Matrici i Determinanti
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1:
1
1. 1. 1.1 1. . m n , P
mxn P. ij , i -
, - ij.
Amxn = [aij]mxn , (i=1,...,m;j=1,...,n)
mxn. (A)ij i- j- A. :
3x2, B 3x3, C 3x1. 2: . , m=n n, n. B 3. mxn = [aij]mxn . m =1, A1xn =[ a12 a12 . . . . a1n] - n=1 ,
- .
=
mnmm
n
n
aaa
aaaaaa
A
..............
......
21
22221
11211
=
1
21
11
1
.
.
m
mx
a
aa
A
+=
=
=
ii
iCBA
2
1,
752021435
,52302/11
1:
2
n=
nnnn
n
n
aaa
aaaaaa
KKKKKKK
21
22221
11211
i=j , aij
i=1,,n, .
, , , .
3. . Amxn = [aij]mxn e . -Amxn = [-aij]mxn .
4. . , mxn Omxn.
5. . Dn = [dij] 0, dij=0 za ij, .
6. . C , d11=d22=. . . =dnn=c C
n.
=
c
cc
Cn
......0.........0...000...00
7. . nxn, 1 n En In E, I .
, 3,
=
100010001
3I .
8. . A = [aij]mxn. () A () A
=
nn
nxn
d
dd
D
0......0..........0...000...00
22
11
1:
3
AT A'. A'=[a'ij]nxm A=[aij]mxn a'ij = aji (AT)ij=(A)ji. :
9. . A = [aij]mxn B = [bij]pxq , m=p n=q aij=bij i=1,2,...,m; j=1,2,...,n. :
2x-y=1 x+y=2 x-y=0 x+3y=4. 1.1 , (AT)T = A . : Amxn=[aij]mxn ATnxm =[aij]nxm a'ij = aji. (AT)T=[aij] mxn, aij=aji=aij. 10. . n, Anxn aij=aji i,j=1,2,...,n; .
1.2 S n, , ST=S. 11. . = [aij] n. aij=-aji . 1.3 An T = -A.
=
=
241302
,210432 'AA
=
=
322212
312111
3231
2221
1211
aaaaaa
Aaaaaaa
A T
=
++
4021
32
yxyxyxyx
1:
4
1.2 . , (+) () : a,b,c 1. a+b, ab ();
2. a+(b+c)=(a+b)+c, a(bc)=(ab)c ( ); 3. a+b=b+a, ab=ba ( ); 3. , 0, (
) a, a+0=0+a=a;
4. a3 b a+b=b+a=0. ( .) a a;
5. , 1, ( ), a, a1=1a=a ;
6. a, a0, b, ab=ba=1. ( 0, .) a-1.
7. a,b,c, a(b+c)=ab+ac ( ).
, (+,), . , P + () (), 1-7 . , (+,) Q(+,), C(+,) . Z2={0,1}, 2, :
1:
5
+ 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1
, . 12. . A=[aij] B=[bij] mxn, C=[cij] mxn cij A B cij = aij + bij (i=1,2, . . . ,m; j=1,2, . . . ,n) A + B = C
:
:
C A B. 13. . A=[aij] mxn 3 C=[cij] mxn
cij = aij (i=1,2, . . . ,m; j=1,2, . . . ,n) C = A.
:
(-1)A -A. 14. A B = A + (-1) B. A, B C ,3 3 :
+++
++++++
=
+
mnmnmmmm
nn
nn
mnmm
n
n
mnmm
n
n
bababa
babababababa
bbb
bbbbbb
aaa
aaaaaa
...............................................
......
........................
......
.........................
......
2211
2222222121
1112121111
21
22221
11211
21
22221
11211
=
=
=3112
,010321
,025432
CBA
=
++=+
015113
001205342312
BA
=
==
=
082016128
042454443424
4,4,025432
AA
1:
6
1.4 A + B = B + A () 1.5 (A + B) + C = A + (B + C) () 1.6 A + O = O + A = A ( O A)
1.7 A + (-A) = (-A) + A = O 1.8 (A + B) = A + B
1.9 ( + ) A = A + A 1.10 () A = (A) 1.11 1 A = A 1.12 (A + B)T = AT + BT 1.13 (A)T = AT 15. A B mxn nxp C mxp,
C= AB. ij i - A j- B. , . a11 a12 . . . a1n b11 . b1j . b1p . . . . . . b21 . b2j . b2p
ai1 ai2 . . . ain . . . . . = . . . . . . . . . . .
am1 am2 . . . amn . . . . . bn1 . bnj . bnp
===
===
===
n
kkpmk
n
kkjmk
n
kkmk
m
kkpik
m
kkjik
n
kkik
m
kkpk
m
kkjk
n
kkk
bababa
bababa
bababa
1111
1111
11
11
111
......
