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