Linear algebra: Chapter Three elementary transformation of matrices and the rank _ linear algebra of elementary transformation matrices of linear equations (1) matrices

The elementary transformation of the Matrix one. Mathematical concepts

The nature of an equivalence relationship:

(i) reflexive a~a;

(ii) If the symmetry of  is a~b, then b~a;

(iii) If the transitivity is A~b, the b~c is a~c; Two. focus, difficulty analysis

The focus of this section is to use matrix Elementary transformations to transform matrices into row (column) ladder-shaped matrices, minimalist matrices, and standard-form matrices. Row (column) step-form matrices are essential for the rank of our next learning matrix, and the minimalist matrix is very useful for solving future learning equations. As long as mastering the matrix into the ladder-shaped, the simplest type and the standard form of the general law, it will make it difficult to easily, quickly get the desired results. Three. Typical examples

Example Set matrix

Will be a row of ladder-shaped, line of the most simple and standard form matrix.

Solution: to matrix A for elementary row transformation,

Obviously, the B1 is a row-step-form matrix characterized by: the number at the bottom of the ladder line is 0; each step has only one line, the number of steps is a non-zero number of rows, the vertical line of the step lines (each line is a row) the first element after is not 0 yuan, that is, not 0 rows of the first non-0 yuan.

To continue the elementary row transformation, the B1

It is obvious that B2 is the simplest type of matrix, which is characterized by the following: the first 0 yuan of 0 rows is 1, and the other elements of the column that are not 0 yuan are 0.

The primary column transformation is applied to the simplest form matrix, and it is:

It is obviously a standard form matrix, which is characterized in that the upper-left corner of the matrix is a unit matrix, and the other elements are all zeros.

Note: The matrix can be converted to a standard form matrix by a primary row transformation to a row-step matrix. Then it becomes the simplest matrix, on the basis of which the Elementary column transformation is used to finalize the standard form matrix, and it can be transformed into a column-step-form matrix by the Elementary column transformation, and then the Elementary column transformation is used to form the column's simplest matrix. Finally, the elementary row transformation is used to transform it into a standard form matrix, and it can be transformed into a standard form matrix by a primary row and column transformation. However, considering the need of solving linear equations, we must be proficient in using elementary row transformations to transform matrices into line-shape matrices.

Rank one of the second matrix. Mathematical Concepts

Definition 2.1 in matrix A, any K-row and K-column (k≤m,k≤n) are located at the intersection of these rows of K2 elements, which do not change the order of their position in the matrix, the K-order determinant, called the K-order subtype of matrix A.

The definition 2.1 is set in matrix A to have a not equal to 0 of R order D, and all r+1 order (if present) is equal to 0, then the end D is called matrix A of the highest order not 0 Zi, the number r is called the rank of matrix A, recorded as

R (A).

1. The rank of the 0 matrix is 0;

2.;

3. The reversible matrix is called the full rank matrix;

4. The irreversible matrix is called descending rank matrix. Two. Principle Formula and law

Theorem 2.1 If a~b, then R (A) = R (B).

According to this theorem, in order to find the rank of the matrix, it is easy to see the order of the highest order of the Matrix, not 0, as long as the matrix is transformed into the ladder-shaped matrix by elementary row. It is obvious that the row number of the 0 rows in the ladder-shaped matrix is the rank of the matrix. This gives the method of finding the rank of the matrix. Three. Analysis of key points and difficulties

The focus of this section is to find out the rank of the matrix with elementary transformation, and it is very useful to solve the linear equations by grasping the matrix into the ladder-shaped matrix with elementary row transformation and then finding the rank of the matrix. The difficulty is how to correctly understand the proof of the theorem and then use the elementary transformation of the matrix into the ladder of methods and skills, mastered the method and skills, will be simple and quick to find the rank of the matrix, otherwise grasp the law, the calculation will be very complicated. Four. Typical examples

Example 1 set the matrix

The rank of matrix A is obtained and a maximum non 0 subtype of a is obtained.

Solution: First of all, the rank of a, for which a primary row transformation into the ladder-shaped matrix:

Because the row-step matrix has 3 non-0 rows, R (A) =3.

and a maximum order of 0 of a. Because R (a) = 3, the highest order of the known A is not 0 sub is 3 order. A 3-order subtype is common (a), it is more troublesome to find a non-0 subtype from the 40-subtype. By examining the row-ladder matrix of A, the row-step-form matrix of a matrix is

Know R (b) = 3, so B must have 3-order non-0 zi. B's 3-order subtype has 4, it is more convenient to find a non-0 subtype in the 4 3-order subtype of B than to find a non 0 subtype in a. The first three lines of B are now calculated as a subtype

，

So this subtype is a group of a high order 0-subtype.

From:http://dec3.jlu.edu.cn/webcourse/t000022/teach/index.htm