.................
......
..................
......
:
),...,2,1;,...2,1(1
pjmibacn
kkjikij ===
=
++++++=
232213212222122121221121
231213112212121121121111
232221
131211
2221
1211
babababababababababababa
bbbbbb
aaaa
1:
7
: 1.14 (AB)C = A(BC) () : A mxn, B nxp C pxq . AB mxp, (AB)C mxq, BC nxq, A(BC) mxq. .
ij
n
ssjis
n
s
p
kkjskis
p
k
n
skjskis
p
kkjikij
BCABCA
CBACBACABCAB
))(()(
)()()())((
1
1 11 11
==
===
=
= == ==
:
1.15(A + B)C = AC + BC A(B + C) = AB + AC () ! AB BA. , . 16. . AB = BA A B . AB BA , C=AB-BA A B. 1.16 A = [aij] mxn , a D =[dii] nxn C = AD jjijij dac = . . A = [aij] mxn , a D =[dii] mxm C = DA cij=diiaij. , . : A = [aij] mxn , a D =[dii] nxn
C = AD jjijn
kkjikij dadac ==
=1,
dkj=0 kj. A = [aij] mxn , a D =[dii] mxm , C = DA ijii
n
kkjikij adadc ==
=1,
dik=0 ik.
=
++++++=
=
=
15125672
023512250215033213220312
,010321
,2532
BABA
1:
8
1.17 A e mxn , a In Im , AIn = ImA = A.
Ap = A A ...A (p=1,2,. . . ). 0=I. p
1.18 Ap Aq = Ap+q
1.19 (AB)T = BT AT. : A mxn, B nxp, AB mxp, (AB)T pxn . BT pxn, AT nxm, BTAT pxm . ,
ijTT
n
kkj
Tik
Tn
kik
Tkj
Tn
kkijkjiij
T BAABBABAABAB )()()()()()()()())((111
===== ===
17. . A () B
AB=BA=I. . :
.
1.20 . : AB=BA=I AC=CA=I.
B=BI=B(AC)=(BA)C=IC=C.
B=C, A . , A-1. ,
=
=
5273
3275
,1001
3275
5273
,1001
5273
3275
imatricite
1:
9
21 (A-1)-1=A 22 A B , AB
(AB)-1=B-1A-1. : A B .
(AB)(B-1A-1)=A(BB-1)A-1=AA-1=I, (B-1A-1)(AB)=B-1(A-1A)B=B-1B=I.
2.2 2.2.1 mxn :
1. () . 2. ()
. 3. ()
(). 2.2.2 . B A A B A~B. 2.2.3 . In . Eij i- j- . Ei() i- . Eij() j- i- . Eij, Ei() Eij() , , , : Eij = Eij, Ei() = Ei() Eij() = Eji() 2.2.1 3:
=
=
=
100010201
200010001
100001010
)2(13)2(312 EEE
1:
10
2.2.2.
=
100001010
1E
. A B
E1A E1B. :
2.2.3
=
=
=
100010201
200010001
100001010
' )2(13)2(312 EEE
2.2.1 Eij, Ei() Eij(), Eij, Ei() Eij().
: A mxn, Im m. Eks k- s- . ,
ik,s eij=
=
jiji
,1,0 ekj=
=
sjsj
,1,0
i esj=
=
kjkj
,1,0
.
B=EksA, B, j{1,,n} :
ik,s bij= ijijiim
lljil aaeae ==
=1,
sjsjks
m
lljklkj aaeaeb ===
=1, kjkjsk
n
lljslsj aaeaeb ===
=1
B k- s- A. Ek() k- . ,
1:
11
ik, eij=
=
jiji
,1,0 ekj=
=
kjkj
,,0
. B=Ek() A, B j{1,,n} :
ik, bij= ijijiim
lljil aaeae ==
=1, kjkjkk
m
lljklkj aaeaeb ===
=1,
B k- A . Eks() s- k- . , ,
ik eij=
=
jiji
,1,0 ekj=
==
sjkj
skj
,,1
,,0
.
B=Eks() A, B j{1,,n} : ik, bij= ijijii
m
lljil aaeae ==
=1,
sjkjsjkskjkk
m
lljklkj aaaeaeaeb +=+==
=1,
B s- k- A. A nxn . 2.2.2 A B mxn , A ( )
B 1, 2, . . . , k B = EkEk-1E2E1A, ( B = A E1E2Ek).
