Transformation and application of Jordon standard form in matrix theory 6

The sixth lecture is about the transformation and application of the Jordon standard form.

 

1. Jordon standard Transformation Matrix
   

Write P in the form of column blocks in J Structure

SolutionRMatrix Equations

SetRSynthetic transformation matrices

★Equation
Solution

Two specific practices :(I), First obtain the feature vector, and then obtain ,,...; (Ii) first obtain the feature vector and then obtain it directly.

Since the previous method is a singular matrix, each step has multiple solutions and no solution problems, so the steps cannot be completely independent, and the previous step still needs to depend on the next step, then step,..., Some undetermined coefficients can only be completely determined in the last step. In the latter case, only one equation is solved, while the others are directly assigned values, without the above problems. However, it may cause the low-level zero vector problem.

Because

Therefore, the following conditions should be met:

The same feature value may appear in differentJordanBlock. In this caseJordanThe number of blocks is processed at a high or low level. The Higher Order is processed first, the lower order is processed later, and the same order is processed at the same time.

1. The highest order (not having the same feature valueJordanBlocks of the same order) can be obtained by the following method, even if the order is used.
   

Then, the equation is obtained until andJEqual to the order of the next Jordan block that belongs to the same feature value.

1. For the new Jordan block, itsNot only meet
   

However,

It should also be linear independent of the above.

1. When processing other Jordan blocks with the same feature value, follow the (2) principle.
   
2. When multiple Jordan blocks with the same feature value are in the same order, the linear independence problem should also be considered.
   

 

For example, evaluate the Jordan standard form and its transformation matrix.

[Solution]: The Jordan standard form has been obtained in the previous lecture. You can also obtain it using the following method.

() Elementary transformations can be used

According to this, the Jordon standard form is obtained.

It can be seen at the same time that the uniform is a triple feature value.

Find the transformation matrix P below

(1) There is only one Jordon matrix,

Please meet

Configuration

The solution is

The solution is

Optional =

(2) There are two Jordan blocks,,,

Respectively,

Start :,

=

=

Fetch

: Unrelated to the linearity. Optional =

(3) synthesis Transformation Matrix

Exist

Verifiable:

 

Ii. Power and polynomial of the Jordan Standard Form

, That is,

It is also similar to the upper triangle strip matrix. Each element on the diagonal line parallel to the main diagonal line is equal. The elements in the first line are

Having polynomial.

Again

......

Hence

That is to say, it is still the upper triangle matrix, and all elements on the diagonal lines parallel to the main diagonal lines are equal, while

The first line of elements is

If so, the calculation is very convenient, and the matrix product is no longer required.

 

 

Job: p107 11