Matrix theory basics 2.4 Matrix partitioning

Section 4 Matrix partitioning

 

When performing matrix operations, if the matrix is large, it may be cumbersome to perform various matrix operations. You can use the Matrix partitioning method, A series of horizontal and vertical straight lines are used to divide matrix A into several small matrices. Each small matrix is called A sub-block. A matrix in the form of sub-blocks is called A block matrix, operations on the matrix after a block greatly reduce the calculation workload and simplify the calculation process. This method is called the block method of a matrix.

For example,

Divide the rectangle into six blocks using the watermark and vertical straight lines

In the form matrix, A is originally A 3×4th level matrix. After A chunk is divided, A is represented as A 2×3rd level matrix with sub-blocks.

Obviously, there are multiple Block Methods for a given matrix, such

The partition matrix is divided into two rows and two columns. The following partition method can be used:

There are eight methods.

Generally, the matrix should be reasonably segmented according to the specific needs of the problem. The following describes several Special Partitioning methods:

(1) multipart by row: each row of the matrix is used as a sub-block and is recorded

In this method, the Order Matrix of m x n is converted into the m element column matrix.

(2) partitioning by column: each column in the matrix is used as a sub-block and recorded

This method converts the Order Matrix of m x n into a row matrix of n.

(3) diagonal matrix: For An n-order matrix, if it is converted into a non-zero subblock only on the primary diagonal, and all are square arrays, the remaining subblocks are zero Matrices, that is

Where Ai is a non-zero square matrix (I = ,..., R), called A as A block diagonal matrix.

This block method simplifies complex square matrix operations. Block Diagonal Matrix A has the following properties:

(1) |A| = |A1 |A2 | ×
×
× |Ar|.

(2) If |AI| Limit 0 (I= 1, 2,
×
×
×R), Then |A| Defaults 0,

.

Example 16 is set.A-1.

Solution,

A11 = (3 ),
;

,
;

So.

The operation of the block matrix is the same as that of the general matrix. The difference is that the operation of the general matrix is performed between elements, the block matrix operation is performed between the sub-block and the sub-block.

MatrixAAndBIs the same type matrix, and the same block method is used for A and B.

,
,

WhereAIJAndBIJSame type (I = 1, 2 ,..., S; j = 1, 2 ,..., R) So

(1 ).

(2 ).

(3)

(4) SetUIsM'PMatrix,
VIsP'NMatrix, U and V are segmented

,
,

WhereUi1,
Ui2,
×
×
×,
UitThe number of columns is equalV1J,
V2J,
×
×
×,
VTJ, The number of rows, then

,

Where (I= 1, 2,
×
×
×,
S;
J= 1, 2,
×
×
×,
R).

In Example 17, if A and B are n-order reversible phalanx, the block matrix is also reversible. And

.

Proof: A and B are n-order reversible phalanx, which is obtained by the necessary conditions of the reversible matrix. | A | ≈ 0, | B | ≈ 0,

So it is reversible.

Set

,

Then

.

Therefore, the token,

So
.