: A B, B A k . . B = EkEk-1E2E1A 1, 2, . . . , k B = EkEk-1E2E1A B A . 2.2.3 ()
,
1:
12
(Eij)-1 = Eij, (Ei())-1 = Ei(1/) (Eij())-1= Eij(-). 2.2.4 A=[aij] mxn
() mxn
=
0..0...000........0..0...000
......000.........
......0......
222322
11131211
rnrr
nr
nr
bb
bbbbbbbbb
B bii0 i=1,. . . , r.
2.2.5 A=[aij] mxn mxn
)()()(
)(
rnxrmxrrm
rnrxr
OOOI
., Ir r. 2.2.6
, .
2.2.7
. 2.2.8 .
, , -1.
: A () . , Ek . . . E2 E1 A = I, Ei i{1, 2, . . . ,k} , -1 Ek . . . E2 E1 AA-1 = I A-1 Ek . . . E2 E1 I = A-1
A. , , , A.
1:
13
2.2.4: =
011111121
.
: , [A|I] :
-1=
110011
111. -1=I
.
1:
14
2.3 2.3.1 .
A=[aij] n, tr A,
, : 2.3.1 tr A = trA
, 2.3.2 tr AT = tr A.
A .
a11a22-a12a21
2.3.1
. A 3.
=
=n
iiiaAtr
1
=2221
1211
aaaa
A
103)2(414321
det,4321 ====
= AAA
=
333231
232221
131211
aaaaaaaaa
A
211222112221
1211det aaaaaaaa
AA ===
1:
15
. , , -.
, , , . , . . 123, , . 123, 132, 231, 213, 312 321. , , 123, 231 312 , , 132, 231 321, . . , . 2.3.2 . S n .
S S . , S={a,b,c}
S. S n 1,2,...,n,
=
=
abccba
gacbcba
f ,
.
det
312213322113332112
312312322311332211
333231
232221
131211
aaaaaaaaa
aaaaaaaaaaaaaaaaaa
A
+
+==
.332112322311312213
322113312312332211
3231
2221
1211
333231
232221
131211
aaaaaaaaa
aaaaaaaaaaaaa
aa
aaaaaaaaa
++=
1:
16
S, S 1,2,...,n. ,
, I= (i1, i2, i3, . . . , in), ik {1,2,3,...,n}. 1 n. 2.3.3 n
123n=n! 2.3.3 . i, j
, i j, i>j.
, 2413 2, 1 , 4 , 1 4, 3 . , ? . . 2 2, 2. .. . , 35142 2 , 3 ( ), 0 , 1 , 2+3+0+1=6. 2.3.4. .
, . I= (i1, i2, i3, . . . , in) Ip .
, 35142 , 6 , 12345 0 , , 21345 . 2.3.4 ,
, , , .
niiiin
......321
321
1:
17
, n. 2.3.5 .
n, A n , ( ). ,
+, (i1, i2, ..., in) -, .
I=(i1, i2, . . . ,in) {1,2,...,n}, a Ip . det A. , , n!, n!/2 n!/2 . , , , .
=
mnmm
n
n
aaa
aaaaaa
A
..............
......
21
22221
11211
nniiiaaa ...
21 21
=
==),...,,(
21
21
22221
11211
21
21...)1(
............
...
...
detn
n
p
iiiIniii
I
nnnn
n
n
aaa
aaa
aaaaaa
A
1:
18
2.3.5 det AT = det A.
: . , . 2.3.6 ( ) ,
. : , 1 ( ) . . .
A. (i1, i2, ..., ik, ..., ij, ..., in), . 1 (i1, i2, ..., ik, ..., ij, ..., in), ik ij, - - . , 1 . det A1 = -det A. 2.3.2:
2.3.7 ( )
.
: , , , . , det A = -detA, , det A=0. 2.3.8 - n- ,
, .. akj=bkj+ckj, (j=1,2,...,n), 1
njk nijikiiiaaaaa .........
21 21
362141051
141362051
=
1:
19
2 , - , 1 bk1, bk2, ..., bkn, 2 ck1, ck2, ..., ckn.
: ,
A1 A2. 2.3.9 ( )
, .
: A=[aij] 1 - .
2.3.10 ( ) .
: , . 2.3.11 D , det D = a11a22...ann.
2.3.12
, . : 2.3.8,
nknk
nkknk
nikiiinikiii
nikikiiinikiii
acaaabaa
acbaaaaaa
............
)......(......
