In mathematics, two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that

where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation.

Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases.