Matrix theory basics 2.2 matrix operations

I. Addition of Matrices

Set A and B to m rows, and the same type matrix of n columns

,

The matrix obtained by adding the elements at their corresponding locations, called the sum of A and B, is recorded as A + B.

For example 1, if the matrix is known, evaluate A + B.

Solution: A + B = + =

Note: Only the same type matrix can be used for addition.

2. Multiplication of numbers and Matrices

The matrix obtained by multiplying the number l by each element of matrix A is called the product of l and A. It is recorded as lA or Al and specified as lA = (laij). That is

In particular, when l =-1,

,

This matrix is called the negative matrix of matrix.

The subtraction operation of two identical matrices can be considered as A A-B = A + (-B)

Addition and subtraction of a matrix and multiplication of numbers and matrices are collectively referred to as linear operations of a matrix.

Linear operations meet the following calculation rules:

(1) addition exchange law:

(2) Addition law:

(3)

(4)

(5)

(6)

(7)

Example 2 known matrix

(1) A + B; (2) 3A; (3) 3A-2B

Solution (1)

(2)

(3)

Iii. Matrix Multiplication

There are two linear transformations:

Set x1, x2, and y1, y2, and y3 as offline relationships:

-- (1)

Coefficient Matrix

The relationship between y1, y2, y3 and z1 and z2 is as follows:

. -- (2)

Coefficient Matrix

 

If you want to obtain the linear transformation from x1, x2, z1, and z2, you can substitute (2) into (1 ).

. -- (3)

Coefficient Matrix

Linear transformation (3) can be seen as the result of first linear transformation (2) And then linear transformation (1. we call linear transformation (3) the product of linear transformation (1) and (2). Correspondingly, we define the corresponding matrix as (1) and (2) the product of the corresponding matrix, that is

.

 

Define 4. Set Am x p = (aij) m x p to an m' p matrix, and B p x n = (Bij) p x n to a P' n matrix, then, the product of matrix A and matrix B is counted as AB, which is defined as m 'n' matrix Cm 'n' = (cij) m'n', where

(I = 1, 2,
×
×
×, M; j = 1, 2,
×
×
× N ).

 

 

 

 

Example 3

Solution:

Note: Only when the number of columns in the left matrix is equal to the number of rows in the right matrix can the two matrices be multiplied.

Special situations of matrix multiplication:

(1) if A is an n-element row matrix and B is an n-element column matrix, AB is A numerical value and BA is an n-order matrix.

,

,

(2) If A is A matrix of order m x n,

B is the n-element column vector, so

(3) The unit matrix multiplied by any matrix is equal to the original matrix.

Likewise

The unit matrix is equivalent to "1" in the number ".

The multiplication of matrices satisfies the following calculation rules:

(1) Combination Law (AB) C = A (BC );

(2) l (AB) = (lA) B = A (lB). (l is the number );

(3) distribution law A (B + C) = AB + AC, (B + C) A = BA + CA.

In Example 4, set matrix A = [1, 3, 5], and calculate AB, BA, AC

Solution:

Example 5,
, Find AB and BA.

Solution,

The following conclusions can be drawn from the above example:

1. The multiplication of matrices generally does not meet the exchange law, that is, AB =ba, which can be described in the following aspects:

(1) When AB can be multiplied, BA may not be multiplied, such as A2 × 3 and B3 × 4. AB makes sense, but BA is meaningless.

(2) even if A = Am x n, B = Bn x m, AB and BA make sense, AB is A matrix of the m order and BA is A matrix of the n order, so AB is equal to BA.

(3) Even if A and B are all matrix arrays of order n, AB is not necessarily equal to BA. See example 5.

2. A non-zero matrix can be A zero matrix, that is, A =o, B =o, but AB = O. Otherwise, even if AB = O, nor can we draw the conclusion that A = O or B = O.

Iv. Transpose operations of Matrices

Define 3. Replace the rows of level m x n matrix A with columns of the same ordinal number to obtain A new n x m matrix, which is called A transpose matrix and is recorded as AT or ′

,

Example 6 set a Matrix

The transpose operation of a matrix has the following calculation rules:

(1) (O) T = O ,;

(2) (A + B) T = AT + BT;

(3) (AT) T =;

(4) (kA) T = kAT; (k is a constant)

(5) (AB) T = BTAT.

Proof below (5)

Set

Reset

According to the multiplication rule of the matrix, fij is equal to column j of column I of BT multiplied by column j of AT, and line I of BT is column I of Column B.

Column j of AT is the row j of.

Because

So

That is, (AB) T = BTAT.

Example 7 matrix settings

Solution:

Define 4. Set A to A matrix of n order. If AT = A is satisfied, that is, aij = aji (I, j = 1, 2,
×
×
*, N) is called A symmetric matrix.

For example, it is a second-order symmetric matrix.

Is a third-order symmetric matrix.

Is a fourth-order symmetric matrix.

Symmetric arrays have the following characteristics: Except for elements on the primary diagonal, all other elements are equal to the primary diagonal axis.

In Example 8, where A and B are symmetric matrices of order n, it is proved that AB is A sufficient and necessary condition for symmetric arrays, that is, AB = BA.

Proof: A and B are symmetric matrices, where AT = A, BT = B,

Necessity of precertification:

If AB is a symmetric array, (AB) T = BTAT = BA = AB

Adequacy of further evidence:

Set AB = BA, so (AB) T = BTAT = BA = AB, so AB is a symmetric array.