Hadamard's 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

:"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")| ≤ $prod\_\{i=1\}^n$ ||"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.