Similarity can lead to equivalence, and vice versa.

A contract can derive equivalence, and vice versa.

On the premise of true symmetry, similarity can lead to contraction, and vice versa.

Matrix similarity:

Let a and b be matrices of order n. If there is an N-order invertible matrix P, so that P (- 1) AP = b, then the matrix A is similar to B, and is denoted as A ~ B. ..

Matrix contract:

In linear algebra, especially in the theory of quadratic form, the contractual relationship between matrices is often used. The two matrices a and b are contractive. If and only if there is an invertible matrix c, so that c tac = b, then square a shrinks to matrix b. ..

Matrix equivalence:

In linear algebra and matrix theory, there are two m×n matrices A and B. If these two matrices satisfy B=Q- 1AP(P is an n×n invertible matrix and Q is an m×m invertible matrix), then the relationship between them is equivalent. That is to say, there is an invertible matrix, and A gets B through finite elementary transformation.

Extended data:

Matrix is a common tool in applied mathematics such as advanced algebra and statistical analysis. In physics, matrices have applications in circuit science, mechanics, optics and quantum physics. In computer science, three-dimensional animation also needs matrix.

Special categories of matrices include:

1, symmetric matrix

In linear algebra, a symmetric matrix is a square matrix, and its transposed matrix is equal to itself.

2. Similarity matrix

In linear algebra, similar matrix refers to a matrix with similar relationship. Similarity relation is the equivalent relation between two matrices. Two n×n matrices, A and B, are similar matrices if and only if there is an n×n invertible matrix P, so: AP=PB.

3. Diagonal matrix

For the matrix of m×m, when I is not equal to J, there is aij=0? At this time, all the elements on the off-diagonal line are 0, and the matrix at this time is called diagonal matrix.

4. Block matrix

Block matrix is to divide the matrix into smaller matrices, and these smaller matrices are called sub-blocks.

5, rotation matrix)

Rotation matrix is a kind of matrix, which, when multiplied by a vector, has the effect of changing the direction of the vector without changing its size. The rotation matrix does not contain inversion, it can change the right-handed coordinate system into the left-handed coordinate system or vice versa. All rotations are added and inverted to form a set of orthogonal matrices.

References:

Baidu encyclopedia-matrix similarity