Prove that each of the following is an equivalence relation.
Similarity on \(M_n(R)\): \(A\sim B\) if there exists \(P\in\operatorname{GL}_n(R)\) such that \[ B=PAP^{-1}. \]
Matrix equivalence on \(M_{m\times n}(R)\): \(A\approx B\) if there exist \[ P\in\operatorname{GL}_m(R), \qquad Q\in\operatorname{GL}_n(R) \] such that \[ B=PAQ. \]
Solution
1. Similarity is reflexive, symmetric, and transitive.
Reflexivity: \(A=I_nAI_n^{-1}\).
Symmetry: if \(B=PAP^{-1}\), then \[ A=P^{-1}BP. \]
Transitivity: if \[ B=PAP^{-1}, \qquad C=QBQ^{-1}, \] then \[ C=(QP)A(QP)^{-1}. \] Thus similarity is an equivalence relation.
2. Matrix equivalence is reflexive, symmetric, and transitive.
Reflexivity: for \(A\in M_{m\times n}(R)\), \[ A=I_m A I_n. \]
Symmetry: if \(B=PAQ\) with \(P\in\operatorname{GL}_m(R)\) and \(Q\in\operatorname{GL}_n(R)\), then \[ A=P^{-1}BQ^{-1}. \]
Transitivity: if \[ B=PAQ, \qquad C=P'BQ' \] with \(P'\in\operatorname{GL}_m(R)\) and \(Q'\in\operatorname{GL}_n(R)\), then \[ C=(P'P)A(QQ'), \] and \(P'P\), \(QQ'\) are invertible. Thus matrix equivalence is an equivalence relation.