Linear Algebra: Chapter 4 moment matrix 2

Section 5 of Matrix

In this section, we will introduce a commonly used method for processing matrices with higher levels, that is, matrix blocks. sometimes, we regard a large matrix as composed of small matrices, just as the matrix is composed of numbers. especially in operations, these small matrices are processed as numbers. this is the so-called matrix block.

To illustrate this method, let's look at an example.

, Indicating the level unit matrix, and

.

In the matrix

Medium,

.

During computing, we regard all the operations as composed of these small matrices, that is, two-level matrices.

,

Where

,

.

Cause

.

It is not difficult to verify that the result is the same as the four-level matrix product.

Generally, let's divide the data into small matrices.

, (1)

, (2)

Each is a small matrix, and each is a small matrix.

, (3)

Where

. (4)

This result is directly verified by the definition of the matrix product.

It should be noted that in blocks (1) and (2), the column division of the matrix must be consistent with the row division of the matrix.

As you can see in the following, block multiplication has many conveniences. After a block is often used, the relationship between matrices is clearer.

In fact, when proving the rank theorem of the matrix product, we have used the idea of Matrix partitioning, where we use the represented row vector, so

,

This is a type of multipart.

.

Using this formula, we can easily see that the row vector is a linear combination of the row vector. We will carry out another type of block multiplication, and the column vector we can see from the results is a linear combination of the column vector.

As an example, we calculate the matrix.

The inverse matrix of which are level and level reversible matrices, which are matrices and zero matrices.

First, because

,

So when it is reversible, it is also reversible.

,

Therefore

,

Here, we represent the level matrix and the level unit matrix respectively. multiply and compare the two sides of the equation.

Obtained from the first and second Formulas

,

In the fourth formula

,

In the third formula

.

Therefore

.

In particular, at that time

.

Format:

Is a number, usually called a diagonal matrix, in the form

A matrix, usually called a quasi-diagonal matrix. Of course, a quasi-diagonal matrix includes a diagonal matrix as a special case.

For two quasi-diagonal matrices with the same block

,,

If the corresponding chunks are of the same level, obviously there are

They are still quasi-diagonal matrices.

Secondly, if all are reversible matrices

.

 

 

 

 

 

 

 

 

 

 

§ 6 elementary matrix

In this section, we will establish the relationship between the elementary transformation of the matrix and the matrix multiplication. Based on this, we will provide a method to calculate the inverse matrix using the elementary transformation.

I. Elementary Matrix

Definition 10The matrix obtained from an elementary transformation of the unit matrix is called an elementary matrix.

Obviously, the primary matrix is a square matrix, and each primary matrix has a corresponding primary matrix.

A non-zero number multiplication row in the number field.

,

Add the number of rows in the Matrix to the row.

The primary matrix corresponding to the column transformation can also be obtained. it should be pointed out that the matrix obtained from an elementary column transformation for the unit matrix is also included in the three types of matrices listed above. for example, we still get the double value of the column added to the column. as a result, these three types of matrices are all elementary matrices.

TheoremAn initial transformation of a matrix is equivalent to multiplying the corresponding elementary matrix on the left. An initial transformation of an equal column is equivalent to multiplying the corresponding elementary matrix on the right.

It is not hard to see that elementary matrices are reversible, and their inverse matrices are still elementary matrices. In fact

.

In chapter 2, section 5, we can see that the matrix can be reduced by using the elementary transformation of rows and columns. If the primary transformation of rows and columns is used at the same time, the matrix can be further reduced.

Ii. Method for Finding reversible matrices and Their Inverse Matrices

Definition 11Matrices are equivalent. If they can be obtained through a series of elementary transformations.

Equivalence is a relationship between matrices. It is not difficult to prove that it has the characteristics of inversion, symmetry, and transmission.

Theorem 5Any matrix is in the same format

It is called the standard form of a matrix. The number of 1 is equal to the rank (the number of 1 can be zero ).

Example 1Use Elementary Transformations to convert the following matrices into standard shapes,

According to the theorem, the elementary transformation of a matrix is equivalent to the multiplication of this matrix using the corresponding elementary matrix. Because of this, the necessary and sufficient condition for Matrix equivalence is that an elementary matrix is used.

. (1)

The rank of the graded reversible matrix is, so the standard form of the reversible matrix is the unit matrix. This is also true.

Theorem 6When a level matrix is reversible, it can be expressed as the product of some elementary matrices:

. (2)

Inference 1The equivalent conditions of the two matrices are: Reversible level matrices and reversible level matrices.

.

Rewrite (2 ).

. (3)

Because the inverse matrix of the elementary matrix is still an elementary matrix, and the multiplication of the elementary matrix on the left of the matrix is equivalent to the transformation of the primary row, so (3) the description

Inference 2The reversible matrix can be transformed into a matrix of units through a series of elementary rows.

