx74x26x1bx25,x4afii1;fii2;￠￠￠ ;fiirx13fi1;fi2;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42.
x3x4,x2fx22x1fx20x42x15x7cx11x42,fii1;￠￠￠ ;fiir x62x11x3ex24x49fi1;￠￠￠ ;fisx74x26x1bx1c,x21x4fx77x37x66x1fx20x42x56
x15,x43x25x47x65x43x15x9r,x3cx13x4ex37x611.9x3exbx1fx20x42fii1;￠￠￠ ;fiirx74x26x2cx2a,x4ex5ex231.8x3exbx38x13x1cx3bx74x26
x2cx2ax42.
4,x18fi1;fi2;￠￠￠ ;fitx42fi1;fi2;￠￠￠ ;fit;fit+1;fit+2;￠￠￠ ;fisx47x65x43x15x9,x53x54,fi1;fi2;￠￠￠ ;fitx42fi1;fi2;
￠￠￠ ;fis x56x15.
x3x4,x3dx3ex31x18,x47
L(fi1;￠￠￠ ;fit) L(fi1;￠￠￠ ;fit;fit+1;￠￠￠ ;fis);
x51x21x77x37x66x1fx20x42x47x65x43x15x9,x21x4fx38x15x2ex2ax15x74x26x0x70x71x47x65x43x15x46xe,x43x25x65x56.x12x33x1ax37x611.1,
x3exbx77x37x66x1fx20x42x74x26x56x15.
5,x2x1dx1ex1fx20x42,x76fi1x1dx30x2ax1fx20x42x15x48x66x1cx3bx2cx2ax42:
(1) fi1 = (1;?1;2;4),fi2 = (0;3;1;2),fi3 = (3;0;7;14),fi4 = (1;?1;2;0),fi5 = (2;1;5;6);
(2) fi1 = (1;?1;0;1;1),fi2 = (2;1;3;?1;0),fi3 = (3;0;3;0;1),fi4 = (1;?1;1;?1;1),fi5 =
(?1;?5;?6;5;3),fi6 = (2;1;2;1;0).
x5,(1) fi1;fi2;fi4.
(2) fi1;fi2;fi4.
6,x18x1fx20x42ffi1;fi2;￠￠￠ ;fisg,ffl1;fl2;￠￠￠ ;fltg,ffi1;fi2;￠￠￠ ;fis;fl1;fl2;￠￠￠;fltgx15x9x11x12x13r1,r2,r3.
x53x54:
maxfr1;r2g6 r3 6 r1 +r2:
x3x4,x4ex3cx1fx20x42ffi1;;￠￠￠ ;fisg,ffl1;￠￠￠ ;fltgx6dx3ex4ex1fx20x42ffi1;￠￠￠ ;fis;fl1;￠￠￠ ;fltgx74x26x1bx1c,x21
r1 6 r3; r2 6 r3;
x43x25
maxfr1;r2g6 r3:
x18fii1;￠￠￠ ;fiir1x13fi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,flj1;￠￠￠ ;fljr2x13fl1;￠￠￠ ;fltx15x48x66x1cx3bx74x26x2c
x2ax42,x4a
ffi1;;￠￠￠ ;fis;fl1;￠￠￠ ;fltg?= ffii1;￠￠￠ ;fiir1;flj1;￠￠￠ ;fljr2g;
x23x24
r3 = rankffii1;￠￠￠ ;fiir1;flj1;￠￠￠ ;fljr2g6 r1 +r2:
7,x18x1fx20x42ffi1;fi2;￠￠￠ ;fisg,ffl1;fl2;￠￠￠ ;flsg,ffi1 + fl1;fi2 + fl2;￠￠￠ ;fis + flsgx15x9x11x12x13r1,r2,
r3,x53x54,r3 6 r1 +r2.
x3x4,x21x22ffi1 +fl1;￠￠￠ ;fis +flsgx3ex4effi1;￠￠￠ ;fis;fl1;￠￠￠ ;flsgx74x26x1bx1c,x21x4fx38x15x9
r3 6 rankffi1;￠￠￠ ;fis;fl1;￠￠￠ ;flsg6 rankffi1;￠￠￠ ;fisg+rankffl1;￠￠￠ ;flsg = r1 +r2:
￠ 2 ￠
8,x18x1fx20x42fi1;fi2;￠￠￠ ;fisx15x9x22r,fii1;fii2;￠￠￠ ;fiimx22x38x15x48x66x7cx11x42.x53x54:
rankffii1;fii2;￠￠￠ ;fiimg> r +m?s:
x3x4,x18fii1;￠￠￠ ;fiimx15x9x56x3ct,x4ax38x15x48x66x1cx3bx2cx2ax42fij1;￠￠￠ ;fijtx13fi1;￠￠￠ ;fisx15x74x26x2cx2ax42,
x38x3ex49x1dx30x22fi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,x25x77x48x1dx30x15x1fx20x55x3ex63x13fii1;￠￠￠ ;fiim x15x1fx20,x29
x4ax42x1cx3bx2cx2ax42x34x35,x25fi1;￠￠￠ ;fisx16x28x47s?mx66x55x4x3cfii1;￠￠￠ ;fiimx15x1fx20,x3cx16x7bx25r?tx66x55x43
x15x1fx20x54x17x8fij1;￠￠￠ ;fijtx24x7x2afi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,x43x25
r?t 6 s?m;
x39x3ax50
rankffii1;￠￠￠ ;fiimg = t > r +m?s:
9,xaxbx37x66x1fx20x42x47x65x43x15x9,x3fx3cx16x48x66x3ex24x49x16x48x66x74x26x1bx1c,x53x54:x77x37x66x1fx20x42x56x15.
x3x4,x18x1fx20x42(I)x3ex49x1fx20x42(II)x74x26x1bx1c,x38x15x7x2ax15x74x26x0x70x71x11x12x1fx22L1;L2,x4aL1 L2.
x51x21x38x15x47x65x43x15x9,x21x4fx38x15x7x2ax15x74x26x0x70x71x47x65x43x15x46xe,x43x25L1 = L2,x4(I)x42(II)x56x15.
x1 x2 4–2
1,x73x1dx1ex5dx5ex15x9:
(1)
0
BB
B@
1 4 10 0
3 2 4 2
4 1 1 3
2 3 7 1
1
CC
CA (2)
0
BB
B@
2 1 11 2
1 0 4?1
1?1 1?5
2 0 8?2
1
CC
CA
(3)
0
BB
BB
B@
1 0 0 1 1
0 1 0?1 1
1?1 1 3 1
0 0 1 1 1
1 1 1 2 3
1
CC
CC
CA (4)
0
BB
B@
2 0 3 1?1
1 2 1 2 1
3?2 5 0?3
1 1 0 2 3
1
CC
CA
x5,(1) 2; (2) 2; (3) 4; (4) 3.
2,x73x1dx1ex1fx20x42x15x9x42x1cx3bx74x26x2cx2ax42:
(1) fi1 = (3;2;?1;?3;?2),fi2 = (2;?1;3;1;?3),fi3 = (1;?4;7;5;4),fi4 = (1;?7;11;9;5);
(2) fi1 = (1;?1;1;1;1),fi2 = (1;1;?1;1;1),fi3 = (1;1;1;?1;1),fi4 = (1;1;1;1;?1),fi5 =
(1;1;1;1:1);
(3) fi1 = (2;?1;3;?2;4),fi2 = (4;?2;5;1;7),fi3 = (2;?1;1;8;2),fi4 = (2;?1;2;3;3);
(4) fi1 = (1;3;3;5),fi2 = (3;2;?5;1),fi3 = (2;3;0;4),fi4 = (5;4;?7;1),fi5 = (3;5;1;7).
x5,(1)x94,fi1;fi2;fi3;fi4.
(2)x95,fi1;fi2;fi3;fi4;fi5.
(3)x92,fi1;fi2.
(4)x93,fi1;fi2;fi4.
3 x73x1fx20x42fi1 = (?3;1;1;1);fi2 = (1;?3;1;1);fi3 = (1;1;?3;1),fi4 = (1;1;1;?3)x15x23x47x1cx3bx74
x26x2cx2ax42.
x5,x0x13x66x1fx20x6dx75x2ax1cx3bx74x26x2cx2ax42.
4,x73x1dx1ex1fx20x42x23x2ex2ax15x0x70x71x15x7ax42x46xe:
(1) fi1=(4;?5;2;6),fi2=(2;1;3;2),fi3=(2;?6;?1;4),fi4=(2;13;5;?6);
￠ 3 ￠
(2) fi1 = (1;0;0;1;?1),fi2 = (0;1;0;2;1),fi3 = (0;0;1;?1;?2),fi4 = (1;1;1;2;?2).
x5,(1)x46xe3,x7afi1;fi2;fi4.
(2)x46xe3,x7afi1;fi2;fi3.
5,x73x1dx1ex5dx5ex15x9:
(1)
0
BB
B@
a1b1 a1b2 ￠￠￠ a1bn
a2b1 a2b2 ￠￠￠ a2bn
:::::::::::::::::::::::
anb1 anb2 ￠￠￠ anbn
1
CC
CA; (2)
0
BB
B@
1 a a ￠￠￠ a a
a 1 a ￠￠￠ a a
::::::::::::::::::::
a a a ￠￠￠ a 1
1
CC
CA.
x5,(1)x21x22x4fx5dx5ex15x0x1x37xdx6dx74x26x65x2a,x21x4fx96 1,x25x4fx5dx5ex15x9x56x3c0x15x30x11x40x26x31x32x13x23
x47x15aibj = 0,x8(a1;￠￠￠ ;an) 6= 0,x4ax40x47(b1;￠￠￠ ;bn) = 0,x8(b1;￠￠￠ ;bn) 6= 0,x4ax40x47(a1;￠￠￠ ;an) = 0.
x21x4fx62(a1;￠￠￠ ;an) = 0x44(b1;￠￠￠ ;bn) = 0x52,x9x220,x29x4a,x9x221.
(2)x62a = 1x52,x9x221;x62a = 11?n x52,x9x22n?1(n > 1);x3cx24xfx36,x9x22n.
6,x18
W = f(a1;￠￠￠ ;ar;0;￠￠￠ ;0)T j ai 2 K; i = 1;￠￠￠ ;rg Km
x53x54,dimW = r.
x3x4,x18
fi1 =
0
BB
B@
1
0.
..
0
1
CC
CA;fi2 =
0
BB
B@
0
1.
..
0
1
CC
CA;￠￠￠ ;fir =
0
BB
BB
BB
@
0.
..
1 (r),
...
0
1
CC
CC
CC
A
x4afi1;fi2;￠￠￠ ;firx74x26x2cx2a,x3fx2x0x1x15
fi =
0
BB
BB
BB
BB
@
a1
...
ar
0.
..
0
1
CC
CC
CC
CC
A
2 W
x47fi = a1fi1 +￠￠￠+arfir,x23x24dimW = r.
7,x18fi1;fi2;￠￠￠ ;firx74x26x2cx2a,flj =
rP
i=1
aijfii (j = 1;￠￠￠ ;s),x49A = (aij),x53x54:
rankffl1;fl2;￠￠￠ ;flsg = rankA:
x3x4,(i)x18flj1;￠￠￠ ;fljtx13fl1;￠￠￠ ;flsx15x48x66x1cx3bx74x26x2cx2ax42.xbx3Ax15x1ex1fx20x42 1;￠￠￠ ; s,x4a
(flj1 ￠￠￠ fljt) = (fi1 ￠￠￠ fir)( j1 ￠￠￠ jt):
x8x1f
tP
i=1
ki ji = 0,x4a
(flj1 ￠￠￠ fljt)
0
B@
k1
...
kt
1
CA = (fi
1 ￠￠￠ fir)( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = 0;
x4
tP
i=1
kiflji = 0,x4ex3cflj1;￠￠￠ ;fljtx74x26x2cx2a,x21x4fk1 = ￠￠￠ = kt = 0,x4 j1;￠￠￠ ; jtx74x26x2cx2a,x23x24
rank(A) > t = rankffl1;￠￠￠ ;flsg:
￠ 4 ￠
(ii)x18 j1;￠￠￠ ; jtx13Ax15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42,x4ax4e
tP
i=1
kiflji = 0x3ex50
(fi1 ￠￠￠ fir)( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = (fl
j1 ￠￠￠ fljt)
0
B@
k1
...
kt
1
CA = 0;
x4ex3cfi1;￠￠￠ ;firx74x26x2cx2a,x40x41x47
( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = 0;
x4e j1;￠￠￠ ; jtx15x74x26x2cx2ax26x3ex50k1 = ￠￠￠ = kt = 0,x4flj1;￠￠￠ ;fljtx74x26x2cx2a,x21x25
rankffl1;￠￠￠ ;flsg> t = rank(A):
x6x2x50x8
rankffl1;￠￠￠ ;flsg = rank(A):
8,x18A 2 Mm;n(K),xaxbAx15x3di1;i2;￠￠￠ ;ir xdx42x2aAx15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42,Ax15x3d
j1;j2;￠￠￠ ;jrx1ex42x2aAx15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42.x53x54:fl
flfl
flfl
flfl
flfl
ai1;j1 ai1;j2 ￠￠￠ ai1;jr
ai2;j1 ai2;j2 ￠￠￠ ai2;jr
::::::::::::::::::::::::
air;j1 air;j2 ￠￠￠ air;jr
flfl
flfl
flfl
flfl
fl
6= 0:
x3x4,x3bx62x3x4ax5dx5ex15xdx42x1e,x3ex18x5dx5ex15x4frxdx42x4frx1ex11x12x22x5dx5ex15xdx1fx20x42x42x1ex1fx20x42x15
x1cx3bx74x26x2cx2ax42.x43x25x5dx5ex3ex6ax5cx56xdx3dx4ax4cx22
B =
0
BB
BB
BB
B@
a1;1 a1;2 ￠￠￠ a1;n
::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;n
0 0 ￠￠￠ 0
::::::::::::::::::::
0 0 ￠￠￠ 0
1
CC
CC
CC
CA;
x21x5dx5ex15x5cx56xdx3dx4ax55x3cx3dx5dx5ex15x1ex1fx20x15x74x26x2ax6a,x21x5dx5eBx15x4frx1exex22Bx15x1ex1fx20x42x15x1cx3bx74
x26x2cx2ax42.x43x25Bx3ex6ax5cx56x1ex3dx4ax4cx22
C =
0
BB
BB
BB
B@
a1;1 a1;2 ￠￠￠ a1;r 0 ￠￠￠ 0
::::::::::::::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;r 0 ￠￠￠ 0
0 0 ￠￠￠ 0 0 ￠￠￠ 0
::::::::::::::::::::::::::::::::
0 0 ￠￠￠ 0 0 ￠￠￠ 0
1
CC
CC
CC
CA
:
x21x22rank(C) = rank(B) = r,x23x24
flfl
flfl
flfl
a1;1 a1;2 ￠￠￠ a1;r
::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;r
flfl
flfl
flfl6= 0:
x1 x2 4–3
1,x55x23x1dx1exax5ex20x74x26x40x41x42x15x2dx15xfx21,x57x73x2d.
￠ 5 ￠
(1)
8>
<
>:
ax1 +bx2 +x3 = 1
x1 +abx2 +x3 = b
x1 +bx2 +ax3 = 1;
(2)
8>
<
>:
(?+3)x1 +x2 +2x3 =?
x1 +(1)x2 +x3 = 2?
3(?+1)x1 +?x2 +(?+3)x3 = 5;
(3)
8>
<
>:
ax1 +bx2 +2x3 = 1
ax1 +(2b?1)x2 +3x3 = 1
ax1 +bx2 +(b+3)x3 = 2b?1:
x5,(1)x62b(a?1)(a+2) 6= 0x52x47x2d,x1= a?b(a?1)(a+2),x2= ab+b?2b(a?1)(a+2),x3 = a?b(a?1)(a+2) ;
x62a = b =?2x52,x47x2dx1 = x3 =?1?2x2;
x62a = b = 1x52,x47x2dx1 = 1?x2?x3;
x3cx24xfx36x2cx2d;
(2)x62? 6= 0,? 6= 1x52x47x2d,x1 =?2 +415?2,x2 =?2 +?+15?2,x3 =?4?2 +?+15?2 ;
x62? = 1x52x47x2d,x1 = 2?x3,x2 =?7+2x3;
x62? = 0x52x2cx2d;
(3)x62a 6= 0,b 6= §1x52x47x2d,x1 = 5?ba(b+1),x2 =?2b+1,x3 = 2(b?1)b+1 ;
x62b = 1x52x47x2d,x2 = 1?ax1,x3 = 0;
x62a = 0,b = 5x52x47x2d,x2 =? 13,x3 = 43,x1x22x0x1xe;
x3cx24xfx36x2cx2d.
2,x33x1ax74x26x40x41x42x15x5x23x53x54:x8x1fx2ex74
L1,
( A
1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x42x2ex74
L2,
( A
3x+B3y +C3z +D3 = 0
A4x+B4y +C4z +D4 = 0
x65x3,x20x30 fl
flfl
flfl
flfl
flfl
A1 A2 A3 A4
B1 B2 B3 B4
C1 C2 C3 C4
D1 D2 D3 D4
flfl
flfl
flfl
flfl
fl
= 0:
x5,x3dx3ex6a3.3x15x2d,x8x1fL1x42L2x65x3,x20x30x74x26x40x41x428
>>>>
<
>>>>
:
A1x+B1y +C1z =?D1
A2x+B2y +C2z =?D2
A3x+B3y +C3z =?D3
A4x+B4y +C4z =?D4
x47x2cx48x2d,x43x25rank(A) = rank(?A) = 3,x77x1eAx42?Ax11x12x13x79x53x40x41x42x15x6axex5dx5ex42x16x1fx5dx5e,x21x4fxd
x1ex29j?Aj = 0,fl
flfl
flfl
flfl
flfl
A1 A2 A3 A4
B1 B2 B3 B4
C1 C2 C3 C4
D1 D2 D3 D4
flfl
flfl
flfl
flfl
fl
=?
flfl
flfl
flfl
flfl
fl
A1 B1 C1?D1
A2 B2 C2?D2
A3 B3 C3?D3
A4 B4 C4?D4
flfl
flfl
flfl
flfl
fl
= 0:
3,x73x34x66xcxfAix+Biy +Ciz +Di = 0(i = 1;2;3)x11x12x2dx2ex1dx1ex2ax6ax15x30x26x31x32.
(1)x47x48x66x66x28x17; (2)x47x48x31x66x28x2ex74;
￠ 6 ￠
(3)x34x66xcxfxcxd; (4)x34x66xcxfx75x2ax34x14x20.
x5,xbx3x6ex3x48x74x26x40x41x42 8
><
>:
A1x+B1y +C1z =?D1
A2x+B2y +C2z =?D2
A3x+B3y +C3z =?D3
(?)
x38x15x6axex5dx5ex42x16x1fx5dx5ex11x12x1fx22Ax42?A.
(1)x34x66xcxfx47x48x66x66x28x17 () x40x41x42(?)x47x2cx48x2d () rank(A) = rank(?A) = 3 () jAj6= 0.
(2)x34x66xcxfx47x48x31x66x28x2ex74 () x40x41x42(?)x47x2d,x25x3f(?)x15xcx25x40x41x42x15x7ax8x2dx6ax7bxax48x66
x1fx20 () rank(A) = rank(?A) = 2.
(3)x34x66xcxfxcxd () AiA
j
= BiB
j
= CiC
j
6= DiD
j
1 6 i < j 6 3.
(4) x34x66xcxfx75x2ax34x14x20 () x40x41x42 (?) x2cx2d,x25 (?) x15xcx25x40x41x42x15x7ax8x2dx6axax48x66x1fx20
() rank(A) = 2,rank(?A) = 3,x25x3fAx16x0x1x37xdx6dx55x2ax65x6a.
x1 x2 4–4
1,x7cx12x1dx1ex47x48x46x5dx22x74x26x46x5d?
(1)x6bx1fx20x70x71V x16,x41(?) = fi,x3cx16fix22x49x1dx1fx20;
(2) x41, K2?! K3
(x;y) 7?! (?1;2;3)
(3) x41, K3?! K3
(x1;x2;x3) 7?! (2x1+x2?x3;?x2+x3;x1+2x2?x3)
(4) x41, K3?! K2
(x;y;z) 7?! (x2 +y2?z;xy)
(5) x41(x"1 +y"2 +z"3) = (x+y)"1 +(x?y +z)"2 +(y?z)"3,x3cx16"1;"2;"3x22x74x26x70x71V x15x7a;
(6)x27x8x70x71R2x16,x52x22xcxfx5bxcx52x21x40x1fx22x4bx17x1bx1c45–x15x3dx4a.
x5,(1)x8fi = 0,x13;x8fi 6= 0,x55x13.
(2)x55x13.
(3)x13.
(4)x55x13.
(5)x13.
(6)x13.
2,x2x3cx79x61x16x15x74x26x46x5d,x73x25x38x15x6bx65x2cx7ax1dx15x5dx5e(x8x7ax2dx54x7a,x4ax7ax67x11x7a).
x5,(1) fi = 0x52x22x6fx5dx5e.
(3)
0
@
2 1?1
0?1 1
1 2?1
1
A.
(5)
0
@
1 1 0
1?1 1
0 1?1
1
A.
(6)
cos45–?sin45–
sin45– cos45–
.
￠ 7 ￠
3,x18x41x22x1fx20x70x71V1x8x1fx20x70x71V2x15x74x26x46x5d,fi1;fi2;￠￠￠ ;fim 2 V1,x41(fii) = fli,i = 1;2;￠￠￠ ;m.
x53x54:x8x1ffl1;fl2;￠￠￠ ;flmx74x26x2cx2a,x4afi1;fi2;￠￠￠ ;fimx67x74x26x2cx2a.
x3x4,x18k1fi1 +k2fi2 +￠￠￠+kmfim = 0,x4a
x41(k1fi1 +k2fi2 +￠￠￠+kmfim) = 0
)k1x41(fi1)+k2x41(fi2)+￠￠￠+kmx41(fim) = 0
)k1fl1 +k2fl2 +￠￠￠+kmflm = 0;
x4ex3cfl1;fl2;￠￠￠ ;flmx74x26x2cx2a,x3ex50k1 = k2 = ￠￠￠ = km = 0,x43x25fi1;fi2;￠￠￠ ;fimx74x26x2cx2a.
4,x1dxfx9x16x15(1)–(7)x6dx13x9(0)x6ax4ex2bx6axex5dx5ex15x74x26x3dx4ax25x50x8x15,x9(0)x16x55x25x72x4bx17Ox68
x7ax1fx20·1;·2,x19x72x4ex64x1dx7ax1fx20x6bx9(1)–(7)x16x15x23x24x68x38x15x2ax3c·1;·2x15x57x55(x4bx22x2bxe)x24x7ex25x65x2c
x74x26x3dx4ax15x5dx5e.
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
O ·1
·2
(0)
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(1)
x8x20x50x72x18x18x3c
x16x3
x0x10x30x70 x8x20x50x72x18x18x3c
x16x3
x0x10x30x70
x8x20x50x72x18x18x3c
x16x3
x0x10x30x70 x8x20x50x72x18x18x3c
x16x3
x0x10x30x70x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(2)
x0x10x30x70x28x28x44x1ax5x8x20x50x72 x0x10x30x70x28x28x44x1ax5x8x20x50x72
x0x10x30x70x28x28x44x1ax5x8x20x50x72 x0x10x30x70x28x28x44x1ax5x8x20x50x72x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(3)
x0x10x30x70xbx26x5cx58x58x72x50x20x8 x0x10x30x70xbx26x5cx58x58x72x50x20x8
x0x10x30x70xbx26x5cx58x58x72x50x20x8 x0x10x30x70xbx26x5cx58x58x72x50x20x8x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(4)
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
x5x71x71
x71x71
x71x20
x18x4
x71x71
x71x71
x71x20
x18x4
x71x71
x71x71
x71x20
x18x4x73x73
x73x73
x73x60
x28xc
x73x73
x73x73
x73x60
x28xc
x73x73
x73x73
x73x60
x28xc
(5)
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x5
xcx28x60x73
x73x73
x73x73
xcx28x60x73
x73x73
x73x73
xcx28x60x73
x73x73
x73x73
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
(6)
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x5
x73x73
x73x73
x73x60
x28xcx73
x73x73
x73x73
x60x28xcx73
x73x73
x73x73
x60x28xc
x10x30x70
x74
x74
x74
x74
x74
x10x30x70
x74
x74
x74
x74
x74
x10x30x70
x74
x74
x74
x74
x74
(7)
x04x1x2
x5,(1)
1 0
0 1
.
(2)
0?1
1 0
.
(3)
1 0
0?1
.
(4)
1 0
0?1
.
(5)
1?1
1 1
.
(6)
1 1
2?1
.
(7)
0 1
2?1
.
5,x47x48x66x38x3bx221x15x2bx40x10x15x73x66x1bxfx6dx24x79x72x65x43x15x25x26x27x15xcxfx9,x1fx28x18x78x29x2ax1dx2bx1a
x3dx34x2cx8x23x53x15x2dx62x2ex16x17x58x25x38x15x2bx10x9(x8x5ax9),x2fx30x31x7bx26x2x72xfx15x9x36x2fx37x66x74x26x3dx4ax6fx63
￠ 8 ￠
x50x8x7exfx3ax44xfx15x37x66x9x36(x22x2fx30?),x8x1fx4ex73x66x44xfx15x1dx1dx35x7ax22x4bx17,x1ax7ex25x7exfx3ax1ex44xfx15x9
x36x2x2cx15x3dx4ax5dx5e.
x74
x74
x74
x74
x74
x74
x70x10x0
x74
x74
x74
x74
x74
x74
x70x10x0
x75x75x75x75x75x75x50x20x8
x75x75x75x75x75x75x50x20x8
x1fx1fx1e
x1fx1fx1f
x1ex1fx1f
x4x74
x74
x74
x74
x74
x74
x70x8x20x50x75x75x75x75x75x75
x4x18x40x71
x71x71
x71
x71x71
x71x71
x40x18x4
x05x1x2
x5,x7exf:
1 p2
4
0
p2
4
!
,x1ex44xf:
p2
4 0p
2
4 1
!
.
x1 x2 4–5
1,x6bx27x8x70x71x16x7ax2ex35x55x56[O;?!i ;?!j ;?!k ],x41;x42;x43x11x12x1bx1cx70x71x5bx1ex32x6ax22xx0 yx0 zx45x1bx1c45–
x15x3dx4a.
(1)x24x57x55x15x36x29x7ex25x41;x42;x43x15x1bx50x29;
(2)x73x41;x42;x43x6bx7a?!i ;?!j ;?!k x1dx15x5dx5e;
(3)x73x41x42,x42x41,x41x42x43,x41 + x42,x414x424x6bx7a?!i ;?!j ;?!k x1dx15x5dx5e;
(4)x53x54,x418 = x428 = x438 = x45,x77x1ex45x1bx1cx36x43x46x5d.
x5,(1) x41(x;y;z) =
x;
p2
2 y?
p2
2 z;
p2
2 y +
p2
2 z
,
x42(x;y;z) =
p
2
2 x+
p2
2 z;y;?
p2
2 x+
p2
2 z
,
x43(x;y;z) =
p
2
2 x?
p2
2 y;
p2
2 x+
p2
2 y;z
.
(2) A =
0
B@
1 0 0
0
p2
2?
p2
2
0
p2
2
p2
2
1
CA,
B =
0
B@
p2
2 0
p2
20 1 0
p2
2 0
p2
2
1
CA,C =
0
BB
@
p2
2?
p2
2 0p
2
2
p2
2 00 0 1
1
CC
A.
(3) AB =
0
BB
B@
p2
2 0
p2
2
1
2
p2
2?
1
2
12
p2
2
1
2
1
CC
CA,BA =
0
BB
B@
p2
2
1
2
1
2
0
p2
2?
p2
2
p2
2
1
2
1
2
1
CC
CA,ABC =
0
BB
B@
1
2?
1
2
p2
2
1
2 +
p2
4
1
2?
p2
4?
1
2
1
2?
p2
4
1
2 +
p2
4
1
2
1
CC
CA,A+B =
0
BB
B@
1+
p2
2 0
p2
2
0 1+
p2
2?
p2
2
p2
2
p2
2
p2
1
CC
CA,
￠ 9 ￠
A4B4 =
0
@
1 0 0
0?1 0
0 0 1
1
A.
(4)x69.
2,x78x67x1dx1ex5dx5ex15x33x67:
(1) A =
0
@
1 1 3
2 1 2
2 3 1
1
A,B =
0
@
1?1 1
0 2?1
1 2 0
1
A;
(2) A =
0
@
a b c
b c a
c a b
1
A,B =
0
@
c b a
a c b
b a c
1
A;
x73AB,AB?BA,(A?B)2.
x5,(1) AB =
0
@
4 7 0
4 4 1
3 6?1
1
A,AB?BA =
0
@
3 4?2
2 5?2
2 3?8
1
A,(A+B)2 =
0
@
6 0 8
1 8 4
3 2 6
1
A.
(2) AB =
0
@
ac+ba+cb ac+ba+cb a2 +b2 +c2
ac+ba+cb a2 +b2 +c2 ac+ba+cb
a2 +b2 +c2 ac+ba+cb ac+ba+cb
1
A,
AB?BA =
0
@
(b?c)(a?b)?(a?c)(a?b) (a?b)2
(a?c)(a?b) (a?b)2 (b?c)(a?b)
(a?b)2 (b?c)(a?b)?(a?c)(a?b)
1
A,
(A+B)2 =
0
@
(a?c)(a+b?2c) 0?(a?c)(a+b?2c)
(a?b)(a+b?2c) 0 (a?b)(a+b?2c)
(b?c)(a+b?2c) 0 (b?c)(a+b?2c)
1
A.
3,x78x67:
(1)
0
@
2 2 1
2 1 2
1 2 2
1
A
2; (2)
0
@
1?1 1
0 1?1
1 0 1
1
A
3;
(3)
0 1
1 1
5; (4)
cos?sin
sin cos
n;
(5) (a b c)
0
@
a
b
c
1
A; (6)
0
@
a
b
c
1
A(a b c);
(7)
0
@
1 0
0? 1
0 0?
1
A
n; (8) (?En +A)n,A =
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1
:::::::::::::
1 1 ￠￠￠ 1
1
CC
CA.
x5,(1)
0
@
9 8 8
8 9 8
8 8 9
1
A.
(2)
0
@
3?2 5
3 0?2
2 3?3
1
A.
(3)
3 5
5 8
.
(4)
cosn?sinn
sinn cosn
.
(5) (a2 +b2 +c2).
￠ 10 ￠
(6)
0
@
a2 ab ac
ab b2 bc
ac bc c2
1
A.
(7)
0
B@?
n n?n?1 n(n?1)
2?
n?2
0?n n?n?1
0 0?n
1
CA.
(8)?n
E? 1n A
·
+ 1n (?+n)nA.
4,x78x67x5dx5ex9x3ax29,x18
(1) f(?) =?3?3?2?2,A =
0
@
3 1 1
3 1 2
1?1 0
1
A;
(2) f(?) =?3?2?2 +1,A =
0
@
1 2 0
1?1 1
0 1 2
1
A.
x5,(1)
0
@
12 2 4
11 3 3
1 1 1
1
A.
(2)
0
@
1 2 0
1?1 1
0 1 2
1
A.
5,x8x1fAB = BA,x75x5dx5eAx42Bx3ex3x4a.x18
(1) A =
1 0
1 2
; (2) A =
0
@
1 1 0
1 1 0
1 1 1
1
A;
(3) A =
0
@
1 1 1
0 1 1
0 0 1
1
A.
x73x23x47x42Ax3ex3x4ax15x5dx5e.
x5,(1)
a 0
b a+b
.
(2)
0
@
a b 0
b a 0
c c a+b?c
1
A.
(3)
0
@
a b c
0 a b
0 0 a
1
A.
6,x18
A = diag(a1;a2;￠￠￠ ;an); x3cx16ai 6= aj,8i 6= j.
x53x54:x42Ax3ex3x4ax15x5dx5ex7bx63x13x2x35x5dx5e.
x3x4,x18B = (bij)x42Ax3ex3x4a,x4a
aibij = bijaj; i;j = 1;￠￠￠ ;n:
x3cx13
(ai?aj)bij = 0; i;j = 1;￠￠￠ ;n:
￠ 11 ￠
x71x62i 6= jx52x47ai 6= aj,x23x24x2x3ci 6= jx47bij = 0,x4
B =
0
BB
B@
b11 0 ￠￠￠ 0
0 b22 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ bnn
1
CC
CA:
7,x53x54:x42x23x47x5dx5ex3ex3x4ax15x5dx5ex7bx63x13x55x20x5dx5e.
x3x4,x10x11x55x20x5dx5ex42x23x47x5dx5ex3ex3x4a.x18Bx42x23x47x5dx5ex3ex3x4a,x4ax4ex79x61xbB = diag(b1;￠￠￠ ;bn).
x51x2x0x1x15i 6= jx47BEij = EijB,x21x4fx2i 6= jx47bi = bj,x4b1 = b2 = ￠￠￠ = bn = b,x23x24B = bEn.
8,x53x54:x55x31x6bx5dx5eA,B,x27AB?BA = En.
x3x4,AB?BAx15x2x35x74xbx58x39x3a=
nP
i=1
nP
k=1
aikbki
nP
i=1
nP
k=1
bikaki
= 0,x25Enx15x2x35x74xb
x58x39x3a= n,x3ex5fAB?BA 6= En.
9,x18A = B +E,x53x54,A2 = 2Ax62x3fx63x62B2 = E.
x3x4,()) B2 = (A?E)2 = A2?2A+E = E.
(() A2 = (B +E)2 = B2 +2B +E = 2(B +E) = 2A.
10,xaxbxex9Kx79x15x37x66x40x5eAx42Bx3ex3x4a.x53x54:
(1) (A+B)2 = A2 +2AB +B2;
(2) (A+B)(A?B) = A2?B2;
(3) (A+B)n =
nP
k=0
CknAkBn?k.
x3x4,x69.
11,x53x54:x79(x1d)x34x35x36x5dx5ex15xfx10x5fx13x79(x1d)x34x35x36x5dx5e.
x3x4,x18A = (aij)x42B = (bij)x6dx13x79x34x35x36x5dx5e,x4x2i > jx47aij = 0x24x68bij = 0,x3cx13x62i > j
x52x47
nX
k=1
aikbkj =
i?1X
k=1
aikbkj +
nX
k=i
aikbkj
=
i?1X
k=1
0￠bkj +
nX
k=i
aik ￠0 = 0;
x21x4fABx13x79x34x35x36x5dx5e,x2x3cx1dx34x35x36x5dx5ex67x3ex24x6bx6cx6dx53x54.
12,x18Ax22m￡nx5dx5e,x53x54,rankA = 1x15x30x11x40x26x31x32x13x31x6bmx46x6ex6fx1fx20fi = (a1;a2;￠￠￠ ;am)
x42nx46x6ex6fx1fx20fl = (b1;b2;￠￠￠ ;bn),x27A = fiTfl.
x3x4,()) x18rank(A) = 1,x4aAx40x47x48xd(x18x22fl = (b1;￠￠￠ ;bn))x55x56x3c0,x25x3cx24x28xdx6dx13x77x48
xdx15x18xe,x43x25
A =
0
BB
B@
a1b1 a1b2 ￠￠￠ a1bn
a2b1 a2b2 ￠￠￠ a2bn
...,..,..,..
amb1 amb2 ￠￠￠ ambn
1
CC
CA =
0
BB
B@
a1
a2
...
am
1
CC
CA(b1 b2 ￠￠￠ bn) = fiTfl:
(()x18fi;flx13x37x66x6ex6fx1fx20,x4ax40x47x6ex66ai 6= 0,bj 6= 0,x43x25aibj 6= 0,x27x50A = fiTfl = (aibj) 6= 0.
x3cx13
1 6 rank(A) 6 minfrank(fi);rank(fl)g = 1:
13,x73x25xcx40x22x6fx15x23x47x62x79x40x5e.
