In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||.
Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then
and equality is achieved if and only if the vectors are orthogonal or at least one of the columns is 0.
Read more about Hadamard's Inequality: Alternate Forms and Corollaries, Proof
Famous quotes containing the word inequality:
“The doctrine of equality!... But there exists no more poisonous poison: for it seems to be preached by justice itself, while it is the end of justice.... “Equality for equals, inequality for unequals”Mthat would be the true voice of justice: and, what follows from it, “Never make equal what is unequal.””
—Friedrich Nietzsche (1844–1900)