Hadamard's inequality

Not to be confused with the Hermite–Hadamard inequality.

In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants[1]) is a result first published by Jacques Hadamard in 1893.[2] 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[3] vi, then

If the n vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal.

  1. ^ "Hadamard theorem - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-06-15.
  2. ^ Maz'ya & Shaposhnikova
  3. ^ The result is sometimes stated in terms of row vectors. That this is equivalent is seen by applying the transpose.