Matrix congruence

Summary

Two square matrices $A$ and $B$ are congruent when $B = P^{T} A P$ for some invertible matrix $P$. Congruent matrices are the matrices of one bilinear form in two different bases. Congruence is an equivalence relation, and its classification for real symmetric matrices is Sylvester's law of inertia. It differs from similarity, which uses $P^{-1}$ in place of $P^{T}$.