￠ 12 ￠
x5,x18A =
a
11 a12
a21 a22
,A2 = 0,x8x1fb12 = 0,x4a
A2 =
a2
11 (a11 +a22)a12
0 a222
= 0;
x3cx13a11 = a22 = 0,A =
0 a
0 0
.
x12x18a12 = b 6= 0,a11 = a,x4ex3c0 < rank(A) = 1 < 2,x4ax47A =
a ka
b kb
,x3cx13
A2 =
a(a+kb) ka(a+kb)
b(a+kb) kb(a+kb)
= 0:
x4ex3cb 6= 0,x3ex50a+kb = 0,k =?ab, x21x4fx5dx5eAx15x3ex63x36x29x13
0 a
0 0
x44
a?a2
bb?a
:
14,x18Ax22m￡nx5dx5e,x53x54:x31x6bn￡sx6ex6fx5dx5eB,x27AB = 0x15x30x11x40x26x31x32x13rankA < n.
x3x4,())x18x47x6ex6fx5dx5eBx27x50AB = 0,x4aBx15x1ex1fx20x6dx13x3x48x74x26x40x41x42AX = 0x15x2d,x25x3f
x3cx16x47x6ex6fx2d,x21x4frankA < n.
(()x18rankA < n,x4ax3x48x74x26x40x41x42AX = 0x47x6ex6fx2d0
B@
b1
...
bn
1
CA6= 0:
x49
B =
0
B@
b1 0 ￠￠￠ 0
...,..,..,..
bn 0 ￠￠￠ 0
1
CA
n￡s
6= 0;
x4aAB = 0.
15,x18A,Bx11x12x22m￡nx42n￡sx5dx5e,x53x54:x8x1fAB = 0,x4a
rankA+rankB 6 n:
x3x4,Bx15x1ex1fx20x6dx13x3x48x74x26x40x41x42AX = 0x15x2d,x25x77x66x3x48x74x26x40x41x42x15x2dx70x71x6x9xax47
n?rankAx66x74x26x2cx2ax15x1fx20,x43x25
rankB 6 n?rankA;
x39x3ax50rankA+rankB 6 n.
16,x18Ax22n￡rx5dx5e,Bx22r￡sx5dx5e,rankB = r,x53x54:
(1)x8x1fAB = 0,x4aA = 0;
(2)x8x1fAB = B,x4aA = E.
x3x4,(1)x4ex79x61,rankA+rankB 6 r,x4erankB = rx3ex50rankA = 0,x43x25A = 0.
(2)x21x22(A?E)B = 0,x4e(1)x50A?E = 0,A = E.
17,x18Ax22m￡nx5dx5e,x53x54:x8x1fx2x23x47x15nx46x1fx20X = (x1 x2 ￠￠￠ xn)Tx6dx47AX = 0,x4aA = 0.
x3x4,x4ex31x18xbx14x2fx5dx5ex15x1ex1fx20x67x13AX = 0x15x2d,x21x4fA = AE = 0.
x1 x2 4–6
1,x78x67x1dx1ex5dx5ex15xcx5dx5e:
￠ 13 ￠
(1)
1 2
4 3
; (2)
2 5
3 7
;
(3)
0
@
1 1 0
0 1 1
0 0 1
1
A; (4)
0
@
2 0 1
1 1 0
3 2 1
1
A;
(5)
0
@
2 1 0
1 2 0
0 0 1
1
A; (6)
0
@
4 2 1
1 0 3
2 1 2
1
A.
x5,(1)? 15
3?2
4 1
.
(2)
7 5
3?2
.
(3)
0
@
1?1 1
0 1?1
0 0 1
1
A.
(4) 13
0
@
1?2 1
1 1 1
5 4?2
1
A.
(5) 13
0
@
2?1 0
1 2 0
0 0 3
1
A.
(6) 13
0
@
3 3?6
4?6 11
1 0 2
1
A.
2,x73x1dx1ex5dx5ex15x34x35x5dx5e:
(1)
0
@
2 3 1
1 2 3
3 2 1
1
A; (2)
0
@
2 1?2
2 2 1
1 2 1
1
A;
(3)
0
@
2 0 0
1 2 0
1 2?3
1
A; (4)
0
@
2 3 4
0?1 2
3?2 1
1
A.
x5,(1)
0
@
4?1 7
8?1?5
4 5 1
1
A.
(2)
0
@
0?5 5
3 4 2
6?3 6
1
A.
(3)
0
@
6 0 0
3?6 0
4?4 4
1
A.
(4)
0
@
3?11 10
6?14 4
3 5 2
1
A.
3,x18A 2 Mn(K),x53x54:x8x1fx31x6bx36xex3ax6ex6fx15x9x3ax29f(x),x27f(A) = 0,x4aAx3exc.
x3x4,x18f(x) = a0xm +a1xm?1 +￠￠￠+am?1x+am,am 6= 0,x27
f(A) = a0Am +a1Am?1 +￠￠￠+am?1A+amE = 0;
￠ 14 ￠
x4a
A
a0a
m
Am?1? a1a
m
Am?2?￠￠￠? am?1a
m
E
= E;
x21x4fAx3exc.
4,x18B3 = 0,x53x54,E?Bx3exc,x57x73E?Bx15xc.
x3x4,x21x22(E?B)(E +B +B2) = E?B3 = E,x23x24E?Bx3exc,x3f(E?B)?1 = E +B +B2.
5,x18A3 = 2E,B = A2 +2A?E,x73B?1.
x3x4,x4ex5dx5ex40x41x42 8
><
>:
B = A2 +2A?E
AB = 2A2?A+2E
A2B =?A2 +2A+4E
x72x4ex17x5x6x7x44x27x56x29x1ex38x7bxaE,x3ex50(5A2 +4A?3E)B = 31E,x21x4f
B?1 = 131 (5A2 +4A?3E):
6,x18A2 = A,x53x54,E +Ax3exc,x57x73(E +A)?1.
x3x4,x18B = E +A,x4aA = B?E,x21x4f(B?E)2 = B?E,B2?3B =?2E,B(3E?B) = 2E.
x21x4fB?1 = 12 (3E?B) = 12 (3E?E +A) = E? 12 A.
7,x18A;B 2 Mn(K),x53x54:x8x1fAB = kEn (k 6= 0),x4aBA = kEn.
x3x4,x4eAB = kEn (k 6= 0)x3ex50A?1 = 1k B,B = kA?1,x21x4fBA = kA?1A = kE.
8,x53x54,(1)x79(x1d)x34x35x36x5dx5ex15x34x35x5dx5ex5fx13x79(x1d)x34x35x36x5dx5e;
(2)x3excx15x79(x1d)x34x35x36x5dx5ex15xcx5dx5ex5fx13x79(x1d)x34x35x36x5dx5e.
x3x4,(1)x18
A =
0
BB
B@
a11 a12 ￠￠￠ a1n
0 a22 ￠￠￠ a2n
...,..,..,..
0 0 ￠￠￠ ann
1
CC
CA:
x4ax62j > ix52,aij x15x24x0x29Mij x5fx13x79x34x35x36x15,x3fMij x15(i;i)xbx58= Ax15(i + 1;i)xbx58= 0,x23x24
Mij = 0(j > i),x43x25x62j > ix52x47Aij = 0,x21x4fx34x35x5dx5eA?x13x79x34x35x36x5dx5e,x6bx6cx3ex53x1dx34x35x36x15xf
x36.
(2)x8Ax3exc,x4aA?1 = 1jAj A?xex22x34x35x36x5dx5e.
9,x53x54:x2x0x8nx79x40x5eA,x40x31x6b?0 2 K,x27x50?0En?Ax13x3excx5e.
x3x4,x21x22j?E?Ajx13x51x3ax6axex221x15nx48x9x3ax29,x25nx48x9x3ax29x6bKx79x6x9x47nx66x3d,x21x40x31x6b
0 2 Kx27x50j?0E?Aj6= 0,x43x25?0E?Ax3exc.
10,x18A =
1 0
2 1
,x73x9x3ax29f(x),x27f(A) = A?.
x5,A? =
1 0
2 1
=
1 0
2 1
4
0 0
1 0
= A?4
h1
2 (A?E)
i
= 2E?A,x23x24f(x) =?x+2.
11,x53x54:x2x23x47x15A 2 M2(K),x31x6bf(?) = a?+b,x27f(A) = A?.
x3x4,x18A =
a b
c d
,x4aA? =
d?b
c a
,x23x24A + A? = (a + d)E,A? =?A + (a + d)E,x21
f(?) =+(a+d).
￠ 15 ￠
12,x18A 2 Mn(K),x53x54:
rankA? =
8>
<
>:
n; rankA = n
1; rankA = n?1
0; rankA < n?1
x3x4,(i)x62rankA = nx52,Ax3exc,jAj6= 0,x25AA? = jAjE,x21A?x67x3exc,x43x25rankA? = n.
(ii)x62rankA < n?1x52,Ax15n?1x79x0x29x6dx56x3c0,x21x4fAij = 0,A? = 0,x23x24rankA? = 0.
(iii)x62rankA = n?1x52,Ax45x46x47x48x66n?1x79x0x29x55x56x3c0,x23x24A? 6= 0,x58x54rankA? > 1,x16
x48x40xfx47AA? = jAjE = 0,x23x24rankA + rankA? 6 n,x4ex3crankA = n?1,x3ex50rankA? 6 1,x21x4f
rankA? = 1.
13,x18A 2 Mn(K) (n > 2),x53x54:
(1) (A?)? = jAjn?2A;
(2) jA?j = jAjn?1.
x3x4,x62rankA = nx52,AA? = jAjE,x23x24jAjjA?j = jAjn,jA?j = jAjn?1,x3cx13A?(A?)? = jA?jE =
jAjn?1E,x4eA?1 = 1jAj A?x3ex50A? = jAjA?1,x21(A?)? = jAjn?1(A?)?1 = jAjn?2A.
x62rankA = n?1x52,rankA? = 1,x23x24(A?)? = 0 = jAjn?2A,jA?j = 0 = jAjn?1.
x62rankA < n?1x52,A? = 0,x79x53x56x29x67x2ax2b.
x1 x2 4–7
1,x18A,Bx22x37x66x43x79x5dx5e,x53x54:
rank(A+B) 6 rank(A j B) 6 rankA+rankB:
x3x4,x18Ax15x1ex1fx20x42x22fi1;￠￠￠ ;fin,Bx15x1ex1fx20x42x22fl1;￠￠￠ ;fln,x4aA + Bx15x1ex1fx20x42x22fi1 +
fl1;￠￠￠ ;fin +fln,(A j B)x15x1ex1fx20x42x22fi1;￠￠￠ ;fin;fl1;￠￠￠ ;fln,x43x25x4ex60x614–1.7,
rank(A+B) = rankffi1 +fl1;￠￠￠ ;fin +flng
6 rankffi1;￠￠￠ ;fin;fl1;￠￠￠ ;flng = rank(A j B)
6 rankffi1;￠￠￠ ;fing+rankffl1;￠￠￠ ;flng
= rankA+rankB:
2,x18
A =
0
BB
B@
0 0 1 1
0 0 0 1
2 1 1?3
3 2 1 2
1
CC
CA;
x73A?1.
x5,x18
A?1 =
B
11 B12
B21 B22
;
￠ 16 ￠
AA?1 =
0 A
12
A21 A22
B
11 B12
B21 B22
=
A
12B21 A12B22
A21B11 +A22B21 A21B12 +A22B22
=
E
2 0
0 E2
:
x23x24
B21 = A?112 =
1?1
0 1
;
B22 = 0; (x21A12x3exc)
B12 = A?121 =
2?1
3 2
;
B11 =?A?121 A22B21 =
1 9
1?14
:
x21x4f
A?1 =
0
BB
@
1 9 2?1
1?14?3 2
1?1 0 0
0 1 0 0
1
CC
A:
3,x18Ax22x3excx15nx79x40x5e,
D =
0 A
a 0
; a 6= 0;
x73D?1.
x5,D?1 =
0 a?1
A?1 0
.
4,x18Aix22rix79x3excx40x5e(i = 1;2;￠￠￠ ;s),
A =
0
BB
B@
A1
A2
...
As
1
CC
CA
x73A?1.
x5,A?1 =
0
BB
B@
A?1s
A?1s?1
...
A?11
1
CC
CA.
5,x18Eix22ri (i = 1;2;￠￠￠ ;s)x79x14x2fx5dx5e,x25
A =
0
BB
B@
a1E1
a2E2
...
asEs
1
CC
CA; ai 6= aj; i 6= j;
x53x54:x42Ax3ex3x4ax15x5dx5ex7bx63x13x11x37x2x35x5dx5e.
x3x4,x18x11x37x5dx5eB = (Bij)x42Ax3ex3x4a,x25x3fBx15x11x37x40x29x42Ax65x43.x4ax4eAB = BAx50
aiBij = Bijaj; i;j = 1;￠￠￠ ;s:
x3cx13
(ai?aj)Bij = 0; i;j = 1;￠￠￠ ;s:
￠ 17 ￠
x71x62i 6= jx52x47ai 6= aj,x23x24x2x3ci 6= jx47Bij = 0,x4
B =
0
BB
B@
B11 0 ￠￠￠ 0
0 B22 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ Bss
1
CC
CA:
6,x18
A =
0
BB
BB
BB
@
0 0 ￠￠￠ 0 a0
a1 0 ￠￠￠ 0 0
0 a2 ￠￠￠ 0 0
...,..,..,..,..
0 0 ￠￠￠ an 0
1
CC
CC
CC
A
ai 6= 0; i = 0;1;2;￠￠￠n;
x73A?1.
x5,A?1 =
0
BB
BB
BB
@
0 a?11 0 ￠￠￠ 0
0 0 a?12 ￠￠￠ 0
...,..,..,..,..
0 0 0 ￠￠￠ a?1n
a?10 0 0 ￠￠￠ 0
1
CC
CC
CC
A
.
7,x18x5dx5eAm￡s,Bt￡nx15x9x11x12x22rA,rB,Cx22x0x1x15m￡nx5dx5e,x25
D =
A C
0 B
x53x54:x5dx5eDx15x9rD > rA +rB.
x3x4,x18Ax15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42x22fii1;￠￠￠ ;fiirA,B x15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42x22
flj1;￠￠￠ ;fljrB,x4a
1 = (fii1;?￠￠￠?| {z }
n
); 2 = (fii2;?￠￠￠?| {z }
n
);￠￠￠ ; rA = (fiirA;?￠￠￠?| {z }
n
)
x74x26x2cx2a.
–1 = (0￠￠￠0| {z }
s;flj1);–2 = (0￠￠￠0| {z }
s;flj2);￠￠￠ ;–rB = (0￠￠￠0| {z }
s;fljrB)
x74x26x2cx2a,x10x11
1; 2;￠￠￠ ; rA;–1;–2;￠￠￠ ;–rB;
x74x26x2cx2a,x23x24
rD >f 1; 2;￠￠￠ ; rA;–1;–2;￠￠￠ ;–rBg = rA +rB:
8,x18Ax22x62x79x40x5e,x53x54:x8x1fAk = 0,x4aA2 = 0.
x3x4,x4eAk = 0x3ex50jAj = 0,x21rankA 6 1,x8x1frankA = 0,x4aA = 0,x22x23x10x11x2ax2b,x8x1f
rankA = 1,x4a
A =
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn); (x60x614–5.12)
A2 =
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn)
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn) =
nX
i=1
aibiA;
Ak =
nX
i=1
aibi
!k?1
A = 0:
￠ 18 ￠
x4ex3cA 6= 0,
nP
i=1
aibi = 0,x23x24A2 = 0.
9,x18A,Bx22x37x66nx79x40x5e,x53x54:x5dx5ex40x41AX = Bx47x2dx15x30x11x40x26x31x32x13rankA = rank(A j B).
x3x4,())x18x5dx5ex40x41AX = Bx47x2dX = C,x4aAC = B,x43x25Bx15x1ex1fx20x42x3ex4eAx15x1ex1fx20x42
x74x26x1bx1c,x23x24
rank(A j B) = rankA:
(()x8x1frank(A j B) = rankA,x4aBx15x1ex1fx20x42x3ex4eAx15x1ex1fx20x42x74x26x1bx1c,x4x31x6b(c1j;￠￠￠ ;cnj)T
x27x50
A
0
B@
c1j
...
cnj
1
CA =
0
B@
b1j
...
bnj
1
CA; j = 1;￠￠￠ ;n:
x49
C =
0
B@
c11 ￠￠￠ c1n
...,..,..
cn1 ￠￠￠ cnn
1
CA;
x4aAC = B.
10,x18A,Bx22x37x66nx79x40x5e,x53x54:x3x48x74x26x40x41x42AX = 0x42x3x48x74x26x40x41x42BAX = 0x43x2dx15
x30x11x40x26x31x32x13rankA = rankBA.
x3x4,x51xa,AX = 0x15x2dx6dx13BAX = 0x15x2d,x43x25BAX = 0x15x7ax8x2dx6ax45x46xax47n?rankAx66
x2d,x51x21x22rankA = rankBA,x23x24BAX = 0x15x7ax8x2dx6ax38xax47n?rankAx66x2d,x21AX = 0x15x7ax8x2d
x6ax67x13BAX = 0x15x7ax8x2dx6a,x21x4fAX = 0x42BAX = 0x43x2d.
x4dx39,x8x1fAX = 0x42BAX = 0x43x2d,x4ax38x15x15x7ax8x2dx6axax47x65x43x66xex15x2d,x21x4fn?rankA =
n?rankBA,rankA = rankBA.
x1 x2 4–8
1,x1ax5cx56x3dx4ax73x1dx1ex5dx5ex15xcx5dx5e:
(1)
0
@
1?2 1
2 1 0
1 2 0
1
A; (2)
0
@
2 3?2
1 0?1
1?2 1
1
A;
(3)
0
BB
B@
1 2 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA; (4)
0
BB
B@
1 1 1 1
1?1 1 1
1 1?1 1
1 1 1?1
1
CC
CA.
x5,(1) 13
0
@
0 2?1
0?1 2
3?4 5
1
A.
