Лекции по ЛинАл

–

	a x		+ a x + + a x = b ,
		11 1	12 2		1n n		1
	a x + a			x + + a		x = b ,		(1.1)
		21 1		22 2		2n n	2
			+ am 2 x2 + ... + amn xn = bm .
	am1 x1
m n , aij					– , bi			–

, x1 ,..., xn – ( ).

– .

. .

! (1.1) . " #

:		a		a	a		b
			11	12	1n		1
A	=	a21		a22	a2n		b2		.
				am 2	amn
		am1					bm

$ :

a	a	a
11	12	1n
A = a21	a22	a2 n .
am1	am 2	amn

% (1.1)

.

§1.

, . & :

a11

A = a21

am1

! m n . ' aij

a	a
12	1n
a22	a2n = A = (a ) .
( m×n)			ij
			( m×n)
am 2	amn
= ai, j	A .

, ,

.
& .
1)	( : A + B = C . !					A , B ,	C . % cij		= aij + bij .
						( m×n) ( m×n) ( m×n)
.		1	2 0	−1	1	1
			+	−2	=	.
		3	4 2		5	2
2)	)	:			A α = B . !		A ,	B , α .	% bij = α aij .
							( m×n)	( m×n )

	1		2		3		6
.		0	−3			0	−9
	3				=			.
		1	1			3	3

3) * : AT = B . ! A ,	B . % b = a		ji	.
( m×n)	( n×m)	ij

% . + B

A , B

A .

		T	1	−1	1 3		T	1		−1
.		1 2					4		3	0
			=					=
		−1 3	2	3		−1 0	−1		4	−1

		x				y	T
		1				1
%, ( x1	x2	xn )T = x2		,		y2		= ( y1 y2 yn ) .
		xn				yn

4) % :	AB = C . ! A , B , C .
	( m×n) ( n× p ) ( m× p )
n
% cij = aik bkj = ai1b1 j	+ ai 2b2 j + + ainbnj .
k =1

% « »: i

A j B . %

. & cij . %

A										B .
.
1		0		1	0	1					3			1	0	1					0
	2			0	1	1		=		1 2		;		0	1	1			1	2		– ;
										−1 3
0	1	1		−1 2		0					1			−1 2		0			0	1	1

1		1 1		1		1
	(1 1 1)						= (3).
1		= 1	1	1	;	(1 1 1) 1
			1
1		1		1		1
%		,

, – .

& . !

.

A B =		C		B A =		D ,	C D .
( n×n) ( n×n)	( n×n)			( n×n) ( n×n)		( n×n)
		1	0 0 1	0	1		0	1	1	0 0	0
.				=		,				=	.
		0	0 0 0	0	0		0	0	0	0 0	0
%:		A B ≠ B A
. AB ≠ BA , . . .
, A0					B0 - A0 B0 = B0 A0 ,
A0			B0 ( ).

– , .

:

( AB) C = A (BC) ;

A (B + C) = AB + AC ;

( A + B) C = AC + BC ;

. .

+

( A + B )T = BT + AT , ( A B )T

= BT AT .

: ( AB)C = A(BC) .

, ,

.

.

(( AB)C)D = ( A(BC))D = A((BC)D) = A(B(CD)) = ABCD .

.

.

.

% γ kp – k = 1, , m ,

p = 1, , n .

m

n

n

m

*

γ kp

= γ kp

(1.2) ,

. . .

k =1 p =1

p =1 k =1

(1.2).

γ11

γ12

γ1n

m n

: = γ 21

γ 22

γ 2n ,

" : γ kp ≡ α .

k =1 p =1

γ m1

γ m 2

γ mn

: ,

.

n

m

γ kp ≡ β

– ,

p =1 k =1

.

, . ( α = β .

.

/

, γ kp

, ( ,

), . % (1.2) -

.

.

.

. A =

A , B =

B , C = C . ( AB)C = A(BC) .

( m×n )

( n× p )

( p×q )

.

A B = F = F .

( m× p )

: ( AB) C = F C = U =

U .

( m×q )

0: BC = G = G ,

A(BC) = A G = V = V .

( n×q )

( m×q )

, . +

p

p

n

p

n

uij = fik ckj =

( ais bsk ) ckj =

ais bsk

ckj

, / i, j – , /.

k =1

k =1 s=1

k =1 s=1

n

n

p

n

p

vij = ais gsj =

ais ( bsk ckj ) =

ais bsk ckj .

s=1

s=1

k =1

s=1 k =1

% (1.2) uij = vij ( i, j ).

* , , ( AB)C = A(BC) .

§2.

& n × n .

