Matrix Similarity is Equivalence Relation

Theorem
Matrix similarity is an equivalence relation.

Proof
Follows directly from Matrix Equivalence is an Equivalence.

Alternatively, checking in turn each of the critera for equivalence:

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

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

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

$$ $$ $$

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

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