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In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893,〔Maz'ya & Shaposhnikova〕 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〔The result is sometimes stated in terms of row vectors. That this is equivalent is seen by applying the transpose.〕 ''vi'', then : and equality is achieved if and only if the vectors are orthogonal or at least one of the columns is 0. ==Alternate forms and corollaries== 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〔Garling〕 : 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.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hadamard's inequality」の詳細全文を読む スポンサード リンク
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