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.