Definition:Matrix Similarity/Definition 2

Definition
Let $R$ be a ring with unity.

Let $n \in \N_{>0}$ be a natural number.

Let $\mathbf A, \mathbf B$ be square matrices of order $n$ over $R$.

$\mathbf A$ and $\mathbf B$ are similar they are the relative matrices, to (possibly) different ordered bases, of the same linear operator.

We write:
 * $\mathbf A \sim \mathbf B$

Also see

 * Equivalence of Definitions of Matrix Similarity