= Determinant of Hermitian congruence
{title2=$\det(A\mapsto BAB^*)=|\det B|^{2n}$}
For a complex $n\times n$ matrix $B$, the real <linear map> $A\mapsto BAB^*$ on the $n^2$-dimensional real <vector space> of <Hermitian matrices> has <determinant> $|\det B|^{2n}$. A real <basis> of <Hermitian matrices> is also a complex <basis> of all matrices, so the matrix of this restriction equals the matrix of the complex-linear congruence map. Left and right multiplication separately have <determinants> $(\det B)^n$ and $(\overline{\det B})^n$.
Back to article page
