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 $$\mathbf A$$.

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

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

$$ $$ $$

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

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