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Hadamard's inequality
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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.
A corollary is that if the entries of an n by n matrix N are bounded by B, so |Nij| ≤ 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.
More generally, suppose that N is a complex matrix of order n, whose entries are bounded by |Nij| ≤ 1, for each i, j between 1 and n. Then Hadamard's inequality states that
Equality in this bound is attained for a real matrix N if and only if N is a Hadamard matrix.
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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.
A corollary is that if the entries of an n by n matrix N are bounded by B, so |Nij| ≤ 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.
More generally, suppose that N is a complex matrix of order n, whose entries are bounded by |Nij| ≤ 1, for each i, j between 1 and n. Then Hadamard's inequality states that
Equality in this bound is attained for a real matrix N if and only if N is a Hadamard matrix.