# Matrix Similarity is Equivalence Relation

## Proof 1

Follows directly from Matrix Equivalence is Equivalence Relation.

$\blacksquare$

## Proof 2

Checking in turn each of the criteria for equivalence:

### Reflexive

$\mathbf A = \mathbf{I_n}^{-1} \mathbf A \mathbf{I_n}$ trivially, for all order $n$ square matrices $\mathbf A$.

$\Box$

### Symmetric

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

As $\mathbf P$ is invertible, we have:

 $\ds \mathbf P \mathbf B \mathbf P^{-1}$ $=$ $\ds \mathbf P \mathbf P^{-1} \mathbf A \mathbf P \mathbf P^{-1}$ $\ds$ $=$ $\ds \mathbf{I_n} \mathbf A \mathbf{I_n}$ $\ds$ $=$ $\ds \mathbf A$

$\Box$

### Transitive

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

Then:

 $\ds \mathbf C$ $=$ $\ds \mathbf P_2^{-1} \mathbf P_1^{-1} \mathbf A \mathbf P_1 \mathbf P_2$ $\ds$ $=$ $\ds \paren {\mathbf P_1 \mathbf P_2}^{-1} \mathbf A \paren {\mathbf P_1 \mathbf P_2}$ Inverse of Matrix Product

By Product of Matrices is Invertible iff Matrices are Invertible, $\paren {\mathbf P_1 \mathbf P_2}$ is invertible .

$\Box$

So, by definition, matrix similarity is an equivalence relation.

$\blacksquare$