# Матрицы как линейные операторы

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• 102

, 6 , 1 , 2 0 0 0

. .

-

- -. , . - -, , -. , . . , , - .

. . ,

y

=

kx

, - : , , , - . . . - -: , , - ..

1.

. (

O

;

i

,

j

),

O

- (-)

i

,

j

1

[1, 2]. , - .

- :

X

, -

O

i

,

Y

,

O

j

(. 1).

M

(

x

,

y

)

1

. .

MATRICES AS LINEAR OPERATORS

V. A. BRUSIN

Basic information about second and third ordermatrices is presented. Their connection withlinear transformations of planes and spaces isexplained. Examples of such transformationshave been considered.

. . - .

..,

2000

• . .

103

M

. [1, 2], - .

M

-

r

,

O

M

. - (

x

,

y

)

M

r

(

i

,

j

),

r

=

x

i

+

y

j

[1, 2]. ( -

O

, ,

O

.)

, - (

x

,

y

). -

R

2

.

1.

,

T

(

R

2

R

2

),

M

( )

M

' .

M

'

M

T

; :

M

' =

T

(

M

).

- , , , [1, 3]

1

.

2.

T

-

, - .

- .

-,

OY

(. 2). - ,

Y

.

3.

T

-

, - .

- .

1

T

,

M

(

x

,

y

)

M

'(

x

,

y

), - (

k

0), -

M

'(

kx

,

ky

).

4.

-

T

M

*, :

T

(

M

*) =

M

*. , ,

. - . , -, .

5.

I

, .

6.

,

T -

(

)

T

1

T

2

(:

T

=

T

1

T

2

,

T

=

T

1

T

2

),

M

T

M

' - :

M

T

2

,

N

',

N

'

T

1

,

M

'. , -

T

T

2

T

1

. . 3.

1.

O

.

2.

O

+

.

, -

T

1

T2 , , , T1T2 T2T1 . ( :

YM2

j

O i M1 X

M

r

. 1

O X

M2M1M'

Y

. 2

M M'

N'T2 T1

T1T2

. 3

• , 6 , 1 , 2 0 0 0104

T1 O, T2 .)

7. T - K, - TK = KT = I.

T K - T 1. : 1) T - M M ', T 1 M ' - M; 2) T 1 , T , , . -, J, - J, JJ = J2 = I.

2.

- . T - - . , --, , , -.

T(M) = M ' T(r) = r', r r' - M M '.

8. T -, [4]:

1) T(O) = O ( - );

2) r T(r) = T(r);

3) r1 r2 T(r1) ++ T(r2) = T(r1 + r2).

1. . , ( ), .

- - .

3.

, - [1, 5].

9. m n, m n , m n , , m

n . m- -, m = n ( ). m 1 m, 1 n n.

- . A

( -

, -, ).

.

1. . B

A , ,

2. . k , kA =

.

3. .

:(AB)C = A(BC) ( ,

: ABC).

. , AE = EA = A. - : AB BA.

-

.

1 2, Ab , col(a11b1 + a12b2 a21b1 + a22b2). - A -. a1 , a2 , A = (a1 a2). B = (b1 b2), b1, b2 B. , AB = (Ab1 Ab2). A, B , c , A(Bc) = (AB)c := ABc.

1 , col (column ).

A = a11 a12a21 a22

b11 b12b21 b22

,

A B = a11 b11+ a12 b12+

a21 b21+ a22 b22+

.+

= ka11 ka12ka21 ka22

, 0 A = 0 00 0

= O

A B = AB = a11b11 a12b21+ a11b12 a12b22+

a21b11 a22b21+ a21b12 a22b22+

.

E = 1 0

0 1

b = b1b2

=

= col b1 b2( )

• . . 105

10. D, - AD = DA = E, A A1. A , .

A |A |, , - [1, 5], , |AB | = |A | |B |, |E | = 1.

11. A , |A | = a11a22 a21a12.

10, 11 -

, A - , |A| 0. ( : - , )

. - - [1, 5]. .

