Similar Matrices are Equivalent

Theorem

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

That is:

every equivalence class for the similarity relation on $\map {\MM_R} n$ is contained in an equivalence class for the relation of matrix equivalence.

where $\map {\MM_R} n$ 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

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices