Hadamard's inequality

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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)| ≤ ||v_i||

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.