. - ,, , - - .

. - TA , . M(x, y) ( - r) - M '(x', y') ( - r'), -

(1)

,

(1) :

(2)

, TA -.

2. E I. - -

A 1 = 1A

------ a22 a12

a21 a11

.

A = a11 a12a21 a22

x' = a11x a12y,+

y' = a21x a22y.+

x'

y' x

y

x'

y'

= A x

y

.

, , .

3. TA, -

A, , A .

1.

[3] O k. k > 1, -, k < 1, .

2. -

.

3. -

.

4. -

( ).

5.

- [1]. , S( + ) == S() S(), S(0) = E, S1() = S().

S() [1, 5], 1, () .

6. , a2 + b2 0,

- ax + by = 0 (. 4).

, ,

ax' + by' = 0, x, y,

r' r N ax + by = 0. N (a, b). -, r' r N - [1, 2].

(), X . M - M1, OX,

k 0

0 k

k > 0,,

1 0

0 1

PX = 1 0

0 0

1 0

0 1

S ( ) = cos sinsin cos

P = 1

a2 b2+---------------- b

2 ab

ab a2

x'

y'

= P x

y

• , 6 , 1 , 2 0 0 0106

M ax + by = 0. , M , M1 OX O. , P S()

PX S(). ,

3, 5, - P.

. -, ,. , - , - [5].

4.

, - , - . - .

O - i, j, k [1, 2]. , . : X , O i, Y , O j Z , O k. M - (x, y, z) M. M r, O M,

= b a2 b2+ =sin,cos

= a a2 b2+

(x, y, z) - r - r = x i + y j + zk. ( r, M, -- .) - R3.

, 17. 17 , - . - . 1 - M r -, . , - 8. 1.

9, A -

: , - , - . - . B

-. ai = (ai1 ai2 ai3) i- - A, bj = (b1j b2j b3j) j- B. , ai, bj ,

ai, bj = ai1b1j + ai2b2j + ai3b3j . (3)

(4)

A b -

A = a11 a12 a13a21 a22 a23a31 a32 a33

.

B = b11 b12 b13b21 b22 b23b31 b32 b33

.

AB =

a1 b1, a1 b2, a

1 b3,

a2 b1, a2 b2, a

2 b3,

a3 b1, a3 b2, a

3 b3,

.

Y

X

M

r

r' r

M'

r'

O

ax +

by =

0

. 4

• . . 107

Ab = col(a1, b, a2, b, a3, b). (5)

, .

10.

12. A |A |, [1, 2]

|A | = (a11a22a33 + a12a23a31 + a13a21a32)

(a13a22a31 + a11a23a32 + a21a12a33). (6)

A1 [1, 2]

(7)

Mij , - A i- j-

. ,

A TA - , M(x, y, z) ( - r) M '(x', y', z') ( - r'),

(8)

(8)

(9)

TA - . - (8) - aij. , .

E = 1 0 0

0 1 0

0 0 1

,

A 1 = 1A

------M11 M21 M31M12 M22 M32

M13 M23 M33

,

M32 = a11 a13a21 a23

= a11a23 a21a13.

x' = a11x a12y a13z,+ +

y' = a21x a22y a23z,+ +

z' = a31x a32y a33z.+ +

x'

y'

z' x

y

z

,

x'

y'

z'

= Ax

y

z

.

- 2 3.

1.

XOY.

2.

OZ. - SOX() SOY() OX OY. - - [4].

6 3 - , .

. - - (, , , ). - - , - .

1. .., .. . .: , 1980. 175 .

2. .. - . .: , 1980. 336 .

3. .. -. .: , 1980. 400 .

4. ., . - . .: , 1984. 831 .

5. . . .: , 1982. 270 .

..

* * *

, -- , , . -- , -- . - . 160 - .

1 0 0

0 1 0

0 0 0

SOZ ( ) = cos sin 0sin cos 0

0 0 1