( 2 × 2 ).			2 3				1 0 2 3								1 0		2 3 2 3
								=			,						=			.
			5	−1			0 1 5				−1				0 1		5 −1 5		−1
1 ,											1		0		.
											0		1
			M =		1	0
%							2 × 2 .
					0	1
!:
1,	i = j
δij =	i	≠ j	, i, j = 1, n .
0,
.		" (n × n)
	1	0	0
E = 0 1 0							= (δij ) .
							( n×n )
	0	0			1
. #$ A , . AE = EA = A, A .
.
						n
% AE = B . * bij					= aik			δkj		= aij	δ jj	= aij		. ( B = A .
						k =1
						n
% EA = C . * cij					= δik			akj		= δii	aij	= aij . ( C = A .
						k =1
. :																	AE = EA = A,		A.
% .
. AJ = JA = A,								A , J = E .
.			)			J E .
( J = JE = E . .
.						/ B A , AB = BA = E .
. " , .
.			% AB = BA = E AC = CA = E ( . .
, A ). *: C = CE = C ( AB) = (CA) B = EB = B .
(, .
* ,
: A−1 – .
:														A−1 A = AA−1 = E.
+ ,		A−1	.

( 2 × 2 ).	0	1		A2	0	1 0	1 0		0
	A =		,		=			=		= Θ –	.
	0	0			0	0 0	0	0	0

% AB = BA = E.

* A = AE = A( AB) = A2 B = ΘB = Θ , . %,

.

! !:

1./ ( n × n ), ,

Θ = Θ .

( n×n )

2./ A , AD = Θ D ≠ Θ

CA = Θ C ≠ Θ .

3./ A % ( ), A ≠ Θ , Ap = Θ ,

p ≥ 2

. " A – % ,

.

. & # . % AD = Θ

D ≠ Θ . % , A , AB = BA = E

B . * D = ED = (BA)D = B( AD) = BΘ = Θ ,

D ≠ Θ . 0

.

, ,

. 0

A .

2.1. ( )

& % ' (2% ) A

:

I.;

II. ;

III. , .

% .

2.2.

.

. A = A . .

( n×n )

1.+ A E . %

:

	a		a	a		0	0
					1
		11	12	1n
( A | E) =	a21		a22	a2n	0	1	0
					.
				ann	0	0	1
	an1		an 2

2.- 2% . (

2% (

).

3.!, 2% ,

. (E | B) . *

B = A−1 .

.	0 , A−1 .
	1	1	1
.		2	3		, A−1 = ?
	A = 1
		3	6
	1

1 1 1				0	0		1 1			1		1 0		0		1 1			1		1	0	0
			1
												−1									−1 1
1	2	3	0	1	0		~	0	1	2			1	0		~	0	1	2				0		~
	3	6	0	0	1			0	2 5			−1	0	1			0	0 1			1	−2 1
1

1 1 0					2	−1			1	0	0		3	−3	1					3	−3	1
				0
	0	1	0	−3 5		−2			0	1	0		−3 5 −2				A	−1		−3 5 −2
								~								.			=				.
	0	0	1	1	−2 1				0	0	1		1	−2	1					1	−2	1

	1 1 1				3	−3	1			1	0	0
%.	1	2	3		−3	5	−2		=	0	1	0	.
		3	6		1	−2	1			0	0	1
	1

2.3.

&

x1 , , xn

y1 , , yn .

a x + a x + + a

x = y ,

11

1

12 2

1n

n

1

,

! y , , y

x , , x

a

21

x + a x + + a

x = y

1

n

1

n

1

22 2

2n

n

2

. ! A = (aij ) – .

( n×n )

an1 x1 + an 2 x2 + + ann xn = yn .

&

x1 , , xn y1 , , yn . %-

a

a

a

x

y

11

12

1n

1

1

a21

a22

a2 n x2

= y2 .

an1

an 2

ann xn

yn

+ Ax = y , A = (aij ) , x – - x1 , , xn ,

y – - y1 , , yn .

% A−1 .

% - Ax = y :

A−1 ( Ax) = A−1 y

( A−1 A)x = A−1 y Ex = A−1 y x = A−1 y .

.

. $

x y

.

x1

x2

xn

y1

y2

yn

x1

x2

xn

y1

y2

yn

a11

a12

a1n

1 0 0

1 0 1

b11

b12

b1n

a2 n

0 1

0 1 0

a21

a22

0

→ Ax = y x = A−1 y

b21

b22

b2n .

2%

ann

0 0

1

0 0 1

an1

an 2

bn1

bn 2

bnn

2%

x y .

x = By . ( - x = A−1 y ,

, B = A−1.

§3.

. , A−1 . .

%. , .

. / ,

.

3.1.

, , ,

, .

& 1, 2, , n ( n - ).

.

1

2

n

= ( j1 j2

jn ) .

:

j =

j

j

j

1

2

n

1, n

Pn

– n .

Pn

– n , :

Pn

= n! .

$ , (0! = 1! = 1,

2! = 2, 3! = 6 ) ,

(n +1)! = (n +1)n!.

&

j = ( j1

j2 jn ) . (, jk > jm , k < m ,

- - , (

« »).

σ ( j) ( .

(−1)σ ( j )				, .
(−1)σ ( j ) > 0						j	– #,
(−1)σ ( j ) < 0						j	– .
.			)
j =	(				+1 +1 = 4	– .
	(	3 2 4 1 , σ ( j) = 2
j =			)		+ 2 +1 = 6	– .
		4 3 2 1 , σ ( j) = 3
j = (2 1 3 4) , σ ( j) = 1 – .
j =	(	1 2 3 4	)	, σ ( j) = 0	– .
.					(	)
							n . .
% j = 1 2 n

& j = ( j1 j2 jn ) . % % '

j .

j .

