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 $$\mathbf{Q} = \mathbf{P}$$.