Hadamard's inequality

In mathematics, Hadamard's inequality bounds above the volume in Euclidean space of n dimensions marked out by n vectors

v_i for 1 ≤ i ≤ n.

It states, in geometric terms, that this is at a maximum when the vectors are an orthogonal set; the problem is homogeneous with respect to scalar multiplication, so that it is enough to state and prove a result for unit vectors

e_i for 1 ≤ i ≤ n.

In this case it states simply that if M is the n× n matrix with columns the e_i, then

|det(M)| ≤ 1.

The corresponding result for the v_i is therefore

|det(N)| ≤ Failed to parse (Missing texvc executable; please see math/README to configure.): \prod_{i=1}^n

||vi||

with N the matrix having the v_i as columns, and ||v_i|| the Euclidean norm (length) of ||v_i||.

In combinatorics matrices N for which equality holds, and the v_i have entries +1 and −1 only are studied; such an M is called an Hadamard matrix.de:Hadamard-Ungleichung th:อสมการของฮาดามาร์ด