Matrix congruence
In mathematics, two square matrices A {\displaystyle A} and B {\displaystyle B} over a field are called congruent if there exists an invertible matrix P {\displaystyle P} over the same field such that P T A P = B {\displaystyle P^{\mathsf {T}}AP=B} 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. Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors.