Matrix Equivalence is Equivalence Relation

Theorem
Matrix equivalence is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
$\mathbf A = \mathbf{I_m}^{-1} \mathbf A \mathbf{I_n}$ trivially, for all $m \times n$ matrices $\mathbf A$.

Thus reflexivity holds.

Symmetric
Let $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$.

As $\mathbf P$ and $\mathbf Q$ are both invertible, we have:

Thus symmetry holds.

Transitive
Let $\mathbf B = \mathbf Q_1^{-1} \mathbf A \mathbf P_1$ and $\mathbf C = \mathbf Q_2^{-1} \mathbf B \mathbf P_2$.

Then $\mathbf C = \mathbf Q_2^{-1} \mathbf Q_1^{-1} \mathbf A \mathbf P_1 \mathbf P_2$.

Transitivity follows from the definition of invertible matrix, that the product of two invertible matrices is itself invertible.

Hence the result by definition of equivalence relation.