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 $$\mathbf Q = \mathbf P$$.