Definition:Matrix Equivalence/Definition 1

Definition
Let $R$ be a ring with unity.

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

Let there exist:


 * an invertible square matrix $\mathbf P$ of order $n$ over $R$
 * an invertible square matrix $\mathbf Q$ of order $m$ over $R$

such that:
 * $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$

Then $\mathbf A$ and $\mathbf B$ are equivalent.

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

Also see

 * Equivalence of Definitions of Matrix Equivalence