Two square matrices A and B are said to be similar if any invertible matrix P ca

Question

Two square matrices A and B are said to be similar if any invertible matrix P can be found such that AP=PB.

Prove that similarity is an equivalence relation.

Explanation / Answer

AI = IA = A, hence every matrix A is similar to itself, Hence similarity is reflexive AP = PB => P^-1 A = BP^-1 [P^-1 is defined as P is invertible] => BP^-1 = P^-1 A hence A is similar to B => B is similar to A and hence similarity is symmetric AP = PB BQ = QC => PBQ = PQC => PB = PQCQ^-1 so AP = PQCQ^-1 => APQ = PQC => A(PQ) = (PQ) C hence if A and B are similar , B and C are similar, so are A and C hence similarity is transitive Hence similarity is equivalence relation

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