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About: Hadamard's inequality
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class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">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</p> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="cs" >Hadamardova nerovnost je označení pro matematickou nerovnost poprvé zveřejněnou Jacquesem Hadamardem v 1893 a vymezující maximální možnou hodnotu determinantu matice složené z komplexních vektorů. V případě reálných čísel ji je možné v n-rozměrném eukleidovském prostoru interpretovat jako horní mez maximálního možného objemu rovnoběžnostěnu vymezeného n vektory vzhledem k jejich délkám . Hadamardova nerovnost říká, že pokud je V matice se sloupci , pak a rovnosti je dosaženo pouze v těch případech, kdy jsou na sebe vektory kolmé a nebo je některý ze sloupců roven nulovému vektoru.</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >In der Mathematik beschreibt die Hadamard-Ungleichung eine Abschätzung für die Determinante einer quadratischen Matrix. Benannt ist sie nach dem französischen Mathematiker Jacques Salomon Hadamard.</span><small> (de)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >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.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="nl" >In de wiskunde geeft de ongelijkheid van Hadamard een bovengrens voor de absolute waarde van de determinant van een vierkante matrix. Ze is genoemd naar de Franse wiskundige Jacques Hadamard.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="uk" >Нерівність Адамара (також теорема Адамара про визначники), визначає верхню межу об'єму паралелепіпеда в -вимірному евклідовому просторі, заданого векторами.Названа на честь Жака Адамара.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="zh" >数学中的阿达马不等式給出一個基於n维複矩陣列向量的行列式值上界。當僅套用於實數時，其可以在歐幾里得空間中，由n支向量, , 标出的体积。' 这不等式的几何意义是当向量为正交集时体积最大。这结果相对于标量乘法齊次，所以只需证明单位向量, , 的结果。在这情况，不等式指出：若是以为列向量的n× n 矩阵，则 。 因此，向量的相应结果是 ， 其中是以为列向量的矩阵，而是的歐幾里得范数（长度）。（就是說若，則 。） 在组合数学中，使等式成立以及列向量的元素为+1和−1的矩阵是研究对象，它们称为阿达马矩阵。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ru" >Нера́венство Адама́ра (также теорема Адамара об определителях), определяет верхнюю границу объёма тела в -мерном евклидовом пространстве, заданного векторами.Названо в честь Жака Адамара.</span><small> (ru)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink"><small>dbo:</small>wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://archive.org/details/inequalitiesjour00djhg" href="https://archive.org/details/inequalitiesjour00djhg">https://archive.org/details/inequalitiesjour00djhg</a></span></li> <li><span class="literal"><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://archive.org/details/inequalitiesjour00djhg/page/n244" href="https://archive.org/details/inequalitiesjour00djhg/page/n244">https://archive.org/details/inequalitiesjour00djhg/page/n244</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" 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text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="cs" >Hadamardova nerovnost je označení pro matematickou nerovnost poprvé zveřejněnou Jacquesem Hadamardem v 1893 a vymezující maximální možnou hodnotu determinantu matice složené z komplexních vektorů. V případě reálných čísel ji je možné v n-rozměrném eukleidovském prostoru interpretovat jako horní mez maximálního možného objemu rovnoběžnostěnu vymezeného n vektory vzhledem k jejich délkám . Hadamardova nerovnost říká, že pokud je V matice se sloupci , pak a rovnosti je dosaženo pouze v těch případech, kdy jsou na sebe vektory kolmé a nebo je některý ze sloupců roven nulovému vektoru.</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >In der Mathematik beschreibt die Hadamard-Ungleichung eine Abschätzung für die Determinante einer quadratischen Matrix. Benannt ist sie nach dem französischen Mathematiker Jacques Salomon Hadamard.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="nl" >In de wiskunde geeft de ongelijkheid van Hadamard een bovengrens voor de absolute waarde van de determinant van een vierkante matrix. Ze is genoemd naar de Franse wiskundige Jacques Hadamard.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="uk" >Нерівність Адамара (також теорема Адамара про визначники), визначає верхню межу об'єму паралелепіпеда в -вимірному евклідовому просторі, заданого векторами.Названа на честь Жака Адамара.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="zh" >数学中的阿达马不等式給出一個基於n维複矩陣列向量的行列式值上界。當僅套用於實數時，其可以在歐幾里得空間中，由n支向量, , 标出的体积。' 这不等式的几何意义是当向量为正交集时体积最大。这结果相对于标量乘法齊次，所以只需证明单位向量, , 的结果。在这情况，不等式指出：若是以为列向量的n× n 矩阵，则 。 因此，向量的相应结果是 ， 其中是以为列向量的矩阵，而是的歐幾里得范数（长度）。（就是說若，則 。） 在组合数学中，使等式成立以及列向量的元素为+1和−1的矩阵是研究对象，它们称为阿达马矩阵。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ru" >Нера́венство Адама́ра (также теорема Адамара об определителях), определяет верхнюю границу объёма тела в -мерном евклидовом пространстве, заданного векторами.Названо в честь Жака Адамара.</span><small> (ru)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >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</span><small> (en)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="cs" >Hadamardova nerovnost</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Hadamard-Ungleichung</span><small> (de)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Hadamard's inequality</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="nl" >Ongelijkheid van Hadamard</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ru" >Неравенство Адамара</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="uk" >Нерівність Адамара</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="zh" >阿达马不等式</span><small> (zh)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs"><small>owl:</small>sameAs</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.02wg7j" href="http://rdf.freebase.com/ns/m.02wg7j"><small>freebase</small>:Hadamard's inequality</a></span></li> <li><span class="literal"><a class="uri" rel="owl:sameAs" resource="http://yago-knowledge.org/resource/Hadamard's_inequality" href="http://yago-knowledge.org/resource/Hadamard's_inequality"><small>yago-res</small>:Hadamard's inequality</a></span></li> <li><span class="literal"><a 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