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EspañolHADAMARD'S INEQUALITY
(Redirected from Hadamard inequality)
In mathematics, 'Hadamard's inequality' bounds above the volume in Euclidean space of ''n'' dimensions marked out by ''n'' vectors
:''vi'' for 1 ≤ ''i'' ≤ ''n''.
It states, in geometric terms, that this is at a maximum when the vectors are an orthogonal set; the problem is homogeneous with respect to scalar multiplication, so that it is enough to state and prove a result for unit vectors
:''ei'' for 1 ≤ ''i'' ≤ ''n''.
In this case it states simply that if ''M'' is the ''n''× ''n'' matrix with columns the ''ei'', then
:|det(''M'')| ≤ 1.
The corresponding result for the ''vi'' is therefore
::|det(''N'')| ≤ ||''vi''||
with ''N'' the matrix having the ''vi'' as columns, and ||''vi''|| the Euclidean norm (length) of ||''vi''||.
In combinatorics matrices ''N'' for which equality holds, and the ''vi'' have entries +1 and −1 only are studied; such an ''M'' is called an Hadamard matrix.
In mathematics, 'Hadamard's inequality' bounds above the volume in Euclidean space of ''n'' dimensions marked out by ''n'' vectors
:''vi'' for 1 ≤ ''i'' ≤ ''n''.
It states, in geometric terms, that this is at a maximum when the vectors are an orthogonal set; the problem is homogeneous with respect to scalar multiplication, so that it is enough to state and prove a result for unit vectors
:''ei'' for 1 ≤ ''i'' ≤ ''n''.
In this case it states simply that if ''M'' is the ''n''× ''n'' matrix with columns the ''ei'', then
:|det(''M'')| ≤ 1.
The corresponding result for the ''vi'' is therefore
::|det(''N'')| ≤ ||''vi''||
with ''N'' the matrix having the ''vi'' as columns, and ||''vi''|| the Euclidean norm (length) of ||''vi''||.
In combinatorics matrices ''N'' for which equality holds, and the ''vi'' have entries +1 and −1 only are studied; such an ''M'' is called an Hadamard matrix.
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