Hadamard's Inequality - Alternate Forms and Corollaries

Alternate Forms and Corollaries

A corollary is that if the entries of an n by n matrix N are bounded by B, so |N_ij|≤B for all i and j, then

In particular, if the entries of N are +1 and −1 only then

In combinatorics, matrices N for which equality holds, i.e. those with orthogonal columns, are called Hadamard matrices.

A positive-semidefinite matrix P can be written as N*N, where N* denotes the conjugate transpose of N (see Cholesky decomposition). Then

So, the determinant of a positive definite matrix is less than or equal to the product of its diagonal entries. Sometimes this is also known as Hadamard's inequality.