Matrices a and a are equivalent (reflexivity); Matrices a and b are equivalent, then b and a are also equivalent (equivalent); Matrices a and b are equivalent, and matrices b and c are equivalent, so a and c are equivalent (transitivity); Matrices a and b are equivalent, then IAI=KIBI. (k is a non-zero constant) The linear equations corresponding to the matrix with equivalent rows have the same solution.

Related knowledge:

Matrix equivalence means that in linear algebra and matrix theory, there are two m×n matrices A and B. If these two matrices satisfy B=QAP(P is an n×n invertible matrix and Q is an m×m invertible matrix), then these two matrices are equivalent. That is to say, there is an invertible matrix (p, q), which makes A get B through finite elementary transformation.

Matrix, matrix. Mathematically, a matrix refers to a two-dimensional data table arranged vertically and horizontally, which originated from a square matrix composed of coefficients and constants of an equation. This concept was first put forward by British mathematician Kelly in19th century. 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. Matrix operation is an important problem in the field of numerical analysis. Decomposition of a matrix into a combination of simple matrices can simplify the operation of the matrix in theory and practical application.

For some widely used and special matrices, such as sparse matrix and quasi-diagonal matrix, there are concrete fast operation algorithms. For the development and application of matrix related theory, please refer to matrix theory. Infinite-dimensional matrices will also appear in astrophysics, quantum mechanics and other fields, which is the generalization of matrices.

A performs a series of elementary transformations until B, then A and B are equivalent, that is, there is an inverse matrix PQ, so that B = PAQ, then AB ranks are the same. The similarity of AB exists, but the inverse matrix P makes B = P- 1AP, so the similarity conclusion is stronger than equivalence. They have more eigenvalues with the same properties and determinants.

Equivalence usually means that you can transform it into another matrix by elementary transformation, which is essentially by having the same rank as another matrix. This is a very broad condition. Not suitable for many places. A and B are very similar, and there is an invariant matrix P that makes PAP- 1 = B, which is the most important relationship in linear algebra or higher algebra, and half of higher algebra is dealing with this relationship. Similarity leads to equivalence.