Definition:Matrix Congruence

This page is about Matrix Congruence. For other uses, see Congruence.

Definition

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 $\mathbf B = \mathbf P^\intercal \mathbf A \mathbf P$

where $\mathbf P^\intercal$ denotes the transpose of $\mathbf P$.

Also see

• Definition:Matrix Equivalence
• Definition:Matrix Similarity
• Matrix Congruence is Equivalence Relation

• Results about matrix congruence can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruent matrices
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): matrix (plural matrices): $(2)$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruent matrices
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): matrix (plural matrices): $(2)$