Determinants | Inequalities

Hadamard's inequality

In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It 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 If the n vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. (Wikipedia).

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Related pages

Inequality of arithmetic and geometric means | Definite matrix | Complex number | Mathematics | Fischer's inequality | Combinatorics | Conjugate transpose | Determinant | Hermite–Hadamard inequality | Jacques Hadamard | Matrix (mathematics) | Hadamard matrix | Euclidean space | Trace (linear algebra) | Volume | Invertible matrix