Matrix theory basics 3.2 elementary phalanx

Section 2 primary phalanx

 

Definition 3 the square matrix obtained from an elementary transformation of the unit matrix is called an elementary square matrix.

The primary transformations of the three pairs correspond to the three primary squares.

(1) convert the I and j matrices of E

(2) use any k = 0 to multiply the I-th row of E to get the matrix, which is recorded

(3) Multiply row I of E by k and add it to the matrix of Row j.

Similarly, the primary transformations of the three columns correspond to three primary matrices, respectively recorded

 

 

 

 

 

 

 

 

 

 

An arbitrary matrix using the left (or right) of the primary square matrix can achieve the same effect of performing the transformation of the same type of primary rows (or columns, this section uses any 3×4th matrix as an example.

Set

The primary matrix of the three level 3 rows is set:

,

The second and second rows of A are swapped.

The 3rd rows of A are multiplied by k.

The 3rd rows of A are multiplied by k and added to the first row.

Similarly, set the primary phalanx of three 4-level columns

,

Then

The second and second columns of A are swapped.

The 2nd Column multiplication k of A is realized.

The 1st column multiplication k of column A is added to 3rd columns.

The following important theorem can be obtained from the above example.

Theorem 1 sets A as an arbitrary m×n matrix, and applies some elementary row transformation to the row of A, which is equivalent to removing left by A with the same m-order elementary square matrix; the result of applying certain elementary column transformations to column A is equivalent to using the same n-order elementary square matrix to go right to. That is:

Note: The left multiplication Am × n refers to the m-level elementary phalanx, which implements the elementary row transformation, and the right multiplication Am × n refers to the n-level elementary phalanx, which implements the elementary column transformation.

The following inference obtained from Theorem 1

It is inferred that if A undergoes several elementary row transformations to obtain B, then there must be several elementary phalanx E1, E2 ,..., Ek, so that B = E1E2... EkA;

If A undergoes several primary column transformations to obtain B, then there must be several primary phalanx C1, C2 ,..., Cl, so that B = AC1C2... Cl,

If ~ B, then there must be a finite elementary square matrix E1, E2 ,..., Ek, C1, C2 ,..., Cl, so that B = E1E2... Ek AC1C2... Cl.

Theorem 2 any elementary matrix has an inverse, and its inverse is an elementary matrix of the same type.

Proof:

Likewise:

Theorem 3 if A is A reversible matrix, it is always the product of several elementary square arrays.

Proof: because A is A reversible matrix, ~ En, obtained from inference 1: there must be a finite elementary matrix E1, E2 ,..., Ek, C1, C2 ,..., Cl to make

En = E1E2... Ek AC1C2... Cl

Given by Theorem 2: Each elementary matrix can be inverse, so

,

It is also an elementary square matrix, so A is always equal to the product of several primary square arrays.

According to Theorem 1 and Theorem 3, a method for finding inverse matrices can be obtained.

Because A reversible, the A-1 is also reversible, so there are primary phalanx L1, L2 ,..., Lr makes

A-1 = L1L2... Lr.

Observe the following two equations:

A-1 A = L1L2... LrA = E

A-1 E = L1L2... LrE = A-1

The first formula indicates that, after several elementary row transformations, A can be converted into A matrix E. The second formula indicates that E can be converted into A A-1 after several identical elementary row transformations.

Therefore, we can put order n reversible square matrix A and Order n unit matrix E together to obtain an order n × 2n matrix, and then apply the Elementary Line Transformation to it, turn sub-block A into E, and sub-block E into A-1.

This is one of the commonly used methods for matrix inversion.

Example 3: Obtain the inverse of matrix A based on the known matrix.

Solution: Perform elementary row transformation on the matrix

We can also use the elementary column transformation to obtain the inverse matrix. At this time, we apply the elementary column transformation to the 2n × n Order Matrix. When the upper half A is converted into E, the lower half E is converted into A-1.

Example 4 Matrix

Solution: Perform elementary column transformation on the matrix

=