Definition:Matrix Congruence

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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 if and only if there exists an invertible matrix $\mathbf P\in R^{n\times n}$ such that $\mathbf B = \mathbf P^\intercal \mathbf A \mathbf P$.

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