(2) 16
0
@
2?1 3
2?4 0
2?7 3
1
A.
(3)
0
BB
B@
1?2 2?2
0 1?1 1
0 0 1?1
0 0 0 1
1
CC
CA.
(4) 14 A.
￠ 19 ￠
2,x2dx1dx1ex5dx5ex40x41:
(1)
2?3
2 4
X =
4 3
2 2
;
(2) X
0
@
1 1 1
1 2 1
1 0?1
1
A =
0
@
1 0 1
1 1 2
1?1 0
1
A;
(3)
0
BB
BB
B@
1 1 1 ￠￠￠ 1 1
0 1 1 ￠￠￠ 1 1
0 0 1 ￠￠￠ 1 1.
..,..,..,..,..,..
0 0 0 ￠￠￠ 0 1
1
CC
CC
CAX =
0
BB
BB
B@
2 1 0 ￠￠￠ 0 0
1 2 1 ￠￠￠ 0 0
0 1 2 ￠￠￠ 0 0.
..,..,..,..,..,..
0 0 0 ￠￠￠ 1 2
1
CC
CC
CA.
x5,(1) X =
2?3
2 4
1 4 3
2 2
= 12
4 3
2 2
4 3
2 2
=
11 9
6 5
.
(2) X =
0
@
1 0 1
1 1 2
1?1 0
1
A
0
@
1 1 1
1 2 1
1 0?1
1
A
1
= 14
0
@
4?2?2
2 1?5
2?3?1
1
A.
x440
BB
BB
BB
BB
BB
BB
@
1 1 1
1 2 1
1 0?1
1 0 1
1 1 2
1?1 0
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
2 1 1
2 1?1
0?1 1
3?1 2
1?1 0
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
1 0 0
1 2 0
0?1 1
32? 52 12
1
2?
1
2
1
2
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
1 0 0
0 1 0
12? 12 1
1
4?
5
4
1
2
34? 14 12
1
CC
CC
CC
CC
CC
CC
A
:
(3) X =
0
BB
BB
BB
B@
1?1?1 0 ￠￠￠ 0 0 0
1 1?1?1 ￠￠￠ 0 0 0
0 1 1?1 ￠￠￠ 0 0 0
:::::::::::::::::::::::::::::::::
0 0 0 0 ￠￠￠ 1 1?1
0 0 0 0 ￠￠￠ 0 1 2
1
CC
CC
CC
CA
.
3,x1ax9x62x40x44x73
A =
0
BB
B@
1 1 1 1
1?1 1?1
1 1?1?1
1?1?1 1
1
CC
CA
x15xcx5dx5e.
x5,x63x39x3ax37x62x2dx44:
(i)x21x22AA = 4E,x23x24A?1 = 14 A.
(ii)x11x37,A1 =
1 1
1?1
,A?11 = 12
1 1
1?1
= 12 A1.
A
1 A1 E 0
A1?A1 0 E
!
A
1 A1 E 0
0?2A1?E E
!
A1 0 12 E 12 E
0?2A1?E E
!
!
E 0 12 A?11 12 A?11
0 E 12 A?11? 12 A?11
!
:
￠ 20 ￠
A?1 = 12
A?1
1 A
1
1
A?11?A?11
= 14
A
1 A1
A1?A1
= 14 A:
4,x18A;B;C;D 2 Mn(K),jAj6= 0,AC = CA,x53x54:fl
flfl
fl
A B
C D
flfl
flfl = jAD?CBj:
x3x4,x21jAj6= 0,x21Ax3exc,x25
E 0
CA?1 E
A B
C D
=
A B
0 D?CA?1B
;
x23x24 fl
flfl
fl
A B
C D
flfl
flfl =
flfl
flflA B
0 D?CA?1B
flfl
flfl = jAjjD?CA?1Bj
= jAD?ACA?1Bj = jAD?CBj:
5,x18A;B 2 Mn(C),x53x54,fl
flfl
fl
A B
B A
flfl
flfl = jA?iBjjA+iBj:
(x3cx16ix22x3bxex14x2f,i2 =?1.)
x3x4,x21x22
E iE
0 E
A B
B A
E?iE
0 E
=
A?iB
A+iB
:
x23x24 fl
flfl
fl
A B
B A
flfl
flfl = jA?iBjjA+iBj:
6,x18A 2 Mm;r(K),x53x54:
(1) Ax22x1ex2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx3excx5dx5eP 2 Mm(K),x27A = P
E
r
0
;
(2) Ax22x1ex2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bxdx2dx9x5dx5eB 2 Mr;m(K),x27BA = Er.
x3x4,(1)x21Ax1ex2dx9,Ax15x3cx3dx36x22
E
r
0
,x43x25x31x6bx3excx5dx5eP1;Q1,x27
A = P1
E
r
0
Q1:
x49
P = P1
Q
1 0
0 Em?r
;
x4a
A = P1
E
r
0
Q1 = P1
Q
1
0
= P1
Q
1 0
0 Em?r
E
r
0
= P
E
r
0
:
x77x6fx53x54x72x40x26x26,x25x30x11x26x13x10x11x15.
(2)x30x11x26x13x10x11x15,x12x53x40x26x26.
x4e(1)xb,x31x6bx3excx5dx5eP,x27A = P
E
r
0
,x4a
P?1A =
E
r
0
;
x49
P?1 =
B gr
B1
;
x4aBxdx2dx9,x3fBA = Er.
7,x2x3cxdx2dx9x5dx5e,x3ex53x57x53x54x6bx6cx15x22x23.
￠ 21 ￠
x5,(1) Ax22xdx2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx3excx5dx5eQ 2 Mm(K),x27A = (Er 0)Q.
(2) Ax22xdx2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx1ex2dx9x5dx5eB 2 Mm;r(K),x27AB = Er.
(x53x54x69)
8,x18m￡nx5dx5eAx15x9x22r,x53x54:x31x6bx1ex2dx9x5dx5ePx3axdx2dx9x5dx5eQ,x27A = PQ.
x3x4,x31x6bx3excx5dx5eP1;Q1,x27
A = P1
E
r 0
0 0
Q1:
x49
P = P1
E
r
0
; Q = (Er 0)Q1;
x4aPx1ex2dx9,Qxdx2dx9,x3f
A = P1
E
r 0
0 0
Q1 = P1
E
r
0
(Er 0)Q1 = PQ:
9,x18A,Bx11x12x22n￡mx42m￡n(n > m)x5dx5e,? 6= 0,x53x54:
j?En?ABj =?n?mj?Em?BAj:
x3x4,x4ex3c
En 0
B Em
E
n A
B?Em
=
E
n A
0?Em?AB
;
E
n?A
0 Em
E
n A
B Em
=
E
n?AB 0
B?Em
;
x23x24
j?En?ABj =
flfl
flfl?En A
B Em
flfl
flfl;
j?Em?BAj =
flfl
flflEn A
0?Em
flfl
flfl:
x25
mj?En?ABj =?m
flfl
flfl?En A
B Em
flfl
flfl =
flfl
flfl?En A
B?Em
flfl
flfl
=?n
flfl
flflEn A
B?Em
flfl
flfl =?nj?Em?BAj:
x21x4f
j?En?ABj =?n?mj?Em?BAj:
10,x18A,Bx11x12x22s￡nx42n￡mx5dx5e,x53x54:
rank(AB) > rank(A)+rank(B)?n:
x3x4:
AB 0
0 En
!
AB?A
0 En
!
0?A
B En
,x23x24
rank(AB)+n = rank
AB 0
0 En
= rank
0?A
B En
= rank
0 A
B En
> rankA+rankB; (4–7.7)
x21x4frank(AB) > rank(A)+rank(B)?n.
11,x18A,B,Cx11x12x22s￡n,n￡mx42m￡tx5dx5e,x53x54:
rank(ABC) > rank(AB)+rank(BC)?rankB:
￠ 22 ￠
x3x4:
ABC 0
0 B
!
ABC AB
0 B
!
0 AB
BC B
!
0 AB
BC B
:
x23x24
rank(ABC)+rankB > rank(AB)+rank(BC);
rank(ABC) > rank(AB)+rank(BC)?rankB:
12,x18A 2 Mn(K),x53x54:
A2 = En,rank(A?En)+rank(A+En) = n:
x3x4:
E
n A+En
A?En 0
!
E
n A+En
0 En?A2
,x23x24
rank(A?En)+rank(A+En) = n+rank(En?A2):
rank(A?En)+rank(A+En) = n () rank(En?A2) = 0 () A2 = En:
13,x18A 2 Mn(K),x53x54:
A2 = A,rank(A)+rank(A?En) = n:
x3x4:
A 0
0 A?En
!
A?E
n
0 A?En
!
0?E
n
A2?A A?En
!
0 E
n
A2?A 0
:
x23x24
rankA+rank(A?En) = n+rank(A2?A):
rank(A)+rank(A?En) = n,A2 = A:
14,x18A 2 Mn(K)x13x3excx5dx5e,X;Y x22nx46x1ex1fx20,x53x54:fl
flfl
fl
A Y
XT 0
flfl
flfl =?XTA?Y:
x3x4,
A Y
XT 0
!
A Y
0?XTA?1Y
;
)
flfl
flfl A Y
XT 0
flfl
flfl = jAjj?XTA?1Yj =?XTjAjA?1Y =?XTA?Y:
x1 x2 4–9
1,x18x41 x22x1fx20x70x71V1x8V2x15x74x26x46x5d,x41 x6bx67x11x7ax1dx15x5dx5ex13
A =
0
BB
B@
1 0 2 1 3
2?3?1 1?4
1?3 1 2?1
1 1 1 0 2
1
CC
CA:
(1)x73x41 x15x3fx42x23x15x46xex42x7a;
(2)x11x12x76x41 x15x3fx42x23x15x7ax1dx30x22V1x42V2x15x7a.
x5,(1) x21x22rankA = 3,x21x4fx23x70x71x15x46xex223,Ax15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42x75x2ax23x70x71x15
x7a:
x41("1) = (?1;2;1;?1);x41("2) = (0;?3;?3;1);x41("5) = (3;?4;?1;2):
￠ 23 ￠
x3fx15x46xe= 5?3 = 2,AX = 0x15x7ax8x2dx6ax75x2ax3fx70x71x15x7a:
1 = (2;1;1;0;0);?2 = (1;1;0;1;0):
(2)?1;?2x3ex1dx30x22V1x15x7a:
1;?2;"3;"4;"5:
x41("1);x41("2);x41("5)x3ex24x1dx30x22V2x15x7a:
x41("1);x41("2);x41("5);(1;0;0;0):
2,x18x41 x22K3x15x74x26x3dx4a,x27
x41(x;y;z) = (x+y?z;x+y +z;x+y?2z):
(1)x73x41 x15x6fx77x42x9;
(2)x73x41 x15x3fx42x23x70x71.
x5,(1)x6fx77= 1,x9= 2.
(2)x3f= L((1;?1;0)),x23= L((1;1;1);(?1;1;?2)).
3,x18W1,W2x22V x15x37x66x0x70x71,x3fdimW1+dimW2 = n,x53x54:x31x6bx74x26x3dx4ax41,x27Kerx41 = W1,
Imx41 = W2.
x3x4,x18W1x15x7ax22fi1;￠￠￠ ;fir,W2x15x7ax22fl1;￠￠￠ ;fln?r,x76W1x15x7ax1dx30x22V x15x7a,fi1;￠￠￠ ;fir;
fir+1;￠￠￠ ;fin,x2x0x1x15fi =
nP
i=1
aifii 2 V,x1dx4d
x41(fi) =
n?rX
i=1
an+ifli;
x4ax41 x13V x15x74x26x3dx4a,x3fKerx41 = W1,Imx41 = W2.
4,x18x41x22nx46x1fx20x70x71x15x74x26x3dx4a,V1,V2x22V x37x66x74x26x0x70x71,x53x54:x8x1fKerx41 = V1 \V2,x4a
x31x6bx74x26x3dx4ax411,x412,x27V1 Kerx411,V2 Kerx412,x3fx41 = x411 + x412.
x3x4,x18 1;￠￠￠ ; r x22V1 \ V2 x15x7a,x76x38x1dx30x22V1 x15x7a,1;￠￠￠ ; r;fi1;￠￠￠ ;fit,x1dx30x22V2 x15x7a:
1;￠￠￠ ; r;fl1;￠￠￠ ;fls,x4ax4ex46xex66x29xb 1;￠￠￠ ; r;fi1;￠￠￠ ;fit;fl1;￠￠￠ ;flsx22V1 + V2x15x7a,x12x4ex38x1dx30x22
V x15x7a:
1;￠￠￠ ; r;fi1;￠￠￠ ;fit;fl1;￠￠￠ ;fls;·1;￠￠￠ ;·u; (r +t+s+u = n):
x11x12x1dx4dx74x26x3dx4ax8x1d:
x411( i) = 0; x411(fii) = 0; x411(fli) = x41(fli); x411(·i) = x41(·i);
x412( i) = 0; x412(fii) = x41(fii); x412(fli) = 0; x412(·i) = 0:
x4axex53V1 Kerx411,V2 Kerx412,x3fx41 = x411 + x412.
5,x18x41;x42x22nx46x1fx20x70x71V x15x37x66x74x26x3dx4a,x3fx41 2 = x41,x422 = x42,x53x54:
(1) Imx41 = Imx42,x41x42 = x42;x42x41 = x41;
(2) Kerx41 = Kerx42,x41x42 = x41;x42x41 = x42.
x3x4,(1) ())x2x0x1x15fi 2 V x31x6bfl 2 V,x27x50x41(fi) = x42(fl),x23x24
x41(fi) = x42(fl) = x422(fl) = x42(x42(fl)) = x42x41(fi):
x4x41 = x42x41,x43x5x3ex53x42 = x41x42.
(() x42(V) = x41x42(V) x41(V),x41(V) = x42x41(V) x42(V),x23x24Imx41 = Imx42.
￠ 24 ￠
(2) ())x2x0x1x15fi 2 V,x4ex3c
x42[(x42?x45)(fi)] = x422(fi)?x42(fi) = x42(fi)?x42(fi) = 0;
x23x24x41[(x42?x45)(fi)] = 0,x3cx13x41x42(fi) = x41(fi),x41x42 = x41,x43x5x3ex53x42x41 = x42.
(()x2x0x1x15fi 2 Kerx42x47
x41(fi) = x41x42(fi) = x41(0) = 0;
x23x24Kerx42 Kerx41,x43x5x3ex53Kerx41 Kerx42,x21x4fKerx41 = Kerx42.
￠ 25 ￠
x0x15x2 x16x17x18x19x11x16x17x18x1ax1b
x1 x2 4–1
1,x18x1fx20flx3ex4ex1fx20x42fi1;fi2;￠￠￠ ;fisx74x26x1bx1c,x71x55x63x4efi1;fi2;￠￠￠ ;fis?1x74x26x1bx1c,x53x54:x1fx20
x42fi1;fi2;￠￠￠ ;fisx42x1fx20x42fi1;fi2;￠￠￠ ;fis?1;flx56x15.
x3x4,x4ex31x18,x31x6ba1;￠￠￠ ;as 2 Kx27x50
fl = a1fi1 +a2fi2 +￠￠￠+asfis:
x8x1fas = 0,x4aflx3ex24x49fi1;fi2;￠￠￠ ;fis?1x74x26x1bx1c,x42x31x18x34x35,x21x4fas 6= 0,x3cx13
fis = 1a
s
fl? a1a
s
fi1?￠￠￠? as?1a
s
fis?1;
x4fisx3ex24x4efi1;fi2;￠￠￠ ;fis?1;flx74x26x1bx1c,x43x25x1fx20x42fi1;fi2;￠￠￠ ;fisx3ex49fi1;fi2;￠￠￠ ;fis?1;flx74x26x1bx1c.
x16x48x40xf,x3dx3ex31x18,x1fx20x42fi1;fi2;￠￠￠ ;fis?1;flx3ex24x49x1fx20x42fi1;fi2;￠￠￠ ;fisx74x26x1bx1c,x21x4fx77x37x66x1fx20
x42x56x15.
2,(x17x18x19x1a) x18x1fx20x42fi1;fi2;￠￠￠ ;fis x74x26x2cx2a,x3fx3ex4ex1fx20x42fl1;fl2;￠￠￠ ;flt x74x26x1bx1c,x53x54:
x31x6bfl1;fl2;￠￠￠ ;flt x15x48x66x59x4afli1;fli2;￠￠￠ ;flit,x27x1fx20x42fi1;fi2;￠￠￠ ;fir;flir+1;flir+2;￠￠￠ ;flit x42x1fx20x42
fl1;fl2;￠￠￠ ;fltx56x15(r = 1;￠￠￠ ;s).
x3x4,x21x22fi1;fi2;￠￠￠ ;fisx74x26x2cx2a,x3fx3ex4ex1fx20x42fl1;fl2;￠￠￠ ;fltx74x26x1bx1c,x21s 6 t.
x1dxfx1ax2x50x44x53x54x1bx4ax1dx5.
(i)x18s = 1.
x21x22fi1x3ex4efl1;￠￠￠ ;fltx74x26x1bx1c,x21x31x6bai 2 Kx27x50fi1 =
tP
i=1
aifli,x25fi1x74x26x2cx2a,x4fi1 6= 0,x23
x24a1;￠￠￠ ;atx55x33x22x6f,x40x47al 6= 0 (1 6 l 6 t),x4a
fll = 1a
l
fi1?
tX
i=1i6=l
ai
al fli;
x21x4fx1fx20x42fi1;fl1;￠￠￠ ;fll?1;fll+1;￠￠￠ ;fltx42x1fx20x42fl1;fl2;￠￠￠ ;fltx56x15.
x49fli1 = fll,fli2 = fl1,:::,flil = fll?1,flil+1 = fll+1,:::,flit = flt,x4x50x22x23.
