Let m be the set of all nxn matrices. define a relation on won m by a b there exists an invertible matrix p such that a = p bp show that is an equivalence relation.

Recall that a relation is an equivalence relation if and only if is symmetric, reflexive and transitive. In order to simplify the notation we will use A↔B when A is in relation with B.

Reflexive: We need to prove that A↔A. Let us write J for the identity matrix and recall that J is invertible. Notice that . Thus, A↔A.

Symmetric: We need to prove that A↔B implies B↔A. As A↔B there exists an invertible matrix P such that . In this equality we can perform a right multiplication by and obtain . Then, in the obtained equality we perform a left multiplication by P and get . If we write and we have . Thus, B↔A.

Transitive: We need to prove that A↔B and B↔C implies A↔C. From the fact A↔B we have and from B↔C we have . Now, if we substitute the last equality into the first one we get

.

Recall that if P and Q are invertible, then QP is invertible and . So, if we denote R=QP we obtained that

. Hence, A↔C.

Therefore, the relation is an equivalence relation.