Definition:Matrix Equivalence/Definition 2

Definition
Let $R$ be a ring with unity.

Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$.

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

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

Also see

 * Equivalence of Definitions of Matrix Equivalence