". ) % ' %.

. & .

1. 2 –		jk jk +1 .
Τ
j = ( j1 jk jk +1 jn )→ j = ( j1 jk +1 jk jn )	( «…», ).
! jk	jk +1 . σ ( j) 1

(−1)σ ( j ) = −(−1)σ ( j ) .

2.. % # jk jm ,

p .

Τ

j =

j j j

j

j

j j

→

j =

j j j

j

j

1 2

m n

1 2

k k +1

m−1

m k +1

m−1

p

p

jk jn .

!- Τ .

( jk	jk +1	jm−1	p +1 -	jm−1	jm	p -
			jm ) →( jk +1			jk ) →
( jm	jk +1	jm−1	jk ) = j . (2 p +1) - . ! (2 p +1) ,
(−1)σ ( j ) = −(−1)σ ( j ) .

. , n

n .

3.2.		a	a	a
		11	12	1n
& (n × n) :	A = (a ) = a21		a22	a2 n
	ij
	( n×n )
		an1	an 2	ann

% A – ( det A, A ), A :

det A =		(−1)σ ( j ) a1 j			a2 j anj		.
				1	2	n
	j Pn
%		1	2	n		– , a1 j					a2 j	anj – det A ,
	j =	j1	j2
					jn					1	2	n
						j . +		(	)	.
									−1 σ ( j ) –

,

. ' , , j . % j . det A . n! .

det A .
a				a	a			a
11				12	13		14
a21				a22	a23		a24				1	2	3	4 - ,
A =										, j	=
a31				a32	a33		a34				3	2	4	1 - .
				a42	a43		a44
	a41
1 ,							σ ( j) = 2 +1 +1 = 4 – . 0 j → (−1)4 a									a	a	a .
															13	22	34	41
.
		0		2	0		0
1. A =			1	0	0		0			1	2	3	4	σ ( j) = 2 - ,
							−3		,	j =			.	*:	1 (−3) 4		= −24.
			0	0	0					2	1	4	3	det A = 2
			0	0	4		0

2.	.
	1	0	0
E = 0		1	0
					det E = 1 1 1 = 1			det E = 1.
	0	0		1
	a		a	a	a
		11	12	13	1n
		0 a22		a23	a2n
3.	A =	0	0	a	a		– .
				33	3n
		0	0	0
					ann

det A = ann a33 a22 a11 = a11 a22 a33 ann ( ).

,

, .

. ,

.

3.3.

", det A = (−1)σ ( j ) a1 j	a2 j	anj .	(1.2)
1	2	n
j Pn
. . :
fk = (ak1 ak 2 akn ) – k - A.
det A = det[ f1 , f2 , , fn ] – , .
1. , , .
. (1.2)			,
det A = 0 .

2.% ,

: det[ , λ fk , ] = λ det[ , fk , ] .

(1.2).

3., - ,

:

det[ , f (1)	+ f ( 2) , ] = det[ , f (1)		, ] + det[ , f ( 2) , ] .
k	k	k		k
(1.2).

( 2 3 , , (

).

m

#.

det[ , λ1 fk(1)

+ λ2 fk( 2) + + λm fk( m) , ] = λj det[ , fk( j ) , ] .

j =1

4. % :

det[ , fk , , fm , ] = − det[ , fm , , fk , ] .

. % A – ,

B – , A

k - m - ( k < m ). *:

det B = (−1)σ ( j ) b1 j

b2 j

bnj .

3 :

1

2

n

j = ( j1 j2 jn )

j pn

det B , j :

b1 j bkj

bmj

bnj

= a1 j

amj

akj

anj

= a1 j

akj

amj

anj .

1

k

m

n

1

k

m

n

1

m

k

n

det A					j , :
j = ( j j	m	j	k	j ) . 1 , j	j .
1				n
'	j		j	– . , det A

det B . + , . ( det A = − det B .

. ( 4 , .

5. , , .

. & : det[..., fk ,..., fk ,...] .

% k - m - .

* det[..., fk ,..., fk ,...] = − det[..., fk ,..., fk ,...] ( x = −x , x = 0 ),

det[..., fk ,..., fk ,...] = 0 .

6.% , ,

.

.

det[..., fk + λ fm ,..., fm ,...] = det[..., fk ,..., fm ,...] + λ det[..., fm ,..., fm ,...] = det[..., fk ,..., fm ,...] ≡ det A .

7. % . / /

#. ) ( % ( ( (1-6) « » $ '

« ».

( ). & . * ,

. %

. ( ,

.

$ . ,

:

(i). % . (ii). + . (iii). ) : det E = 1

/ , : (i-iii)

(1.2).

, .

. / A (n × n)	,
. A – $ .
". & % ' ( $
( $ ) .
.
/% . §2 ( )/.
% .
.		- .
# !.
. * A = A B =	B , det( AB) = det A det B .
( n×n)	( n×n)

% (2 × 2) . & :