How to define the relations of equivalence, similarity and contract of matrix? What are their conditions

How to define the relations of equivalence, similarity and contract of matrix? What are their conditions

If there is a full rank matrix PQ such that B = PAQ holds, then the matrices A and B are equivalent;
If there is an invertible matrix P such that B = p-1ap holds, then the matrices A and B are similar;
If there is an invertible matrix P such that: B = p'ap holds, then matrices A and B are called congruent

What's the meaning of this equivalence?
If two matrices are equivalent, what quantitative properties do they have

Definition: if matrix B can be obtained by a series of elementary transformations, then a and B are equivalent
If a and B are equivalent, then B and a are equivalent
If a is equivalent to B and B is equivalent to C, then a is equivalent to C
Equivalent rank of a and B (a) = rank (b)
A and B are equivalent. A and B have equivalent canonical forms
There are invertible matrices P, Q such that PAQ = B

What does matrix equivalence mean

The equivalence in a broad sense is that the set remains invariant under certain transformations. For example, a and B are equivalent if B can be obtained by a series of elementary transformations. The matrix is determinant invariant under elementary transformations. In linear algebra, congruence and similarity are equivalent relations