(ii)x31x1dx22x23x2s?1x2ax2b,xbx3sx66x74x26x2cx2ax15x1fx20fi1;fi2;￠￠￠ ;fis.
x21fi1;fi2;￠￠￠ ;fis?1x74x26x2cx2a,x4ex2x50x31x18,x31x6bfl1;￠￠￠ ;fltx15x48x66x59x4aflj1;￠￠￠ ;fljt,x27
ffi1;￠￠￠ ;fir;fljr+1;￠￠￠ ;fljtg?= ffl1;￠￠￠ ;fltg (r = 1;￠￠￠ ;s?1):
x51fisx3ex4efl1;￠￠￠ ;fltx74x26x1bx1c,x23x24fisx3ex24x4efi1;￠￠￠ ;fis?1;fljs;￠￠￠ ;fljt x74x26x1bx1c,x21x31x6bki;lk 2 K,
i = 1;￠￠￠ ;s?1,k = s;￠￠￠ ;t,x27x50
fis =
s?1X
i=1
kifii +
tX
k=s
lkfljk:
￠ 1 ￠
x4ex3cfi1;￠￠￠ ;fis x74x26x2cx2a,x21ls;￠￠￠ ;lt x55x33x22x6f,x18x3dx48x66x55x22x6fx15x13lh,x4ah > s,x43x25fljh x3ex24
x4efi1;￠￠￠ ;fis;fljh+1;￠￠￠ ;fljt x74x26x1bx1c,x49flis = fljh,fli1 = flj1,:::,flis?1 = fljs?1,flis+1 = fljs+1,:::,
flit = fljt,x4a
ffi1;￠￠￠ ;fis;flis+1;￠￠￠ ;flitg?= ffl1;￠￠￠ ;fltg:
x4ex2x50x44x4bx5x3exbx22x23x2ax2b.
3,x18x1fx20x42fi1;fi2;￠￠￠ ;fisx15x9x22r,fii1;fii2;￠￠￠ ;fiirx13x38x15x48x66x7cx11x42.x53x54:x8x1ffi1;fi2;￠￠￠ ;fis
x3ex4efii1;fii2;￠￠￠ ;fiirx74x26x1bx25,x4afii1;fii2;￠￠￠ ;fiirx13fi1;fi2;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42.
x3x4,x2fx22x1fx20x42x15x7cx11x42,fii1;￠￠￠ ;fiir x62x11x3ex24x49fi1;￠￠￠ ;fisx74x26x1bx1c,x21x4fx77x37x66x1fx20x42x56
x15,x43x25x47x65x43x15x9r,x3cx13x4ex37x611.9x3exbx1fx20x42fii1;￠￠￠ ;fiirx74x26x2cx2a,x4ex5ex231.8x3exbx38x13x1cx3bx74x26
x2cx2ax42.
4,x18fi1;fi2;￠￠￠ ;fitx42fi1;fi2;￠￠￠ ;fit;fit+1;fit+2;￠￠￠ ;fisx47x65x43x15x9,x53x54,fi1;fi2;￠￠￠ ;fitx42fi1;fi2;
￠￠￠ ;fis x56x15.
x3x4,x3dx3ex31x18,x47
L(fi1;￠￠￠ ;fit) L(fi1;￠￠￠ ;fit;fit+1;￠￠￠ ;fis);
x51x21x77x37x66x1fx20x42x47x65x43x15x9,x21x4fx38x15x2ex2ax15x74x26x0x70x71x47x65x43x15x46xe,x43x25x65x56.x12x33x1ax37x611.1,
x3exbx77x37x66x1fx20x42x74x26x56x15.
5,x2x1dx1ex1fx20x42,x76fi1x1dx30x2ax1fx20x42x15x48x66x1cx3bx2cx2ax42:
(1) fi1 = (1;?1;2;4),fi2 = (0;3;1;2),fi3 = (3;0;7;14),fi4 = (1;?1;2;0),fi5 = (2;1;5;6);
(2) fi1 = (1;?1;0;1;1),fi2 = (2;1;3;?1;0),fi3 = (3;0;3;0;1),fi4 = (1;?1;1;?1;1),fi5 =
(?1;?5;?6;5;3),fi6 = (2;1;2;1;0).
x5,(1) fi1;fi2;fi4.
(2) fi1;fi2;fi4.
6,x18x1fx20x42ffi1;fi2;￠￠￠ ;fisg,ffl1;fl2;￠￠￠ ;fltg,ffi1;fi2;￠￠￠ ;fis;fl1;fl2;￠￠￠;fltgx15x9x11x12x13r1,r2,r3.
x53x54:
maxfr1;r2g6 r3 6 r1 +r2:
x3x4,x4ex3cx1fx20x42ffi1;;￠￠￠ ;fisg,ffl1;￠￠￠ ;fltgx6dx3ex4ex1fx20x42ffi1;￠￠￠ ;fis;fl1;￠￠￠ ;fltgx74x26x1bx1c,x21
r1 6 r3; r2 6 r3;
x43x25
maxfr1;r2g6 r3:
x18fii1;￠￠￠ ;fiir1x13fi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,flj1;￠￠￠ ;fljr2x13fl1;￠￠￠ ;fltx15x48x66x1cx3bx74x26x2c
x2ax42,x4a
ffi1;;￠￠￠ ;fis;fl1;￠￠￠ ;fltg?= ffii1;￠￠￠ ;fiir1;flj1;￠￠￠ ;fljr2g;
x23x24
r3 = rankffii1;￠￠￠ ;fiir1;flj1;￠￠￠ ;fljr2g6 r1 +r2:
7,x18x1fx20x42ffi1;fi2;￠￠￠ ;fisg,ffl1;fl2;￠￠￠ ;flsg,ffi1 + fl1;fi2 + fl2;￠￠￠ ;fis + flsgx15x9x11x12x13r1,r2,
r3,x53x54,r3 6 r1 +r2.
x3x4,x21x22ffi1 +fl1;￠￠￠ ;fis +flsgx3ex4effi1;￠￠￠ ;fis;fl1;￠￠￠ ;flsgx74x26x1bx1c,x21x4fx38x15x9
r3 6 rankffi1;￠￠￠ ;fis;fl1;￠￠￠ ;flsg6 rankffi1;￠￠￠ ;fisg+rankffl1;￠￠￠ ;flsg = r1 +r2:
￠ 2 ￠
8,x18x1fx20x42fi1;fi2;￠￠￠ ;fisx15x9x22r,fii1;fii2;￠￠￠ ;fiimx22x38x15x48x66x7cx11x42.x53x54:
rankffii1;fii2;￠￠￠ ;fiimg> r +m?s:
x3x4,x18fii1;￠￠￠ ;fiimx15x9x56x3ct,x4ax38x15x48x66x1cx3bx2cx2ax42fij1;￠￠￠ ;fijtx13fi1;￠￠￠ ;fisx15x74x26x2cx2ax42,
x38x3ex49x1dx30x22fi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,x25x77x48x1dx30x15x1fx20x55x3ex63x13fii1;￠￠￠ ;fiim x15x1fx20,x29
x4ax42x1cx3bx2cx2ax42x34x35,x25fi1;￠￠￠ ;fisx16x28x47s?mx66x55x4x3cfii1;￠￠￠ ;fiimx15x1fx20,x3cx16x7bx25r?tx66x55x43
x15x1fx20x54x17x8fij1;￠￠￠ ;fijtx24x7x2afi1;￠￠￠ ;fisx15x48x66x1cx3bx74x26x2cx2ax42,x43x25
r?t 6 s?m;
x39x3ax50
rankffii1;￠￠￠ ;fiimg = t > r +m?s:
9,xaxbx37x66x1fx20x42x47x65x43x15x9,x3fx3cx16x48x66x3ex24x49x16x48x66x74x26x1bx1c,x53x54:x77x37x66x1fx20x42x56x15.
x3x4,x18x1fx20x42(I)x3ex49x1fx20x42(II)x74x26x1bx1c,x38x15x7x2ax15x74x26x0x70x71x11x12x1fx22L1;L2,x4aL1 L2.
x51x21x38x15x47x65x43x15x9,x21x4fx38x15x7x2ax15x74x26x0x70x71x47x65x43x15x46xe,x43x25L1 = L2,x4(I)x42(II)x56x15.
x1 x2 4–2
1,x73x1dx1ex5dx5ex15x9:
(1)
0
BB
B@
1 4 10 0
3 2 4 2
4 1 1 3
2 3 7 1
1
CC
CA (2)
0
BB
B@
2 1 11 2
1 0 4?1
1?1 1?5
2 0 8?2
1
CC
CA
(3)
0
BB
BB
B@
1 0 0 1 1
0 1 0?1 1
1?1 1 3 1
0 0 1 1 1
1 1 1 2 3
1
CC
CC
CA (4)
0
BB
B@
2 0 3 1?1
1 2 1 2 1
3?2 5 0?3
1 1 0 2 3
1
CC
CA
x5,(1) 2; (2) 2; (3) 4; (4) 3.
2,x73x1dx1ex1fx20x42x15x9x42x1cx3bx74x26x2cx2ax42:
(1) fi1 = (3;2;?1;?3;?2),fi2 = (2;?1;3;1;?3),fi3 = (1;?4;7;5;4),fi4 = (1;?7;11;9;5);
(2) fi1 = (1;?1;1;1;1),fi2 = (1;1;?1;1;1),fi3 = (1;1;1;?1;1),fi4 = (1;1;1;1;?1),fi5 =
(1;1;1;1:1);
(3) fi1 = (2;?1;3;?2;4),fi2 = (4;?2;5;1;7),fi3 = (2;?1;1;8;2),fi4 = (2;?1;2;3;3);
(4) fi1 = (1;3;3;5),fi2 = (3;2;?5;1),fi3 = (2;3;0;4),fi4 = (5;4;?7;1),fi5 = (3;5;1;7).
x5,(1)x94,fi1;fi2;fi3;fi4.
(2)x95,fi1;fi2;fi3;fi4;fi5.
(3)x92,fi1;fi2.
(4)x93,fi1;fi2;fi4.
3 x73x1fx20x42fi1 = (?3;1;1;1);fi2 = (1;?3;1;1);fi3 = (1;1;?3;1),fi4 = (1;1;1;?3)x15x23x47x1cx3bx74
x26x2cx2ax42.
x5,x0x13x66x1fx20x6dx75x2ax1cx3bx74x26x2cx2ax42.
4,x73x1dx1ex1fx20x42x23x2ex2ax15x0x70x71x15x7ax42x46xe:
(1) fi1=(4;?5;2;6),fi2=(2;1;3;2),fi3=(2;?6;?1;4),fi4=(2;13;5;?6);
￠ 3 ￠
(2) fi1 = (1;0;0;1;?1),fi2 = (0;1;0;2;1),fi3 = (0;0;1;?1;?2),fi4 = (1;1;1;2;?2).
x5,(1)x46xe3,x7afi1;fi2;fi4.
(2)x46xe3,x7afi1;fi2;fi3.
5,x73x1dx1ex5dx5ex15x9:
(1)
0
BB
B@
a1b1 a1b2 ￠￠￠ a1bn
a2b1 a2b2 ￠￠￠ a2bn
:::::::::::::::::::::::
anb1 anb2 ￠￠￠ anbn
1
CC
CA; (2)
0
BB
B@
1 a a ￠￠￠ a a
a 1 a ￠￠￠ a a
::::::::::::::::::::
a a a ￠￠￠ a 1
1
CC
CA.
x5,(1)x21x22x4fx5dx5ex15x0x1x37xdx6dx74x26x65x2a,x21x4fx96 1,x25x4fx5dx5ex15x9x56x3c0x15x30x11x40x26x31x32x13x23
x47x15aibj = 0,x8(a1;￠￠￠ ;an) 6= 0,x4ax40x47(b1;￠￠￠ ;bn) = 0,x8(b1;￠￠￠ ;bn) 6= 0,x4ax40x47(a1;￠￠￠ ;an) = 0.
x21x4fx62(a1;￠￠￠ ;an) = 0x44(b1;￠￠￠ ;bn) = 0x52,x9x220,x29x4a,x9x221.
(2)x62a = 1x52,x9x221;x62a = 11?n x52,x9x22n?1(n > 1);x3cx24xfx36,x9x22n.
6,x18
W = f(a1;￠￠￠ ;ar;0;￠￠￠ ;0)T j ai 2 K; i = 1;￠￠￠ ;rg Km
x53x54,dimW = r.
x3x4,x18
fi1 =
0
BB
B@
1
0.
..
0
1
CC
CA;fi2 =
0
BB
B@
0
1.
..
0
1
CC
CA;￠￠￠ ;fir =
0
BB
BB
BB
@
0.
..
1 (r),
...
0
1
CC
CC
CC
A
x4afi1;fi2;￠￠￠ ;firx74x26x2cx2a,x3fx2x0x1x15
fi =
0
BB
BB
BB
BB
@
a1
...
ar
0.
..
0
1
CC
CC
CC
CC
A
2 W
x47fi = a1fi1 +￠￠￠+arfir,x23x24dimW = r.
7,x18fi1;fi2;￠￠￠ ;firx74x26x2cx2a,flj =
rP
i=1
aijfii (j = 1;￠￠￠ ;s),x49A = (aij),x53x54:
rankffl1;fl2;￠￠￠ ;flsg = rankA:
x3x4,(i)x18flj1;￠￠￠ ;fljtx13fl1;￠￠￠ ;flsx15x48x66x1cx3bx74x26x2cx2ax42.xbx3Ax15x1ex1fx20x42 1;￠￠￠ ; s,x4a
(flj1 ￠￠￠ fljt) = (fi1 ￠￠￠ fir)( j1 ￠￠￠ jt):
x8x1f
tP
i=1
ki ji = 0,x4a
(flj1 ￠￠￠ fljt)
0
B@
k1
...
kt
1
CA = (fi
1 ￠￠￠ fir)( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = 0;
x4
tP
i=1
kiflji = 0,x4ex3cflj1;￠￠￠ ;fljtx74x26x2cx2a,x21x4fk1 = ￠￠￠ = kt = 0,x4 j1;￠￠￠ ; jtx74x26x2cx2a,x23x24
rank(A) > t = rankffl1;￠￠￠ ;flsg:
￠ 4 ￠
(ii)x18 j1;￠￠￠ ; jtx13Ax15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42,x4ax4e
tP
i=1
kiflji = 0x3ex50
(fi1 ￠￠￠ fir)( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = (fl
j1 ￠￠￠ fljt)
0
B@
k1
...
kt
1
CA = 0;
x4ex3cfi1;￠￠￠ ;firx74x26x2cx2a,x40x41x47
( j1 ￠￠￠ jt)
0
B@
k1
...
kt
1
CA = 0;
x4e j1;￠￠￠ ; jtx15x74x26x2cx2ax26x3ex50k1 = ￠￠￠ = kt = 0,x4flj1;￠￠￠ ;fljtx74x26x2cx2a,x21x25
rankffl1;￠￠￠ ;flsg> t = rank(A):
x6x2x50x8
rankffl1;￠￠￠ ;flsg = rank(A):
8,x18A 2 Mm;n(K),xaxbAx15x3di1;i2;￠￠￠ ;ir xdx42x2aAx15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42,Ax15x3d
j1;j2;￠￠￠ ;jrx1ex42x2aAx15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42.x53x54:fl
flfl
flfl
flfl
flfl
ai1;j1 ai1;j2 ￠￠￠ ai1;jr
ai2;j1 ai2;j2 ￠￠￠ ai2;jr
::::::::::::::::::::::::
air;j1 air;j2 ￠￠￠ air;jr
flfl
flfl
flfl
flfl
fl
6= 0:
x3x4,x3bx62x3x4ax5dx5ex15xdx42x1e,x3ex18x5dx5ex15x4frxdx42x4frx1ex11x12x22x5dx5ex15xdx1fx20x42x42x1ex1fx20x42x15
x1cx3bx74x26x2cx2ax42.x43x25x5dx5ex3ex6ax5cx56xdx3dx4ax4cx22
B =
0
BB
BB
BB
B@
a1;1 a1;2 ￠￠￠ a1;n
::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;n
0 0 ￠￠￠ 0
::::::::::::::::::::
0 0 ￠￠￠ 0
1
CC
CC
CC
CA;
x21x5dx5ex15x5cx56xdx3dx4ax55x3cx3dx5dx5ex15x1ex1fx20x15x74x26x2ax6a,x21x5dx5eBx15x4frx1exex22Bx15x1ex1fx20x42x15x1cx3bx74
x26x2cx2ax42.x43x25Bx3ex6ax5cx56x1ex3dx4ax4cx22
C =
0
BB
BB
BB
B@
a1;1 a1;2 ￠￠￠ a1;r 0 ￠￠￠ 0
::::::::::::::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;r 0 ￠￠￠ 0
0 0 ￠￠￠ 0 0 ￠￠￠ 0
::::::::::::::::::::::::::::::::
0 0 ￠￠￠ 0 0 ￠￠￠ 0
1
CC
CC
CC
CA
:
x21x22rank(C) = rank(B) = r,x23x24
flfl
flfl
flfl
a1;1 a1;2 ￠￠￠ a1;r
::::::::::::::::::::
ar;1 ar;2 ￠￠￠ ar;r
flfl
flfl
flfl6= 0:
x1 x2 4–3
1,x55x23x1dx1exax5ex20x74x26x40x41x42x15x2dx15xfx21,x57x73x2d.
￠ 5 ￠
(1)
8>
<
>:
ax1 +bx2 +x3 = 1
x1 +abx2 +x3 = b
x1 +bx2 +ax3 = 1;
(2)
8>
<
>:
(?+3)x1 +x2 +2x3 =?
x1 +(1)x2 +x3 = 2?