The above discussion provides a method to calculate the inverse matrix. It is set to a level-1 reversible matrix. From inference 2, there is a series of elementary Matrices

, (4)

Get from (4)

. (5)

(4), (5) two sub-statements describe that if a reversible matrix is converted into a matrix of units using a series of elementary row transformations, this series of elementary row transformations will also be used to describe the matrix of units.

Combine these two matrices to form a matrix.

,

Multiplication by matrix, (4), (5) can be combined and written

. (6)

(6) provides a method to calculate the inverse matrix. As a matrix, the left half of the matrix is converted into an elementary row transformation. At this time, the right half is.

Example 2Set

Please.

Of course, it can also be proved that the reversible matrix can also be transformed into a matrix of units using the primary column, which gives a method to calculate the inverse matrix using the primary column transformation.

 

 

 

 

 

 

 

 

 

 

 

 

 

7. Elementary Transformation and application examples of multipart Multiplication

Combining block multiplication with elementary transformation becomes an extremely important means in matrix operations.

A matrix of units is set as follows:

.

Swap two rows (columns) with one matrix. One row (column) is a left multiplication (right multiplication) with one matrix. One row (column) is added with a (matrix) multiple of the other row (column, you can obtain some matrices of the following types:

.

Similar to the relationship between an elementary matrix and an elementary transformation, we use these matrices to multiply any block matrix left.

,

As long as the multipart Multiplication can be performed, the result is a corresponding Transformation:

, (1)

, (2)

. (3)

Similarly, when using them to right multiply any matrix, there are also corresponding results for the multipart multiplication.

In (3), select appropriate options to enable. For example, if it can be reversed, then the right end of. (3) becomes

This matrix shape is convenient for determining the determinant, inverse matrix, and solving other problems. Therefore, (3) the computation is very useful.

Example 1Set

,

Reversible.

Example 2Set

,

The test evidence exists and is obtained.

Example 3The product formula of the determining factor is proved.

Example 4And

Make

= Upper triangle matrix.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chapter 4 moment array (Summary)

I. Content Overview

1. Matrix Operations

1) addition and subtraction are all matrices.

2) number multiplication, where is the matrix

3) Multiplication

Here, it is a matrix, and if it is a level matrix, then.

4) reversible Matrix

If a level matrix exists.

It is called a reversible matrix. It is called an inverse matrix.

2. matrix operation rules

1) satisfies the exchange law of addition, combination law, multiplication combination law, number multiplication allocation law to addition, multiplication to the left and right distribution law of addition.

.

2) Pay attention to the following characteristics different from numbers

(1)

(2) Possible

3. Several Special Matrices

Number Matrix, diagonal matrix, Triangle Matrix, symmetric matrix, antisymmetric matrix

4. Mandatory and sufficient conditions for Matrix reversible

The level matrix can be converted into the unit matrix through elementary transformation;

Can be written as the product of the elementary matrix;

The rank;

.

Inverse Matrix Method:

(1) Elementary Transformation Method

(2) Adjoint Matrix Method

5. rank of the matrix

6. Elementary Transformation of elementary matrices and Matrices

1) the three primary matrices correspond to three primary transformations respectively.

2) perform elementary row (column) transformation on the matrix, which is equivalent to multiplication by the left (right) of the corresponding elementary matrix.

3) matrix equivalence and standard form.

7. block matrix operation.

 

Ii. Main Contents of this chapter and their internal relationships

 

 

 

 

 

 

This chapterKey PointsIs the problem of matrix multiplication and its inverse operations-the existence and method of the Inverse Matrix

This chapterDifficultiesIs the matrix multiplication and the block multiplication of the matrix.

 

 

 

 

Iii. Solution and Example Analysis

The basic questions in this chapter are: Calculate the sum, difference, and product of the given matrix. find a matrix that can be exchanged with a given matrix, prove the reversible matrix and obtain the inverse matrix, calculate and prove the rank of the matrix, and solve the matrix equation.

1. Sum, difference, product, and hybrid operations on the given matrix

Example 1. Set as a hierarchical real matrix to prove

2. Method and proof of a matrix that can be exchanged with a given matrix

Example 2. It indicates the matrix where the element of the row and column is 1 and the other elements are all 0.

1) if so, then, then;

2) if so, then, then, and;

3) If it is interchangeable with all the level matrices, it must be a Number Matrix, that is.

3. Proof of reversible matrix and method of Inverse Matrix

Example 3.If the level matrix is satisfied, it is proved reversible and its inverse matrix is obtained.

Example 4. Set it as a level integer matrix to prove that the necessary and sufficient conditions for an existing integer matrix are:

5. Calculation and proof of Matrix Rank and Related Problems

Example 5. If the level matrix () is used

6. solving matrix equations

Example 6.Test Matrix Equation

All solutions.

7. the determining factor of the Block Matrix

Example 7. All settings are level matrices, and prove

.