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.
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.
Let We note that Applying the AM-GM inequality to the eigenvalues of
, we see By multiplicativity of determinant, we have In this case, equality holds if and only if M = D that is, all entries of B are 0.
In particular, if the block matrices B and C are also square matrices, then the following inequality by Everett is valid:[2] Thompson's inequality can also be generalized by an inequality in terms of the coefficients of the characteristic polynomial of the block matrices.
Expressing the characteristic polynomial of the matrix A as and supposing that the blocks Mij are m x m matrices, the following inequality by Lin and Zhang is valid:[3] Note that if r = m, then this inequality is identical to Thompson's inequality.