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
- 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)| ≤ Π ||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.