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.

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1. Matrices A and A are equivalent (reflexivity);

2. Matrices A and B are equivalent, so B and A are also equivalent (equivalent);

3. Matrix A and B are equivalent, and matrix B and C are equivalent, so A and C are equivalent (transitivity);

4. Matrices A and B are equivalent, then IAI=KIBI. (k is a non-zero constant)

5. The linear equations corresponding to the matrix with row equivalence relation have the same solution.

6. For two rectangular matrices with the same size, their equivalence can also be characterized by the following conditions:

(1) matrices can be converted to each other by basic row and column operations.

(2) Two matrices are equivalent if and only if they have the same rank. ?

Extended data:

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A 1, a2, ... An, is linearly independent, while a 1, A2, ... an, B, R are linearly related, so there is X 1A 1+X2A2+ ... If y=0, Xnan+X.

If both x and y are not zero, we can get-b = x1/x) a 1+(x2/x) a2+...+(xn/x) an+(y/x) r divided by x, which means that b can be a 1, a2. The comprehensive proposition can prove.

When a and b are isomorphic matrices, and r(A)=r(B), a and b must be equivalent.

Reference: Baidu Encyclopedia-Equivalent Matrix