Matrix Theory-Method of Matrix Functions

8. Method of Matrix Functions

1. evaluate matrix functions using the standard form of Jordan.

For the polynomial of the matrix, we have derived: Polynomial

In fact, the above results not only apply to the polynomial of the matrix, but also to the idempotence of the matrix. This leads to another definition and calculation method of matrix functions.

1. Definition: Set the Jordan Standard of n-level matrix A to J.

,

The non-singular matrix P makes:

For function f (z), if the following functions

The matrix function f (A) is meaningful and

2. Method of Matrix Function (STEP ):

Obtain the Jordan Standard Form and transformation matrix P of,

Calculate the Jordan blocks of J.

And in order,

Synthesis

Matrix Product

It must be noted that the calculation result is irrelevant to the order of the Jordan blocks in the Jordan standard form.

Example 1 (Textbook P176 Example 3-8 ).

[Solution] 1. Obtain J and P.

2. Obtain and construct:

F (1) = 1,

3 Synthesis

4. Please,

Note:

(1 ),

It can be seen that this indeed constitutes an inverse function;

(2) The types of matrix functions are not just described, such as the Xin matrix. To

For example, we use the definition here,

B can also be seen as

Ii. solving matrix functions using zero polynomial.

The method for solving matrix functions using the Jordan standard type is complex. It requires J

And P. Next we will introduce a method for solving matrix functions based on the zero polynomial.

Law: The minimum polynomial of the nth square matrix A is equal to the nth (or last) invariant factor of its feature matrix. (See Zhang Yuanda linear algebra principle P215)

Set the invariant factor of the nth square matrix A to the inverse order, and the primary factors given by them are

;;

Because

1 must appear in the middle;

2.

According to the above theorem, the minimum polynomial of

The smallest polynomial of A is its zero polynomial,

That is

So that it can be represented in a linear way or in a linear way. Therefore, if the matrix function f (A) exists, it must be linearly expressed.

Therefore, we define m-1) with undetermined coefficients. According to the above discussion, f (A) = g (A) can be made by selecting appropriate coefficients ).

Also, assume that J and P are the Jordan standard form of A and the corresponding transformation matrix:

Then

, F (J) = g (J)

Since g (A) is A polynomial of the undetermined coefficient, the above is A linear equations. And the number of equations is equal to the number of unknowns, which can be determined exactly

A general method for finding matrix functions based on the least polynomial is provided.

1. Obtain the least polynomial.

;

(Or feature polynomials)

2. Write the undetermined Polynomial in the form

(Or)

3. Solving Linear Equations

(Or)

4. Find g (A) and then obtain f (A) = g ().

From the process of derivation, it seems that not only the least polynomial can be used for the calculation of matrix functions, but also the general zero polynomial can be used, and the feature polynomial is the most convenient.

Example 2. Functions computed using the new method. ()

[Solution] 1;

2

3. The equations are

4

,

 

 

The method is exactly the same as the Jordan standard method.

 

 

Job: p163 6