Similarity matrix and equivalent matrix

1. Equivalent matrix
Definition: For homotype matrices A and B, there are invertible matrices P and Q, so that B=PAQ.
Necessary and sufficient conditions: A and B are equal in rank
two. Similarity matrix
definition: For homotype square matrices A and B, exist The invertible matrix P makes B=P−1AP
. The properties of the similarity matrix: 1. The eigenvalues ​​of B
and A are equal
2.r(A)=r(B)
3.|A|=|B|
** Three, both Relationship **
Equivalence (only the rank is the same) -> Similar (rank and eigenvalues ​​are the same), the matrix intimacy is deepened step by step.
The similar matrix must be an equivalent matrix, but the equivalent matrix may not be a similar matrix
. The equivalent matrix of PQ=E is a similar matrix