Hadamard's inequality

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Another inequality is called the Hermite–Hadamard inequality.

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

vi 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

ei for 1 ≤ i ≤ n.

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

|det(M)| ≤ 1.

The corresponding result for the vi is therefore

|det(N)| ≤ \prod_{i=1}^n ||vi||

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

In combinatorics matrices N for which equality holds, and the vi have entries +1 and −1 only are studied; such an M is called an Hadamard matrix.