\begin{definition}[Definition:Matrix Congruence]
Let $R$ be a commutative ring with unity.
Let $n$ be a positive integer.
Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$ over $R$.
Then $\mathbf A$ and $\mathbf B$ are '''congruent''' {{Iff}} there exists an invertible matrix $\mathbf P\in R^{n\times n}$ such that $\mathbf B = \mathbf P^\intercal \mathbf A \mathbf P$.
\end{definition}