3(?+1)x1 +?x2 +(?+3)x3 = 5;
(3)
8>
<
>:
ax1 +bx2 +2x3 = 1
ax1 +(2b?1)x2 +3x3 = 1
ax1 +bx2 +(b+3)x3 = 2b?1:
x5,(1)x62b(a?1)(a+2) 6= 0x52x47x2d,x1= a?b(a?1)(a+2),x2= ab+b?2b(a?1)(a+2),x3 = a?b(a?1)(a+2) ;
x62a = b =?2x52,x47x2dx1 = x3 =?1?2x2;
x62a = b = 1x52,x47x2dx1 = 1?x2?x3;
x3cx24xfx36x2cx2d;
(2)x62? 6= 0,? 6= 1x52x47x2d,x1 =?2 +415?2,x2 =?2 +?+15?2,x3 =?4?2 +?+15?2 ;
x62? = 1x52x47x2d,x1 = 2?x3,x2 =?7+2x3;
x62? = 0x52x2cx2d;
(3)x62a 6= 0,b 6= §1x52x47x2d,x1 = 5?ba(b+1),x2 =?2b+1,x3 = 2(b?1)b+1 ;
x62b = 1x52x47x2d,x2 = 1?ax1,x3 = 0;
x62a = 0,b = 5x52x47x2d,x2 =? 13,x3 = 43,x1x22x0x1xe;
x3cx24xfx36x2cx2d.
2,x33x1ax74x26x40x41x42x15x5x23x53x54:x8x1fx2ex74
L1,
( A
1x+B1y +C1z +D1 = 0
A2x+B2y +C2z +D2 = 0
x42x2ex74
L2,
( A
3x+B3y +C3z +D3 = 0
A4x+B4y +C4z +D4 = 0
x65x3,x20x30 fl
flfl
flfl
flfl
flfl
A1 A2 A3 A4
B1 B2 B3 B4
C1 C2 C3 C4
D1 D2 D3 D4
flfl
flfl
flfl
flfl
fl
= 0:
x5,x3dx3ex6a3.3x15x2d,x8x1fL1x42L2x65x3,x20x30x74x26x40x41x428
>>>>
<
>>>>
:
A1x+B1y +C1z =?D1
A2x+B2y +C2z =?D2
A3x+B3y +C3z =?D3
A4x+B4y +C4z =?D4
x47x2cx48x2d,x43x25rank(A) = rank(?A) = 3,x77x1eAx42?Ax11x12x13x79x53x40x41x42x15x6axex5dx5ex42x16x1fx5dx5e,x21x4fxd
x1ex29j?Aj = 0,fl
flfl
flfl
flfl
flfl
A1 A2 A3 A4
B1 B2 B3 B4
C1 C2 C3 C4
D1 D2 D3 D4
flfl
flfl
flfl
flfl
fl
=?
flfl
flfl
flfl
flfl
fl
A1 B1 C1?D1
A2 B2 C2?D2
A3 B3 C3?D3
A4 B4 C4?D4
flfl
flfl
flfl
flfl
fl
= 0:
3,x73x34x66xcxfAix+Biy +Ciz +Di = 0(i = 1;2;3)x11x12x2dx2ex1dx1ex2ax6ax15x30x26x31x32.
(1)x47x48x66x66x28x17; (2)x47x48x31x66x28x2ex74;
￠ 6 ￠
(3)x34x66xcxfxcxd; (4)x34x66xcxfx75x2ax34x14x20.
x5,xbx3x6ex3x48x74x26x40x41x42 8
><
>:
A1x+B1y +C1z =?D1
A2x+B2y +C2z =?D2
A3x+B3y +C3z =?D3
(?)
x38x15x6axex5dx5ex42x16x1fx5dx5ex11x12x1fx22Ax42?A.
(1)x34x66xcxfx47x48x66x66x28x17 () x40x41x42(?)x47x2cx48x2d () rank(A) = rank(?A) = 3 () jAj6= 0.
(2)x34x66xcxfx47x48x31x66x28x2ex74 () x40x41x42(?)x47x2d,x25x3f(?)x15xcx25x40x41x42x15x7ax8x2dx6ax7bxax48x66
x1fx20 () rank(A) = rank(?A) = 2.
(3)x34x66xcxfxcxd () AiA
j
= BiB
j
= CiC
j
6= DiD
j
1 6 i < j 6 3.
(4) x34x66xcxfx75x2ax34x14x20 () x40x41x42 (?) x2cx2d,x25 (?) x15xcx25x40x41x42x15x7ax8x2dx6axax48x66x1fx20
() rank(A) = 2,rank(?A) = 3,x25x3fAx16x0x1x37xdx6dx55x2ax65x6a.
x1 x2 4–4
1,x7cx12x1dx1ex47x48x46x5dx22x74x26x46x5d?
(1)x6bx1fx20x70x71V x16,x41(?) = fi,x3cx16fix22x49x1dx1fx20;
(2) x41, K2?! K3
(x;y) 7?! (?1;2;3)
(3) x41, K3?! K3
(x1;x2;x3) 7?! (2x1+x2?x3;?x2+x3;x1+2x2?x3)
(4) x41, K3?! K2
(x;y;z) 7?! (x2 +y2?z;xy)
(5) x41(x"1 +y"2 +z"3) = (x+y)"1 +(x?y +z)"2 +(y?z)"3,x3cx16"1;"2;"3x22x74x26x70x71V x15x7a;
(6)x27x8x70x71R2x16,x52x22xcxfx5bxcx52x21x40x1fx22x4bx17x1bx1c45–x15x3dx4a.
x5,(1)x8fi = 0,x13;x8fi 6= 0,x55x13.
(2)x55x13.
(3)x13.
(4)x55x13.
(5)x13.
(6)x13.
2,x2x3cx79x61x16x15x74x26x46x5d,x73x25x38x15x6bx65x2cx7ax1dx15x5dx5e(x8x7ax2dx54x7a,x4ax7ax67x11x7a).
x5,(1) fi = 0x52x22x6fx5dx5e.
(3)
0
@
2 1?1
0?1 1
1 2?1
1
A.
(5)
0
@
1 1 0
1?1 1
0 1?1
1
A.
(6)
cos45–?sin45–
sin45– cos45–
.
￠ 7 ￠
3,x18x41x22x1fx20x70x71V1x8x1fx20x70x71V2x15x74x26x46x5d,fi1;fi2;￠￠￠ ;fim 2 V1,x41(fii) = fli,i = 1;2;￠￠￠ ;m.
x53x54:x8x1ffl1;fl2;￠￠￠ ;flmx74x26x2cx2a,x4afi1;fi2;￠￠￠ ;fimx67x74x26x2cx2a.
x3x4,x18k1fi1 +k2fi2 +￠￠￠+kmfim = 0,x4a
x41(k1fi1 +k2fi2 +￠￠￠+kmfim) = 0
)k1x41(fi1)+k2x41(fi2)+￠￠￠+kmx41(fim) = 0
)k1fl1 +k2fl2 +￠￠￠+kmflm = 0;
x4ex3cfl1;fl2;￠￠￠ ;flmx74x26x2cx2a,x3ex50k1 = k2 = ￠￠￠ = km = 0,x43x25fi1;fi2;￠￠￠ ;fimx74x26x2cx2a.
4,x1dxfx9x16x15(1)–(7)x6dx13x9(0)x6ax4ex2bx6axex5dx5ex15x74x26x3dx4ax25x50x8x15,x9(0)x16x55x25x72x4bx17Ox68
x7ax1fx20·1;·2,x19x72x4ex64x1dx7ax1fx20x6bx9(1)–(7)x16x15x23x24x68x38x15x2ax3c·1;·2x15x57x55(x4bx22x2bxe)x24x7ex25x65x2c
x74x26x3dx4ax15x5dx5e.
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x70x30
x10x0x58x58x5cx26xbx8x20x50x72
x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
O ·1
·2
(0)
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x70x30
x10x0x5x1ax44x28x28x72x50x20x8
x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(1)
x8x20x50x72x18x18x3c
x16x3
x0x10x30x70 x8x20x50x72x18x18x3c
x16x3
x0x10x30x70
x8x20x50x72x18x18x3c
x16x3
x0x10x30x70 x8x20x50x72x18x18x3c
x16x3
x0x10x30x70x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(2)
x0x10x30x70x28x28x44x1ax5x8x20x50x72 x0x10x30x70x28x28x44x1ax5x8x20x50x72
x0x10x30x70x28x28x44x1ax5x8x20x50x72 x0x10x30x70x28x28x44x1ax5x8x20x50x72x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(3)
x0x10x30x70xbx26x5cx58x58x72x50x20x8 x0x10x30x70xbx26x5cx58x58x72x50x20x8
x0x10x30x70xbx26x5cx58x58x72x50x20x8 x0x10x30x70xbx26x5cx58x58x72x50x20x8x5
x75x75x72x40x20x8
x75x75x72x40x20x8
x75x75x72x40x20x8
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
x74
x74
x70x0
x10x0
(4)
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
xcx28x60x73x3bx3cx3bx3cx3bx4x18x40x71
x5x71x71
x71x71
x71x20
x18x4
x71x71
x71x71
x71x20
x18x4
x71x71
x71x71
x71x20
x18x4x73x73
x73x73
x73x60
x28xc
x73x73
x73x73
x73x60
x28xc
x73x73
x73x73
x73x60
x28xc
(5)
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x73x60
x28xc
xex2bx66x6bx6c
x6bx6cx6c
x6bx6c
x70x70x70
x70x70x68
x2c
x5
xcx28x60x73
x73x73
x73x73
xcx28x60x73
x73x73
x73x73
xcx28x60x73
x73x73
x73x73
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x68x2c
(6)
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x73x60
x28xc
x39x14x13
x13
x74
x0x30x10
x5
x73x73
x73x73
x73x60
x28xcx73
x73x73
x73x73
x60x28xcx73
x73x73
x73x73
x60x28xc
x10x30x70
x74
x74
x74
x74
x74
x10x30x70
x74
x74
x74
x74
x74
x10x30x70
x74
x74
x74
x74
x74
(7)
x04x1x2
x5,(1)
1 0
0 1
.
(2)
0?1
1 0
.
(3)
1 0
0?1
.
(4)
1 0
0?1
.
(5)
1?1
1 1
.
(6)
1 1
2?1
.
(7)
0 1
2?1
.
5,x47x48x66x38x3bx221x15x2bx40x10x15x73x66x1bxfx6dx24x79x72x65x43x15x25x26x27x15xcxfx9,x1fx28x18x78x29x2ax1dx2bx1a
x3dx34x2cx8x23x53x15x2dx62x2ex16x17x58x25x38x15x2bx10x9(x8x5ax9),x2fx30x31x7bx26x2x72xfx15x9x36x2fx37x66x74x26x3dx4ax6fx63
￠ 8 ￠
x50x8x7exfx3ax44xfx15x37x66x9x36(x22x2fx30?),x8x1fx4ex73x66x44xfx15x1dx1dx35x7ax22x4bx17,x1ax7ex25x7exfx3ax1ex44xfx15x9
x36x2x2cx15x3dx4ax5dx5e.
x74
x74
x74
x74
x74
x74
x70x10x0
x74
x74
x74
x74
x74
x74
x70x10x0
x75x75x75x75x75x75x50x20x8
x75x75x75x75x75x75x50x20x8
x1fx1fx1e
x1fx1fx1f
x1ex1fx1f
x4x74
x74
x74
x74
x74
x74
x70x8x20x50x75x75x75x75x75x75
x4x18x40x71
x71x71
x71
x71x71
x71x71
x40x18x4
x05x1x2
x5,x7exf:
1 p2
4
0
p2
4
!
,x1ex44xf:
p2
4 0p
2
4 1
!
.
x1 x2 4–5
1,x6bx27x8x70x71x16x7ax2ex35x55x56[O;?!i ;?!j ;?!k ],x41;x42;x43x11x12x1bx1cx70x71x5bx1ex32x6ax22xx0 yx0 zx45x1bx1c45–
x15x3dx4a.
(1)x24x57x55x15x36x29x7ex25x41;x42;x43x15x1bx50x29;
(2)x73x41;x42;x43x6bx7a?!i ;?!j ;?!k x1dx15x5dx5e;
(3)x73x41x42,x42x41,x41x42x43,x41 + x42,x414x424x6bx7a?!i ;?!j ;?!k x1dx15x5dx5e;
(4)x53x54,x418 = x428 = x438 = x45,x77x1ex45x1bx1cx36x43x46x5d.
x5,(1) x41(x;y;z) =
x;
p2
2 y?
p2
2 z;
p2
2 y +
p2
2 z
,
x42(x;y;z) =
p
2
2 x+
p2
2 z;y;?
p2
2 x+
p2
2 z
,
x43(x;y;z) =
p
2
2 x?
p2
2 y;
p2
2 x+
p2
2 y;z
.
(2) A =
0
B@
1 0 0
0
p2
2?
p2
2
0
p2
2
p2
2
1
CA,
B =
0
B@
p2
2 0
p2
20 1 0
p2
2 0
p2
2
1
CA,C =
0
BB
@
p2
2?
p2
2 0p
2
2
p2
2 00 0 1
1
CC
A.
(3) AB =
0
BB
B@
p2
2 0
p2
2
1
2
p2
2?
1
2
12
p2
2
1
2
1
CC
CA,BA =
0
BB
B@
p2
2
1
2
1
2
0
p2
2?
p2
2
p2
2
1
2
1
2
1
CC
CA,ABC =
0
BB
B@
1
2?
1
2
p2
2
1
2 +
p2
4
1
2?
p2
4?
1
2
1
2?
p2
4
1
2 +
p2
4
1
2
1
CC
CA,A+B =
0
BB
B@
1+
p2
2 0
p2
2
0 1+
p2
2?
p2
2
p2
2
p2
2
p2
1
CC
CA,
￠ 9 ￠
A4B4 =
0
@
1 0 0
0?1 0
0 0 1
1
A.
(4)x69.
2,x78x67x1dx1ex5dx5ex15x33x67:
(1) A =
0
@
1 1 3
2 1 2
2 3 1
1
A,B =
0
@
1?1 1
0 2?1
1 2 0
1
A;
(2) A =
0
@
a b c
b c a
c a b
1
A,B =
0
@
c b a
a c b
b a c
1
A;
x73AB,AB?BA,(A?B)2.
x5,(1) AB =
0
@
4 7 0
4 4 1
3 6?1
1
A,AB?BA =
0
@
3 4?2
2 5?2
2 3?8
1
A,(A+B)2 =
0
@
6 0 8
1 8 4
3 2 6
1
A.
(2) AB =
0
@
ac+ba+cb ac+ba+cb a2 +b2 +c2
ac+ba+cb a2 +b2 +c2 ac+ba+cb
a2 +b2 +c2 ac+ba+cb ac+ba+cb
1
A,
AB?BA =
0
@
(b?c)(a?b)?(a?c)(a?b) (a?b)2
(a?c)(a?b) (a?b)2 (b?c)(a?b)
(a?b)2 (b?c)(a?b)?(a?c)(a?b)
1
A,
(A+B)2 =
0
@
(a?c)(a+b?2c) 0?(a?c)(a+b?2c)
(a?b)(a+b?2c) 0 (a?b)(a+b?2c)
(b?c)(a+b?2c) 0 (b?c)(a+b?2c)
1
A.
3,x78x67:
(1)
0
@
2 2 1
2 1 2
1 2 2
1
A
2; (2)
0
@
1?1 1
0 1?1
1 0 1
1
A
3;
(3)
0 1
1 1
5; (4)
cos?sin
sin cos
n;
(5) (a b c)
0
@
a
b
c
1
A; (6)
0
@
a
b
c
1
A(a b c);
(7)
0
@
1 0
0? 1
0 0?
1
A
n; (8) (?En +A)n,A =
0
BB
B@
1 1 ￠￠￠ 1
1 1 ￠￠￠ 1
:::::::::::::
1 1 ￠￠￠ 1
1
CC
CA.
x5,(1)
0
@
9 8 8
8 9 8
8 8 9
1
A.
(2)
0
@
3?2 5
3 0?2
2 3?3
1
A.
(3)
3 5
5 8
.
(4)
cosn?sinn
sinn cosn
.
(5) (a2 +b2 +c2).
￠ 10 ￠
(6)
0
@
a2 ab ac
ab b2 bc
ac bc c2
1
A.
(7)
0
B@?
n n?n?1 n(n?1)
2?
n?2
0?n n?n?1
0 0?n
1
CA.
(8)?n
E? 1n A
·
+ 1n (?+n)nA.
4,x78x67x5dx5ex9x3ax29,x18
(1) f(?) =?3?3?2?2,A =
0
@
3 1 1
3 1 2
1?1 0
1
A;
(2) f(?) =?3?2?2 +1,A =
0
@
1 2 0
1?1 1
0 1 2
1
A.
x5,(1)
0
@
12 2 4
11 3 3
1 1 1
1
A.
(2)
0
@
1 2 0
1?1 1
0 1 2
1
A.
5,x8x1fAB = BA,x75x5dx5eAx42Bx3ex3x4a.x18
(1) A =
1 0
1 2
; (2) A =
0
@
1 1 0
1 1 0
1 1 1
1
A;
(3) A =
0
@
1 1 1
0 1 1
0 0 1
1
A.
x73x23x47x42Ax3ex3x4ax15x5dx5e.
x5,(1)
a 0
b a+b
.
(2)
0
@
a b 0
b a 0
c c a+b?c
1
A.
(3)
0
@
a b c
0 a b
0 0 a
1
A.
6,x18
A = diag(a1;a2;￠￠￠ ;an); x3cx16ai 6= aj,8i 6= j.
x53x54:x42Ax3ex3x4ax15x5dx5ex7bx63x13x2x35x5dx5e.
x3x4,x18B = (bij)x42Ax3ex3x4a,x4a
aibij = bijaj; i;j = 1;￠￠￠ ;n:
x3cx13
(ai?aj)bij = 0; i;j = 1;￠￠￠ ;n:
￠ 11 ￠
x71x62i 6= jx52x47ai 6= aj,x23x24x2x3ci 6= jx47bij = 0,x4
B =
0
BB
B@
b11 0 ￠￠￠ 0
0 b22 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ bnn
1
CC
CA:
7,x53x54:x42x23x47x5dx5ex3ex3x4ax15x5dx5ex7bx63x13x55x20x5dx5e.
x3x4,x10x11x55x20x5dx5ex42x23x47x5dx5ex3ex3x4a.x18Bx42x23x47x5dx5ex3ex3x4a,x4ax4ex79x61xbB = diag(b1;￠￠￠ ;bn).
x51x2x0x1x15i 6= jx47BEij = EijB,x21x4fx2i 6= jx47bi = bj,x4b1 = b2 = ￠￠￠ = bn = b,x23x24B = bEn.
