Fischer's inequality

In mathematics, Fischer's inequality gives an upper bound for the determinant of a positive-semidefinite matrix whose entries are complex numbers in terms of the determinants of its principal diagonal blocks. Suppose A, C are respectively p×p, q×q positive-semidefinite complex matrices and B is a p×q complex matrix. Let

${\displaystyle M:=\left[{\begin{matrix}A&B\\B^{*}&C\end{matrix}}\right]}$

so that M is a (p+q)×(p+q) matrix.

Then Fischer's inequality states that

${\displaystyle \det(M)\leq \det(A)\det(C).}$

If M is positive-definite, equality is achieved in Fischer's inequality if and only if all the entries of B are 0. Inductively one may conclude that a similar inequality holds for a block decomposition of M with multiple principal diagonal blocks. Considering 1×1 blocks, a corollary is Hadamard's inequality. On the other hand, Fischer's inequality can also be proved by using Hadamard's inequality, see the proof of Theorem 7.8.5 in Horn and Johnson's Matrix Analysis.