Equivalence Relation on Square Matrices induced by Positive Integer Powers
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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.
$\Box$
Symmetry
\(\ds \mathbf A\) | \(\alpha\) | \(\ds \mathbf B\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf A^r\) | \(=\) | \(\ds \mathbf B^s\) | for some $r, s \in \N$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf B^s\) | \(=\) | \(\ds \mathbf A^r\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf B\) | \(\alpha\) | \(\ds \mathbf A\) |
Thus $\alpha$ is seen to be symmetric.
$\Box$
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:
\(\ds \mathbf A^r\) | \(=\) | \(\ds \mathbf B^s\) | for some $r, s \in \N$ | |||||||||||
\(\ds \mathbf B^u\) | \(=\) | \(\ds \mathbf C^v\) | for some $u, v \in \N$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf A^{r u}\) | \(=\) | \(\ds \mathbf B^{s u}\) | raising both sides to $u$th power | ||||||||||
\(\ds \mathbf B^{s u}\) | \(=\) | \(\ds \mathbf C^{s v}\) | raising both sides to $s$th power | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf A^{r u}\) | \(=\) | \(\ds \mathbf C^{s v}\) |
Thus $\alpha$ is seen to be transitive.
$\Box$
$\alpha$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $6$