8,x53x54:x55x31x6bx5dx5eA,B,x27AB?BA = En.
x3x4,AB?BAx15x2x35x74xbx58x39x3a=
nP
i=1
nP
k=1
aikbki
nP
i=1
nP
k=1
bikaki
= 0,x25Enx15x2x35x74xb
x58x39x3a= n,x3ex5fAB?BA 6= En.
9,x18A = B +E,x53x54,A2 = 2Ax62x3fx63x62B2 = E.
x3x4,()) B2 = (A?E)2 = A2?2A+E = E.
(() A2 = (B +E)2 = B2 +2B +E = 2(B +E) = 2A.
10,xaxbxex9Kx79x15x37x66x40x5eAx42Bx3ex3x4a.x53x54:
(1) (A+B)2 = A2 +2AB +B2;
(2) (A+B)(A?B) = A2?B2;
(3) (A+B)n =
nP
k=0
CknAkBn?k.
x3x4,x69.
11,x53x54:x79(x1d)x34x35x36x5dx5ex15xfx10x5fx13x79(x1d)x34x35x36x5dx5e.
x3x4,x18A = (aij)x42B = (bij)x6dx13x79x34x35x36x5dx5e,x4x2i > jx47aij = 0x24x68bij = 0,x3cx13x62i > j
x52x47
nX
k=1
aikbkj =
i?1X
k=1
aikbkj +
nX
k=i
aikbkj
=
i?1X
k=1
0￠bkj +
nX
k=i
aik ￠0 = 0;
x21x4fABx13x79x34x35x36x5dx5e,x2x3cx1dx34x35x36x5dx5ex67x3ex24x6bx6cx6dx53x54.
12,x18Ax22m￡nx5dx5e,x53x54,rankA = 1x15x30x11x40x26x31x32x13x31x6bmx46x6ex6fx1fx20fi = (a1;a2;￠￠￠ ;am)
x42nx46x6ex6fx1fx20fl = (b1;b2;￠￠￠ ;bn),x27A = fiTfl.
x3x4,()) x18rank(A) = 1,x4aAx40x47x48xd(x18x22fl = (b1;￠￠￠ ;bn))x55x56x3c0,x25x3cx24x28xdx6dx13x77x48
xdx15x18xe,x43x25
A =
0
BB
B@
a1b1 a1b2 ￠￠￠ a1bn
a2b1 a2b2 ￠￠￠ a2bn
...,..,..,..
amb1 amb2 ￠￠￠ ambn
1
CC
CA =
0
BB
B@
a1
a2
...
am
1
CC
CA(b1 b2 ￠￠￠ bn) = fiTfl:
(()x18fi;flx13x37x66x6ex6fx1fx20,x4ax40x47x6ex66ai 6= 0,bj 6= 0,x43x25aibj 6= 0,x27x50A = fiTfl = (aibj) 6= 0.
x3cx13
1 6 rank(A) 6 minfrank(fi);rank(fl)g = 1:
13,x73x25xcx40x22x6fx15x23x47x62x79x40x5e.
￠ 12 ￠
x5,x18A =
a
11 a12
a21 a22
,A2 = 0,x8x1fb12 = 0,x4a
A2 =
a2
11 (a11 +a22)a12
0 a222
= 0;
x3cx13a11 = a22 = 0,A =
0 a
0 0
.
x12x18a12 = b 6= 0,a11 = a,x4ex3c0 < rank(A) = 1 < 2,x4ax47A =
a ka
b kb
,x3cx13
A2 =
a(a+kb) ka(a+kb)
b(a+kb) kb(a+kb)
= 0:
x4ex3cb 6= 0,x3ex50a+kb = 0,k =?ab, x21x4fx5dx5eAx15x3ex63x36x29x13
0 a
0 0
x44
a?a2
bb?a
:
14,x18Ax22m￡nx5dx5e,x53x54:x31x6bn￡sx6ex6fx5dx5eB,x27AB = 0x15x30x11x40x26x31x32x13rankA < n.
x3x4,())x18x47x6ex6fx5dx5eBx27x50AB = 0,x4aBx15x1ex1fx20x6dx13x3x48x74x26x40x41x42AX = 0x15x2d,x25x3f
x3cx16x47x6ex6fx2d,x21x4frankA < n.
(()x18rankA < n,x4ax3x48x74x26x40x41x42AX = 0x47x6ex6fx2d0
B@
b1
...
bn
1
CA6= 0:
x49
B =
0
B@
b1 0 ￠￠￠ 0
...,..,..,..
bn 0 ￠￠￠ 0
1
CA
n￡s
6= 0;
x4aAB = 0.
15,x18A,Bx11x12x22m￡nx42n￡sx5dx5e,x53x54:x8x1fAB = 0,x4a
rankA+rankB 6 n:
x3x4,Bx15x1ex1fx20x6dx13x3x48x74x26x40x41x42AX = 0x15x2d,x25x77x66x3x48x74x26x40x41x42x15x2dx70x71x6x9xax47
n?rankAx66x74x26x2cx2ax15x1fx20,x43x25
rankB 6 n?rankA;
x39x3ax50rankA+rankB 6 n.
16,x18Ax22n￡rx5dx5e,Bx22r￡sx5dx5e,rankB = r,x53x54:
(1)x8x1fAB = 0,x4aA = 0;
(2)x8x1fAB = B,x4aA = E.
x3x4,(1)x4ex79x61,rankA+rankB 6 r,x4erankB = rx3ex50rankA = 0,x43x25A = 0.
(2)x21x22(A?E)B = 0,x4e(1)x50A?E = 0,A = E.
17,x18Ax22m￡nx5dx5e,x53x54:x8x1fx2x23x47x15nx46x1fx20X = (x1 x2 ￠￠￠ xn)Tx6dx47AX = 0,x4aA = 0.
x3x4,x4ex31x18xbx14x2fx5dx5ex15x1ex1fx20x67x13AX = 0x15x2d,x21x4fA = AE = 0.
x1 x2 4–6
1,x78x67x1dx1ex5dx5ex15xcx5dx5e:
￠ 13 ￠
(1)
1 2
4 3
; (2)
2 5
3 7
;
(3)
0
@
1 1 0
0 1 1
0 0 1
1
A; (4)
0
@
2 0 1
1 1 0
3 2 1
1
A;
(5)
0
@
2 1 0
1 2 0
0 0 1
1
A; (6)
0
@
4 2 1
1 0 3
2 1 2
1
A.
x5,(1)? 15
3?2
4 1
.
(2)
7 5
3?2
.
(3)
0
@
1?1 1
0 1?1
0 0 1
1
A.
(4) 13
0
@
1?2 1
1 1 1
5 4?2
1
A.
(5) 13
0
@
2?1 0
1 2 0
0 0 3
1
A.
(6) 13
0
@
3 3?6
4?6 11
1 0 2
1
A.
2,x73x1dx1ex5dx5ex15x34x35x5dx5e:
(1)
0
@
2 3 1
1 2 3
3 2 1
1
A; (2)
0
@
2 1?2
2 2 1
1 2 1
1
A;
(3)
0
@
2 0 0
1 2 0
1 2?3
1
A; (4)
0
@
2 3 4
0?1 2
3?2 1
1
A.
x5,(1)
0
@
4?1 7
8?1?5
4 5 1
1
A.
(2)
0
@
0?5 5
3 4 2
6?3 6
1
A.
(3)
0
@
6 0 0
3?6 0
4?4 4
1
A.
(4)
0
@
3?11 10
6?14 4
3 5 2
1
A.
3,x18A 2 Mn(K),x53x54:x8x1fx31x6bx36xex3ax6ex6fx15x9x3ax29f(x),x27f(A) = 0,x4aAx3exc.
x3x4,x18f(x) = a0xm +a1xm?1 +￠￠￠+am?1x+am,am 6= 0,x27
f(A) = a0Am +a1Am?1 +￠￠￠+am?1A+amE = 0;
￠ 14 ￠
x4a
A
a0a
m
Am?1? a1a
m
Am?2?￠￠￠? am?1a
m
E
= E;
x21x4fAx3exc.
4,x18B3 = 0,x53x54,E?Bx3exc,x57x73E?Bx15xc.
x3x4,x21x22(E?B)(E +B +B2) = E?B3 = E,x23x24E?Bx3exc,x3f(E?B)?1 = E +B +B2.
5,x18A3 = 2E,B = A2 +2A?E,x73B?1.
x3x4,x4ex5dx5ex40x41x42 8
><
>:
B = A2 +2A?E
AB = 2A2?A+2E
A2B =?A2 +2A+4E
x72x4ex17x5x6x7x44x27x56x29x1ex38x7bxaE,x3ex50(5A2 +4A?3E)B = 31E,x21x4f
B?1 = 131 (5A2 +4A?3E):
6,x18A2 = A,x53x54,E +Ax3exc,x57x73(E +A)?1.
x3x4,x18B = E +A,x4aA = B?E,x21x4f(B?E)2 = B?E,B2?3B =?2E,B(3E?B) = 2E.
x21x4fB?1 = 12 (3E?B) = 12 (3E?E +A) = E? 12 A.
7,x18A;B 2 Mn(K),x53x54:x8x1fAB = kEn (k 6= 0),x4aBA = kEn.
x3x4,x4eAB = kEn (k 6= 0)x3ex50A?1 = 1k B,B = kA?1,x21x4fBA = kA?1A = kE.
8,x53x54,(1)x79(x1d)x34x35x36x5dx5ex15x34x35x5dx5ex5fx13x79(x1d)x34x35x36x5dx5e;
(2)x3excx15x79(x1d)x34x35x36x5dx5ex15xcx5dx5ex5fx13x79(x1d)x34x35x36x5dx5e.
x3x4,(1)x18
A =
0
BB
B@
a11 a12 ￠￠￠ a1n
0 a22 ￠￠￠ a2n
...,..,..,..
0 0 ￠￠￠ ann
1
CC
CA:
x4ax62j > ix52,aij x15x24x0x29Mij x5fx13x79x34x35x36x15,x3fMij x15(i;i)xbx58= Ax15(i + 1;i)xbx58= 0,x23x24
Mij = 0(j > i),x43x25x62j > ix52x47Aij = 0,x21x4fx34x35x5dx5eA?x13x79x34x35x36x5dx5e,x6bx6cx3ex53x1dx34x35x36x15xf
x36.
(2)x8Ax3exc,x4aA?1 = 1jAj A?xex22x34x35x36x5dx5e.
9,x53x54:x2x0x8nx79x40x5eA,x40x31x6b?0 2 K,x27x50?0En?Ax13x3excx5e.
x3x4,x21x22j?E?Ajx13x51x3ax6axex221x15nx48x9x3ax29,x25nx48x9x3ax29x6bKx79x6x9x47nx66x3d,x21x40x31x6b
0 2 Kx27x50j?0E?Aj6= 0,x43x25?0E?Ax3exc.
10,x18A =
1 0
2 1
,x73x9x3ax29f(x),x27f(A) = A?.
x5,A? =
1 0
2 1
=
1 0
2 1
4
0 0
1 0
= A?4
h1
2 (A?E)
i
= 2E?A,x23x24f(x) =?x+2.
11,x53x54:x2x23x47x15A 2 M2(K),x31x6bf(?) = a?+b,x27f(A) = A?.
x3x4,x18A =
a b
c d
,x4aA? =
d?b
c a
,x23x24A + A? = (a + d)E,A? =?A + (a + d)E,x21
f(?) =+(a+d).
￠ 15 ￠
12,x18A 2 Mn(K),x53x54:
rankA? =
8>
<
>:
n; rankA = n
1; rankA = n?1
0; rankA < n?1
x3x4,(i)x62rankA = nx52,Ax3exc,jAj6= 0,x25AA? = jAjE,x21A?x67x3exc,x43x25rankA? = n.
(ii)x62rankA < n?1x52,Ax15n?1x79x0x29x6dx56x3c0,x21x4fAij = 0,A? = 0,x23x24rankA? = 0.
(iii)x62rankA = n?1x52,Ax45x46x47x48x66n?1x79x0x29x55x56x3c0,x23x24A? 6= 0,x58x54rankA? > 1,x16
x48x40xfx47AA? = jAjE = 0,x23x24rankA + rankA? 6 n,x4ex3crankA = n?1,x3ex50rankA? 6 1,x21x4f
rankA? = 1.
13,x18A 2 Mn(K) (n > 2),x53x54:
(1) (A?)? = jAjn?2A;
(2) jA?j = jAjn?1.
x3x4,x62rankA = nx52,AA? = jAjE,x23x24jAjjA?j = jAjn,jA?j = jAjn?1,x3cx13A?(A?)? = jA?jE =
jAjn?1E,x4eA?1 = 1jAj A?x3ex50A? = jAjA?1,x21(A?)? = jAjn?1(A?)?1 = jAjn?2A.
x62rankA = n?1x52,rankA? = 1,x23x24(A?)? = 0 = jAjn?2A,jA?j = 0 = jAjn?1.
x62rankA < n?1x52,A? = 0,x79x53x56x29x67x2ax2b.
x1 x2 4–7
1,x18A,Bx22x37x66x43x79x5dx5e,x53x54:
rank(A+B) 6 rank(A j B) 6 rankA+rankB:
x3x4,x18Ax15x1ex1fx20x42x22fi1;￠￠￠ ;fin,Bx15x1ex1fx20x42x22fl1;￠￠￠ ;fln,x4aA + Bx15x1ex1fx20x42x22fi1 +
fl1;￠￠￠ ;fin +fln,(A j B)x15x1ex1fx20x42x22fi1;￠￠￠ ;fin;fl1;￠￠￠ ;fln,x43x25x4ex60x614–1.7,
rank(A+B) = rankffi1 +fl1;￠￠￠ ;fin +flng
6 rankffi1;￠￠￠ ;fin;fl1;￠￠￠ ;flng = rank(A j B)
6 rankffi1;￠￠￠ ;fing+rankffl1;￠￠￠ ;flng
= rankA+rankB:
2,x18
A =
0
BB
B@
0 0 1 1
0 0 0 1
2 1 1?3
3 2 1 2
1
CC
CA;
x73A?1.
x5,x18
A?1 =
B
11 B12
B21 B22
;
￠ 16 ￠
AA?1 =
0 A
12
A21 A22
B
11 B12
B21 B22
=
A
12B21 A12B22
A21B11 +A22B21 A21B12 +A22B22
=
E
2 0
0 E2
:
x23x24
B21 = A?112 =
1?1
0 1
;
B22 = 0; (x21A12x3exc)
B12 = A?121 =
2?1
3 2
;
B11 =?A?121 A22B21 =
1 9
1?14
:
x21x4f
A?1 =
0
BB
@
1 9 2?1
1?14?3 2
1?1 0 0
0 1 0 0
1
CC
A:
3,x18Ax22x3excx15nx79x40x5e,
D =
0 A
a 0
; a 6= 0;
x73D?1.
x5,D?1 =
0 a?1
A?1 0
.
4,x18Aix22rix79x3excx40x5e(i = 1;2;￠￠￠ ;s),
A =
0
BB
B@
A1
A2
...
As
1
CC
CA
x73A?1.
x5,A?1 =
0
BB
B@
A?1s
A?1s?1
...
A?11
1
CC
CA.
5,x18Eix22ri (i = 1;2;￠￠￠ ;s)x79x14x2fx5dx5e,x25
A =
0
BB
B@
a1E1
a2E2
...
asEs
1
CC
CA; ai 6= aj; i 6= j;
x53x54:x42Ax3ex3x4ax15x5dx5ex7bx63x13x11x37x2x35x5dx5e.
x3x4,x18x11x37x5dx5eB = (Bij)x42Ax3ex3x4a,x25x3fBx15x11x37x40x29x42Ax65x43.x4ax4eAB = BAx50
aiBij = Bijaj; i;j = 1;￠￠￠ ;s:
x3cx13
(ai?aj)Bij = 0; i;j = 1;￠￠￠ ;s:
￠ 17 ￠
x71x62i 6= jx52x47ai 6= aj,x23x24x2x3ci 6= jx47Bij = 0,x4
B =
0
BB
B@
B11 0 ￠￠￠ 0
0 B22 ￠￠￠ 0
...,..,..,..
0 0 ￠￠￠ Bss
1
CC
CA:
6,x18
A =
0
BB
BB
BB
@
0 0 ￠￠￠ 0 a0
a1 0 ￠￠￠ 0 0
0 a2 ￠￠￠ 0 0
...,..,..,..,..
0 0 ￠￠￠ an 0
1
CC
CC
CC
A
ai 6= 0; i = 0;1;2;￠￠￠n;
x73A?1.
x5,A?1 =
0
BB
BB
BB
@
0 a?11 0 ￠￠￠ 0
0 0 a?12 ￠￠￠ 0
...,..,..,..,..
0 0 0 ￠￠￠ a?1n
a?10 0 0 ￠￠￠ 0
1
CC
CC
CC
A
.
7,x18x5dx5eAm￡s,Bt￡nx15x9x11x12x22rA,rB,Cx22x0x1x15m￡nx5dx5e,x25
D =
A C
0 B
x53x54:x5dx5eDx15x9rD > rA +rB.
x3x4,x18Ax15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42x22fii1;￠￠￠ ;fiirA,B x15xdx1fx20x42x15x1cx3bx74x26x2cx2ax42x22
flj1;￠￠￠ ;fljrB,x4a
1 = (fii1;?￠￠￠?| {z }
n
); 2 = (fii2;?￠￠￠?| {z }
n
);￠￠￠ ; rA = (fiirA;?￠￠￠?| {z }
n
)
x74x26x2cx2a.
