Matrix theory basics 3.1 Elementary Transformation

Section 1 Elementary Transformation

 

Definition 1 the following three Transformations on the matrix, collectively referred to as the primary Row Transformation of the matrix

(1) If the position of the two rows in the interchange matrix isI,
JTwo rows are exchanged and recordedRi«Rj;

(2) Any numberKLimit 0: the number of the de-multiplication MatrixIAll the elements of the row.K ri);

(3) PlaceIRowKTimes to the nthJRecord the rows (where k is an arbitrary number)KriRj,).

Example 1 sets A matrix and applies an elementary transformation to.

Solution:

From Example 1, matrix A is transformed into A matrix through the Elementary Line.

We call this type of matrix A Row step matrix. Its feature is that the position of the column where the first non-zero element in each row is located increases one by one, and the zero row is under the non-zero row. For example

All are row step matrices. Rather than a row step matrix.

If the row step matrix in Example 1 is further transformed, it can be simplified.

 

Finally, this matrix is called the simplest matrix of rows. Its features are as follows:

(1) The row step matrix is a row step matrix.

(2) The first non-zero element in each row is 1, and the column where the first non-zero element is located, except 1, all other elements are 0, for example:

All are the simplest matrix of rows.

The Elementary Transformations of matrices are described as follows:

(1) elementary row transformation can convert any level m × n matrix into a row step matrix and a row simplest matrix.

(2) The Matrix after the elementary row transformation is generally not equal to the original matrix, so you must use "→" to connect the matrix before and after the transformation.

(3) the transformation of the three primary rows is reversible, that is, the transformation of the matrix after transformation will return to the original matrix. For example

Note: all three transformations are reversible, and their inverse transformations are primary transformations of the same type.Ri«RjThe inverse transformation is itself; TransformationRi'KIs (or recordedRiBytesK); TransformRiKrjToRi(-K)Rj(Or recordedRi-Krj).

(4) If the three transformations of the rows are replaced with those of the columns, the three transformations of the columns are also obtained, respectively recordedCi«Cj(ReconciliationI,
JTwo columns );K ci(In any numberKLimit 0: the number of the de-multiplication MatrixIAll columns );KciCj(Column IKTimes to the column j ). This is the elementary column transformation of the matrix. The primary Row Transformation and primary column transformation of a matrix are collectively referred to as the primary transformation of a matrix.

Define 2 If MatrixAConverted into a matrix through finite elementary transformationsBIs called a matrix.AAndBEquivalent, recorded.

If An is a reversible matrix, An can be converted into a matrix En after a finite elementary transformation. Therefore, An equivalence matrix has the following properties:

(I) anti-bodyA~A;

(Ii) symmetry ifA~B, ThenB~A;

(Iii) transmission ifA~B,
B~C, ThenA~C
.

Example 2 convert the matrix into the simplest matrix of rows

Solution

So