Hadamard's inequality

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 v_i for 1 ≤ i ≤ n in terms of the lengths of these vectors ||v_i||.

Specifically, Hadamard's inequality states that if N is the matrix having columnsv_i, then

If the n vectors are linearly independent, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal.

A corollary is that if the entries of an n by n matrix N are bounded by B, so |N_ij|≤B for all i and j, then

In particular, if the entries of N are +1 and −1 only then

In combinatorics, matrices N for which equality holds, i.e. those with orthogonal columns, are called Hadamard matrices.

A positive-semidefinite matrix P can be written as N^*N, where N^* denotes the conjugate transpose of N (see Cholesky decomposition). Then

So, the determinant of a positive definite matrix is less than or equal to the product of its diagonal entries. Sometimes this is also known as Hadamard's inequality.

The result is trivial if the matrix N is singular, so assume the columns of N are linearly independent. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if e_i are unit vectors and M is the matrix having the e_i as columns then

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