Similar Matrices are Equivalent

Theorem
If two matrices are similar, then they are equivalent.

However, if two square matrices of order $n > 1$ are equivalent, they are not necessarily similar.

It follows directly that every equivalence class for the relation of similarity on $\mathcal M_R \left({n}\right)$ is contained in an equivalence class for the relation of matrix equivalence.

Proof
If $\mathbf A \sim \mathbf B$ then $\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$.

Let $\mathbf Q = \mathbf P$ and the first result follows.

If $\mathbf A \equiv \mathbf B$ then $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$.

It is not necessarily the case that $\mathbf Q = \mathbf P$.