Similar Matrices are Equivalent

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Theorem

If two square matrices over a ring with unity $R$ are similar, then they are equivalent.


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.

Here, $\mathcal M_R \left({n}\right)$ denotes the $n \times n$ matrix space over $R$.


Proof

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

Let $\mathbf Q = \mathbf P$.

Then $\mathbf A$ are equivalent to $\mathbf B$, as:

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

$\blacksquare$


Sources