Matrix Theory 12th full rank decomposition and Singular Value Decomposition

12th full rank decomposition and Singular Value Decomposition

 

I. full rank decomposition of Matrices

1. Definition: set. If a matrix and

Is called a full rank decomposition.

Description: (1) full rank matrix of columns, that is, the number of columns equals the rank; full rank matrix of rows, that is, the number of rows equals the rank.

(2) full rank decomposition is not unique. (Order reversible matrix), then

, And

2. Existence Theorem: all non-zero Matrices have full rank decomposition.

Evidence: constructive proof is used. If this parameter is specified, an elementary transformation matrix exists,

, Where

It will be written and segmented into, where

Is full rank decomposition.

3. Hermite standard form (row STEP standard form)

And meet

1. Each row in the forward direction contains at least one non-zero element (called a non-zero row), and the first non-zero element is 1, and the elements in the next row are all zero (called a zero row );
   
2. If the first non-zero element (1) in the row is in the column
   

;

1. Column, column ,..., The column combination is exactly the top of the square matrix of the order unit (that is, all the column except the aforementioned 1 is 0) is referred to as the Hermite standard form.
   

Example 1 is Hermite standard

 

It is also the Hermite standard form

4. A method for full rank decomposition

Set,

1. The line elementary transformation is used to convert the form into the Hermite standard form. The matrix form is the shape given in the Hermite standard form definition;
   
2. Select replacement Matrix
   

   
   Column, that is, except the first element of the column vector is 1, all other elements are zero;

   
   Other columns only need to be replaced by a matrix (each row in each column has only one non-zero element and is 1 );
   

   
   If you use any right multiplication matrix (when the multiplication property is satisfied), you can replace the column of the Matrix with the column of the new matrix (that is, the product matrix ).
   

   
   Command, that is

(3) Before the order

Proof:, then, known, but, of course, you can obtain the block by finding it, but in this case, there is no need to use the Hermite standard form.

Example 1 full rank decomposition

Solution: (1) rank obtained first. Obviously, the first two columns are independent of each other, and the third row can be obtained from the first row minus the second row.

(2) The primary transformation will be converted to the Hermite standard type.

, That is

,,

(3) Finding and

Visible, so,

Verification:

While

Ii. Euclidean decomposition and Singular Value Decomposition

1. spectral decomposition of the ermi Matrix

If it is an ermi matrix, a normal matrix exists.

Is written as a column vector, that is

2. Non-singular matrix Ry-diagonal Decomposition

Theorem: if it is set to a non-singular matrix of the order, the order matrix and

(If

, Then)

Proof: it is also a non-singular matrix of order, and it is an ermi, positive definite matrix. Therefore, the feature value of the Order Matrix exists.

, Then

, Then

That is, it is also a user matrix, and the certificate is complete

The method used to obtain the diagonal decomposition is as shown in the proof: first obtain the transformation matrix and then the right corner.

3. Singular Value Decomposition of General Matrices

Theorem: If this parameter is set, the Order Matrix and Order Matrix exist.

That is

Certificate: first consider. Because, therefore,

And it is ermi, Which is semi-definite, and has the order matrix.

 

, Then

Order, and

On the basis of the construction of a matrix, that is

This is feasible according to the base expansion theorem.

Hence

It is known

While

Therefore, the theorem is proved.

The method of Singular Value Decomposition can be obtained according to the proof steps.

 

Job: P225 1 (2), 2, 5

P233 1