= Matrix congruence
{title2=$C=P^TAP$}
Two square <matrices> $A,C$ are congruent when $C=P^TAP$ for an <invertible matrix> $P$. This is the transformation of the <matrix of a bilinear form> under <change of basis>, and also transforms a <quadratic form> under an invertible <linear map> on its <coordinates>. The <matrices> need not be symmetric. It differs from <matrix similarity>, which uses $P^{-1}AP$ for a <linear operator>. For real <symmetric matrices>, <completing the square> diagonalizes the corresponding <quadratic form> by <matrix congruence>.
Back to article page
