Equivalence Relation on Square Matrices induced by Positive Integer Powers

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $S$ be the set of all square matrices of order $n$.

Let $\alpha$ denote the relation defined on $S$ by:
 * $\forall \mathbf A, \mathbf B \in S: \mathbf A \mathrel \alpha \mathbf B \iff \exists r, s \in \N: \mathbf A^r = \mathbf B^s$

Then $\alpha$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
We have that for all $\mathbf A \in S$:


 * $\mathbf A^r = \mathbf A^r$

for all $r \in \N$.

It follows by definition of $\alpha$ that:
 * $\mathbf A \mathrel \alpha \mathbf A$

Thus $\alpha$ is seen to be reflexive.

Symmetry
Thus $\alpha$ is seen to be symmetric.

Transitivity
Let:
 * $\mathbf A \mathrel \alpha \mathbf B$ and $\mathbf B \mathrel \alpha \mathbf C$

for square matrices of order $n$ $\mathbf A, \mathbf B, \mathbf C$.

Then by definition:

Thus $\alpha$ is seen to be transitive.

$\alpha$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.