Two symmetric matrices are congruent when for an invertible matrix . This is the transformation of a quadratic form under an invertible linear change of coordinates. It differs from matrix similarity, which uses . Completing the square diagonalizes a real quadratic form by matrix congruence.
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Matrix congruence is a concept in linear algebra that relates to two matrices being similar in a specific way through the use of a non-singular matrix. Specifically, two square matrices \( A \) and \( B \) are said to be congruent if there exists a non-singular matrix \( P \) such that: \[ A = P^T B P \] Here, \( P^T \) denotes the transpose of the matrix \( P \).