Hadamard's inequality
In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||. Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then If the n vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. Alternate forms and corollaries A corollary is that if the entries of an n by n matrix N are bounded by B, so |Nij|≤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 Decomposition of a semidefinite