\begin{definition}[Definition:Matrix Equivalence/Definition 1]
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:
:$\mathbf A \equiv \mathbf B$
\end{definition}