–1 = (0￠￠￠0| {z }
s;flj1);–2 = (0￠￠￠0| {z }
s;flj2);￠￠￠ ;–rB = (0￠￠￠0| {z }
s;fljrB)
x74x26x2cx2a,x10x11
1; 2;￠￠￠ ; rA;–1;–2;￠￠￠ ;–rB;
x74x26x2cx2a,x23x24
rD >f 1; 2;￠￠￠ ; rA;–1;–2;￠￠￠ ;–rBg = rA +rB:
8,x18Ax22x62x79x40x5e,x53x54:x8x1fAk = 0,x4aA2 = 0.
x3x4,x4eAk = 0x3ex50jAj = 0,x21rankA 6 1,x8x1frankA = 0,x4aA = 0,x22x23x10x11x2ax2b,x8x1f
rankA = 1,x4a
A =
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn); (x60x614–5.12)
A2 =
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn)
0
B@
a1
...
an
1
CA(b
1 ￠￠￠ bn) =
nX
i=1
aibiA;
Ak =
nX
i=1
aibi
!k?1
A = 0:
￠ 18 ￠
x4ex3cA 6= 0,
nP
i=1
aibi = 0,x23x24A2 = 0.
9,x18A,Bx22x37x66nx79x40x5e,x53x54:x5dx5ex40x41AX = Bx47x2dx15x30x11x40x26x31x32x13rankA = rank(A j B).
x3x4,())x18x5dx5ex40x41AX = Bx47x2dX = C,x4aAC = B,x43x25Bx15x1ex1fx20x42x3ex4eAx15x1ex1fx20x42
x74x26x1bx1c,x23x24
rank(A j B) = rankA:
(()x8x1frank(A j B) = rankA,x4aBx15x1ex1fx20x42x3ex4eAx15x1ex1fx20x42x74x26x1bx1c,x4x31x6b(c1j;￠￠￠ ;cnj)T
x27x50
A
0
B@
c1j
...
cnj
1
CA =
0
B@
b1j
...
bnj
1
CA; j = 1;￠￠￠ ;n:
x49
C =
0
B@
c11 ￠￠￠ c1n
...,..,..
cn1 ￠￠￠ cnn
1
CA;
x4aAC = B.
10,x18A,Bx22x37x66nx79x40x5e,x53x54:x3x48x74x26x40x41x42AX = 0x42x3x48x74x26x40x41x42BAX = 0x43x2dx15
x30x11x40x26x31x32x13rankA = rankBA.
x3x4,x51xa,AX = 0x15x2dx6dx13BAX = 0x15x2d,x43x25BAX = 0x15x7ax8x2dx6ax45x46xax47n?rankAx66
x2d,x51x21x22rankA = rankBA,x23x24BAX = 0x15x7ax8x2dx6ax38xax47n?rankAx66x2d,x21AX = 0x15x7ax8x2d
x6ax67x13BAX = 0x15x7ax8x2dx6a,x21x4fAX = 0x42BAX = 0x43x2d.
x4dx39,x8x1fAX = 0x42BAX = 0x43x2d,x4ax38x15x15x7ax8x2dx6axax47x65x43x66xex15x2d,x21x4fn?rankA =
n?rankBA,rankA = rankBA.
x1 x2 4–8
1,x1ax5cx56x3dx4ax73x1dx1ex5dx5ex15xcx5dx5e:
(1)
0
@
1?2 1
2 1 0
1 2 0
1
A; (2)
0
@
2 3?2
1 0?1
1?2 1
1
A;
(3)
0
BB
B@
1 2 0 0
0 1 1 0
0 0 1 1
0 0 0 1
1
CC
CA; (4)
0
BB
B@
1 1 1 1
1?1 1 1
1 1?1 1
1 1 1?1
1
CC
CA.
x5,(1) 13
0
@
0 2?1
0?1 2
3?4 5
1
A.
(2) 16
0
@
2?1 3
2?4 0
2?7 3
1
A.
(3)
0
BB
B@
1?2 2?2
0 1?1 1
0 0 1?1
0 0 0 1
1
CC
CA.
(4) 14 A.
￠ 19 ￠
2,x2dx1dx1ex5dx5ex40x41:
(1)
2?3
2 4
X =
4 3
2 2
;
(2) X
0
@
1 1 1
1 2 1
1 0?1
1
A =
0
@
1 0 1
1 1 2
1?1 0
1
A;
(3)
0
BB
BB
B@
1 1 1 ￠￠￠ 1 1
0 1 1 ￠￠￠ 1 1
0 0 1 ￠￠￠ 1 1.
..,..,..,..,..,..
0 0 0 ￠￠￠ 0 1
1
CC
CC
CAX =
0
BB
BB
B@
2 1 0 ￠￠￠ 0 0
1 2 1 ￠￠￠ 0 0
0 1 2 ￠￠￠ 0 0.
..,..,..,..,..,..
0 0 0 ￠￠￠ 1 2
1
CC
CC
CA.
x5,(1) X =
2?3
2 4
1 4 3
2 2
= 12
4 3
2 2
4 3
2 2
=
11 9
6 5
.
(2) X =
0
@
1 0 1
1 1 2
1?1 0
1
A
0
@
1 1 1
1 2 1
1 0?1
1
A
1
= 14
0
@
4?2?2
2 1?5
2?3?1
1
A.
x440
BB
BB
BB
BB
BB
BB
@
1 1 1
1 2 1
1 0?1
1 0 1
1 1 2
1?1 0
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
2 1 1
2 1?1
0?1 1
3?1 2
1?1 0
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
1 0 0
1 2 0
0?1 1
32? 52 12
1
2?
1
2
1
2
1
CC
CC
CC
CC
CC
CC
A
!
0
BB
BB
BB
BB
BB
BB
@
0 0 1
1 0 0
0 1 0
12? 12 1
1
4?
5
4
1
2
34? 14 12
1
CC
CC
CC
CC
CC
CC
A
:
(3) X =
0
BB
BB
BB
B@
1?1?1 0 ￠￠￠ 0 0 0
1 1?1?1 ￠￠￠ 0 0 0
0 1 1?1 ￠￠￠ 0 0 0
:::::::::::::::::::::::::::::::::
0 0 0 0 ￠￠￠ 1 1?1
0 0 0 0 ￠￠￠ 0 1 2
1
CC
CC
CC
CA
.
3,x1ax9x62x40x44x73
A =
0
BB
B@
1 1 1 1
1?1 1?1
1 1?1?1
1?1?1 1
1
CC
CA
x15xcx5dx5e.
x5,x63x39x3ax37x62x2dx44:
(i)x21x22AA = 4E,x23x24A?1 = 14 A.
(ii)x11x37,A1 =
1 1
1?1
,A?11 = 12
1 1
1?1
= 12 A1.
A
1 A1 E 0
A1?A1 0 E
!
A
1 A1 E 0
0?2A1?E E
!
A1 0 12 E 12 E
0?2A1?E E
!
!
E 0 12 A?11 12 A?11
0 E 12 A?11? 12 A?11
!
:
￠ 20 ￠
A?1 = 12
A?1
1 A
1
1
A?11?A?11
= 14
A
1 A1
A1?A1
= 14 A:
4,x18A;B;C;D 2 Mn(K),jAj6= 0,AC = CA,x53x54:fl
flfl
fl
A B
C D
flfl
flfl = jAD?CBj:
x3x4,x21jAj6= 0,x21Ax3exc,x25
E 0
CA?1 E
A B
C D
=
A B
0 D?CA?1B
;
x23x24 fl
flfl
fl
A B
C D
flfl
flfl =
flfl
flflA B
0 D?CA?1B
flfl
flfl = jAjjD?CA?1Bj
= jAD?ACA?1Bj = jAD?CBj:
5,x18A;B 2 Mn(C),x53x54,fl
flfl
fl
A B
B A
flfl
flfl = jA?iBjjA+iBj:
(x3cx16ix22x3bxex14x2f,i2 =?1.)
x3x4,x21x22
E iE
0 E
A B
B A
E?iE
0 E
=
A?iB
A+iB
:
x23x24 fl
flfl
fl
A B
B A
flfl
flfl = jA?iBjjA+iBj:
6,x18A 2 Mm;r(K),x53x54:
(1) Ax22x1ex2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx3excx5dx5eP 2 Mm(K),x27A = P
E
r
0
;
(2) Ax22x1ex2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bxdx2dx9x5dx5eB 2 Mr;m(K),x27BA = Er.
x3x4,(1)x21Ax1ex2dx9,Ax15x3cx3dx36x22
E
r
0
,x43x25x31x6bx3excx5dx5eP1;Q1,x27
A = P1
E
r
0
Q1:
x49
P = P1
Q
1 0
0 Em?r
;
x4a
A = P1
E
r
0
Q1 = P1
Q
1
0
= P1
Q
1 0
0 Em?r
E
r
0
= P
E
r
0
:
x77x6fx53x54x72x40x26x26,x25x30x11x26x13x10x11x15.
(2)x30x11x26x13x10x11x15,x12x53x40x26x26.
x4e(1)xb,x31x6bx3excx5dx5eP,x27A = P
E
r
0
,x4a
P?1A =
E
r
0
;
x49
P?1 =
B gr
B1
;
x4aBxdx2dx9,x3fBA = Er.
7,x2x3cxdx2dx9x5dx5e,x3ex53x57x53x54x6bx6cx15x22x23.
￠ 21 ￠
x5,(1) Ax22xdx2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx3excx5dx5eQ 2 Mm(K),x27A = (Er 0)Q.
(2) Ax22xdx2dx9x5dx5ex15x30x11x40x26x31x32x13x31x6bx1ex2dx9x5dx5eB 2 Mm;r(K),x27AB = Er.
(x53x54x69)
8,x18m￡nx5dx5eAx15x9x22r,x53x54:x31x6bx1ex2dx9x5dx5ePx3axdx2dx9x5dx5eQ,x27A = PQ.
x3x4,x31x6bx3excx5dx5eP1;Q1,x27
A = P1
E
r 0
0 0
Q1:
x49
P = P1
E
r
0
; Q = (Er 0)Q1;
x4aPx1ex2dx9,Qxdx2dx9,x3f
A = P1
E
r 0
0 0
Q1 = P1
E
r
0
(Er 0)Q1 = PQ:
9,x18A,Bx11x12x22n￡mx42m￡n(n > m)x5dx5e,? 6= 0,x53x54:
j?En?ABj =?n?mj?Em?BAj:
x3x4,x4ex3c
En 0
B Em
E
n A
B?Em
=
E
n A
0?Em?AB
;
E
n?A
0 Em
E
n A
B Em
=
E
n?AB 0
B?Em
;
x23x24
j?En?ABj =
flfl
flfl?En A
B Em
flfl
flfl;
j?Em?BAj =
flfl
flflEn A
0?Em
flfl
flfl:
x25
mj?En?ABj =?m
flfl
flfl?En A
B Em
flfl
flfl =
flfl
flfl?En A
B?Em
flfl
flfl
=?n
flfl
flflEn A
B?Em
flfl
flfl =?nj?Em?BAj:
x21x4f
j?En?ABj =?n?mj?Em?BAj:
10,x18A,Bx11x12x22s￡nx42n￡mx5dx5e,x53x54:
rank(AB) > rank(A)+rank(B)?n:
x3x4:
AB 0
0 En
!
AB?A
0 En
!
0?A
B En
,x23x24
rank(AB)+n = rank
AB 0
0 En
= rank
0?A
B En
= rank
0 A
B En
> rankA+rankB; (4–7.7)
x21x4frank(AB) > rank(A)+rank(B)?n.
11,x18A,B,Cx11x12x22s￡n,n￡mx42m￡tx5dx5e,x53x54:
rank(ABC) > rank(AB)+rank(BC)?rankB:
￠ 22 ￠
x3x4:
ABC 0
0 B
!
ABC AB
0 B
!
0 AB
BC B
!
0 AB
BC B
:
x23x24
rank(ABC)+rankB > rank(AB)+rank(BC);
rank(ABC) > rank(AB)+rank(BC)?rankB:
12,x18A 2 Mn(K),x53x54:
A2 = En,rank(A?En)+rank(A+En) = n:
x3x4:
E
n A+En
A?En 0
!
E
n A+En
0 En?A2
,x23x24
rank(A?En)+rank(A+En) = n+rank(En?A2):
rank(A?En)+rank(A+En) = n () rank(En?A2) = 0 () A2 = En:
13,x18A 2 Mn(K),x53x54:
A2 = A,rank(A)+rank(A?En) = n:
x3x4:
A 0
0 A?En
!
A?E
n
0 A?En
!
0?E
n
A2?A A?En
!
0 E
n
A2?A 0
:
x23x24
rankA+rank(A?En) = n+rank(A2?A):
rank(A)+rank(A?En) = n,A2 = A:
14,x18A 2 Mn(K)x13x3excx5dx5e,X;Y x22nx46x1ex1fx20,x53x54:fl
flfl
fl
A Y
XT 0
flfl
flfl =?XTA?Y:
x3x4,
A Y
XT 0
!
A Y
0?XTA?1Y
;
)
flfl
flfl A Y
XT 0
flfl
flfl = jAjj?XTA?1Yj =?XTjAjA?1Y =?XTA?Y:
x1 x2 4–9
1,x18x41 x22x1fx20x70x71V1x8V2x15x74x26x46x5d,x41 x6bx67x11x7ax1dx15x5dx5ex13
A =
0
BB
B@
1 0 2 1 3
2?3?1 1?4
1?3 1 2?1
1 1 1 0 2
1
CC
CA:
(1)x73x41 x15x3fx42x23x15x46xex42x7a;
(2)x11x12x76x41 x15x3fx42x23x15x7ax1dx30x22V1x42V2x15x7a.
x5,(1) x21x22rankA = 3,x21x4fx23x70x71x15x46xex223,Ax15x1ex1fx20x42x15x1cx3bx74x26x2cx2ax42x75x2ax23x70x71x15
x7a:
x41("1) = (?1;2;1;?1);x41("2) = (0;?3;?3;1);x41("5) = (3;?4;?1;2):
￠ 23 ￠
x3fx15x46xe= 5?3 = 2,AX = 0x15x7ax8x2dx6ax75x2ax3fx70x71x15x7a:
1 = (2;1;1;0;0);?2 = (1;1;0;1;0):
(2)?1;?2x3ex1dx30x22V1x15x7a:
1;?2;"3;"4;"5:
x41("1);x41("2);x41("5)x3ex24x1dx30x22V2x15x7a:
x41("1);x41("2);x41("5);(1;0;0;0):
2,x18x41 x22K3x15x74x26x3dx4a,x27
x41(x;y;z) = (x+y?z;x+y +z;x+y?2z):
(1)x73x41 x15x6fx77x42x9;
(2)x73x41 x15x3fx42x23x70x71.
x5,(1)x6fx77= 1,x9= 2.
(2)x3f= L((1;?1;0)),x23= L((1;1;1);(?1;1;?2)).
3,x18W1,W2x22V x15x37x66x0x70x71,x3fdimW1+dimW2 = n,x53x54:x31x6bx74x26x3dx4ax41,x27Kerx41 = W1,
Imx41 = W2.
x3x4,x18W1x15x7ax22fi1;￠￠￠ ;fir,W2x15x7ax22fl1;￠￠￠ ;fln?r,x76W1x15x7ax1dx30x22V x15x7a,fi1;￠￠￠ ;fir;
fir+1;￠￠￠ ;fin,x2x0x1x15fi =
nP
i=1
aifii 2 V,x1dx4d
x41(fi) =
n?rX
i=1
an+ifli;
x4ax41 x13V x15x74x26x3dx4a,x3fKerx41 = W1,Imx41 = W2.
4,x18x41x22nx46x1fx20x70x71x15x74x26x3dx4a,V1,V2x22V x37x66x74x26x0x70x71,x53x54:x8x1fKerx41 = V1 \V2,x4a
x31x6bx74x26x3dx4ax411,x412,x27V1 Kerx411,V2 Kerx412,x3fx41 = x411 + x412.
x3x4,x18 1;￠￠￠ ; r x22V1 \ V2 x15x7a,x76x38x1dx30x22V1 x15x7a,1;￠￠￠ ; r;fi1;￠￠￠ ;fit,x1dx30x22V2 x15x7a:
1;￠￠￠ ; r;fl1;￠￠￠ ;fls,x4ax4ex46xex66x29xb 1;￠￠￠ ; r;fi1;￠￠￠ ;fit;fl1;￠￠￠ ;flsx22V1 + V2x15x7a,x12x4ex38x1dx30x22
V x15x7a:
1;￠￠￠ ; r;fi1;￠￠￠ ;fit;fl1;￠￠￠ ;fls;·1;￠￠￠ ;·u; (r +t+s+u = n):
x11x12x1dx4dx74x26x3dx4ax8x1d:
x411( i) = 0; x411(fii) = 0; x411(fli) = x41(fli); x411(·i) = x41(·i);
x412( i) = 0; x412(fii) = x41(fii); x412(fli) = 0; x412(·i) = 0:
x4axex53V1 Kerx411,V2 Kerx412,x3fx41 = x411 + x412.
5,x18x41;x42x22nx46x1fx20x70x71V x15x37x66x74x26x3dx4a,x3fx41 2 = x41,x422 = x42,x53x54:
(1) Imx41 = Imx42,x41x42 = x42;x42x41 = x41;
(2) Kerx41 = Kerx42,x41x42 = x41;x42x41 = x42.
x3x4,(1) ())x2x0x1x15fi 2 V x31x6bfl 2 V,x27x50x41(fi) = x42(fl),x23x24
x41(fi) = x42(fl) = x422(fl) = x42(x42(fl)) = x42x41(fi):
x4x41 = x42x41,x43x5x3ex53x42 = x41x42.
(() x42(V) = x41x42(V) x41(V),x41(V) = x42x41(V) x42(V),x23x24Imx41 = Imx42.
￠ 24 ￠
(2) ())x2x0x1x15fi 2 V,x4ex3c
x42[(x42?x45)(fi)] = x422(fi)?x42(fi) = x42(fi)?x42(fi) = 0;
x23x24x41[(x42?x45)(fi)] = 0,x3cx13x41x42(fi) = x41(fi),x41x42 = x41,x43x5x3ex53x42x41 = x42.
(()x2x0x1x15fi 2 Kerx42x47
x41(fi) = x41x42(fi) = x41(0) = 0;
x23x24Kerx42 Kerx41,x43x5x3ex53Kerx41 Kerx42,x21x4fKerx41 = Kerx42.
￠ 25 ￠