2121
2121
2121
2121
+=+=
AaaaaaaaaAI
kikiiiI
Inikiii
Ink
p
nk
p det......)1(......)1(det2121 21211
===
1:
20
, . 2.4 2.4.1 , . aij,
i- j- n, Aij n-1 i- j- . ( n-1) aij Aij Mij=detAij. ( ) aij
a*ij = (-1)i+j det Aij= (-1)i+j Mij
2.4.1:
2.4.1 A ,
aij, , = ijij aaAdet .
A
aaa
aaa
aaa
aaaaaa
aaa
aaa
aaa
aaaaaa
aaa
aaa
aaa
aaaaaa
aaa
aaa
aaa
aaaaaa
aaa
aaa
aaaaaa
aaaaaa
nnnn
jnjj
jnjj
n
n
nnnn
jnjj
knkk
n
n
nnnn
jnjj
jnjj
n
n
nnnn
jnjj
knkk
n
n
nnnn
jnjj
jnknjkjk
n
n
det
...........
.............
............
...
...
...........
.............
............
...
...
...........
.............
............
...
...
...........
.............
............
...
...
......................
......................
...................
.....
.......
21
21
21
22221
11211
21
21
21
22221
11211
21
21
21
22221
11211
21
21
21
22221
11211
21
21
2211
22221
11211
=+
=+=+++
3331
2321
3331
23212112
3331
232112
333231
232221
131211
)1(,,aaaa
aaaa
aaaaa
Aaaaaaaaaa
A ==
=
= +
1:
21
: i=j=1.
1111)...(
211)...(
212
2
21
21...)1(...)1(det MaaaaaaaA
n
n
p
n
n
p
iinii
I
iiiniii
I === 11 . i- , j- , aij , i-1 . . , j- j-1 . . 1 ,
AAA jiji det)1(det)1(det 111++ == . 1 aij
(1,1) ,
++ === ijijijijjiji aaMaAA )1(det)1(det 1 .
2.4.2
( ) . ,
: 2.3.8, i- n-1 , aij=0+. . . +0 +aij+0. . . +0,
= ijij
nnnjnn
ij
nj
aa
aaaa
a
aaaa
..............
0...00..00........
......
21
111211
i{1,2,. . . n}. ++= ininii aaaaA ....det 11 .
: aij A Aij n-1, ( i-) ( j-) .
= =
+
= =+
==
==n
j
n
jijij
jiijij
n
i
n
iijij
jiijij
MaaaA
iliMaaaA
1 1
1 1
)1(det
)1(det
1:
22
i- ( j- ) , . , . !
. aij aij* : 2.4.3
: ' i-
k- , 2.3.12, ' . ', i-
2.4.4 A B , det (AB) = det A det B. 2.4.2 . A*
(i,j) aij* aij A. (A*)T A.
2.4.5 A=[aij] n,
A(A*)T=(A*)TA=AIn
==+++
==+++
sjakosjakoA
aaaaaa
ikiakokiakoA
aaaaaa
nsnjsjsij
kninkiki
0det
...
0det
...
221
2211
.0
detdet)('det
1
1111
ikakoaa
AaaAaaaaaaaA
n
jijkj
n
jijkj
n
jijkj
n
jijij
n
jijkjij
=
=+=+=+=
=
=
=
=
=
1:
23
: (*).
bij=ai1aj1*+ai2aj2*+ . . . +ainajn* , 36 bii=A, bij=0 ij. A(A*)T A A(A*)T=AI, I A. (A*)TA=AI. 2.4.2:
, . 2.4.5 A A0.
: A0, . 2.4.5,
A(A*)T=(A*)TA=AIn
, A
A , A_1 AA-1=A-1A=I.
=
=
nnnn
iniji
nj
nnjnn
nj
nj
nnnn
inii
n
T
bbb
bbb
bbb
aaa
aaa
aaa
aaa
aaa
aaa
AA
............
............
......
............
......
......
........
........
...
*)(
21
1
1111
1
2212
1111
21
21
11211
IAAAA
AAAAko
TT
T
4400040004
315222404
212021
211*)(*)(
315222404
*)(,4,212021
211
=
=
==
==
=
IAAA
AA
A TT == )*)(1()*)(1(
TAA
A *)(11 =
1:
24
2.4.4 AA-1=I AA-1=1
A0. . k mxn. 2.4.3 k mxn
k- k- , kmin{n,m}.
2.4.4
r(A).
2.4.6 A n, A n. 2.4.7 mxn, r(A) min{n,m}. : 2.4.8 r(A-1)=r(A)=n. 2.4.9 r(AB) min{r(A),r(B)}.
2.4.10 r(ATA)=r(AAT)=r(A).
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