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/sparql endpoint"> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div>   <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6">About: <a href="http://dbpedia.org/resource/Trace_(linear_algebra)">Trace (linear algebra)</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/class/yago/Abstraction100002137">Abstraction100002137</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n). The trace is related to the derivative of the determinant (see Jacobi's formula). </div> </div> </div> </section>   <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">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">في الجبر الخطي، أثر مصفوفة مربعة A (بالإنجليزية: Trace of matrix)‏ هو مجموع مداخل المصفوفة الواقعة على القطر الرئيسي (القطر الممتد من العنصر الأعلى يسارا إلى العنصر الأسفل يمينا). حيث aii تمثل المدخل على الصف i والعمود i للمصفوفة A.أثر المصفوفة هو مجموع قيمها الذاتية، حقيقية كانت أم عقدية. أثر مصفوفة لا يتغير بتغيير القاعدة، صانعًا منها . (ar)</li> <li style="display:none;">En àlgebra lineal, la traça d'una matriu quadrada A d'nxn es defineix com la suma dels elements de la diagonal principal d'A, és a dir on aij representa l'element que és a la fila i-èsima i a la columna j-èsima d'A. (ca)</li> <li style="display:none;">Stopa je v oboru lineární algebry označení pro součet prvků na hlavní diagonále čtvercové matice. Pro matici: má tedy stopa podobu (cs)</li> <li style="display:none;">Die Spur (Spurfunktion, Spurabbildung) ist ein Konzept in den mathematischen Teilgebieten der Linearen Algebra sowie der Funktionalanalysis und wird auch in der Theorie der Körper und Körpererweiterungen verwendet. (de)</li> <li style="display:none;">En lineara algebro, la spuro de n-per-n kvadrata matrico A estas sumo de la eroj sur la ĉefdiagonalo de A, kio estas kie . (eo)</li> <li style="display:none;">Aljebra linealean, matrize karratu baten aztarna bere diagonal nagusiko elementuen batura da. Hau da, non aij i-garren errenkadan eta j-garren zutabean dagoen elementua den. (eu)</li> <li style="display:none;">En algèbre linéaire, la trace d'une matrice carrée A est définie comme la somme de ses coefficients diagonaux et souvent notée Tr(A). La trace peut être vue comme une forme linéaire sur l'espace vectoriel des matrices. Elle vérifie l'identité : Tr(AB) = Tr(BA), et est en conséquence invariante par similitude. De façon voisine, si u est un endomorphisme d'un espace vectoriel de dimension finie sur un corps commutatif K, on peut définir la trace de l'opérateur u, par exemple comme trace de sa matrice dans n'importe quelle base. Plus généralement, sur une algèbre A, une trace est une forme linéaire λ telle que λ(ab) = λ(ba). Cette définition se rencontre en particulier dans l'étude des algèbres de von Neumann, qui sont des algèbres d'opérateurs sur des espaces de Hilbert. (fr)</li> <li style="display:none;">En álgebra lineal, la traza de una matriz cuadrada A de nxn está definida como la suma de los elementos de la diagonal principal de A. Es decir, donde aij representa el elemento que está en la fila i-ésima y en la columna j-ésima de A.Para cualquier otra matriz, la traza es la suma de sus valores propios. Debido al especial comportamiento de la traza de una matriz al cambiar de base puede definirse unívocamente la traza de una aplicación lineal, independientemente de cual sea la base elegida. Si un espacio vectorial de dimensión finita está dotado de un producto escalar, y se tiene una base ortonormal entonces la traza de un endomorfismo de dicho espacio viene dada por: Puede comprobarse que si Af es la matriz de dicha aplicación respecto a dicha base la cantidad anterior es igual a la traza de la matriz A. Y de hecho si Bf es la matriz de la misma aplicación respecto a cualquier otra base ortonormal se tiene: (es)</li> <li style="display:none;">Dalam aljabar linear, teras (juga disebut dengan trace), dari matriks persegi didefinisikan sebagai jumlah dari setiap elemen pada diagonal utama matriks tersebut. Notasi yang digunakan untuk mewakili teras dari matriks A adalah tr(A). Nilai teras juga sama dengan jumlah nilai eigen (kompleks) berserta kelipatannya dari matriks tersebut; dan tidak bergantung pada basis yang dipakai. Sifat ini dapat digunakan untuk mendefinisikan teras dari operator linear secara umum. Nilai teras hanya terdefinisi untuk matriks persegi (matriks berukuran n × n). Nilai teras suatu matriks juga berhubungan dengan turunan dari determinan (lihat ) (in)</li> <li style="display:none;">数学の線型代数学において、正方行列の跡（せき、英: trace; トレース、独: Spur; シュプール）あるいは対角和（たいかくわ）とは、主対角成分の総和である。つまりを指す。それは基底変換に関して不変であり、また固有値の総和（固有値和）に等しい。即ち、行列の跡は行列の相似を除いて定まり、したがって一般に行列に対応する線型写像の跡として定義することができる。行列の跡は、正方行列に対してのみ定義されることに注意せよ。この語は（この同じ数学的対象を意味する）ドイツ語のSpurからの翻訳借用である。 (ja)</li> <li>In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that tr(AB) = tr(BA) for any two matrices A and B. This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). (en)</li> <li style="display:none;">선형대수학에서 대각합(對角合, 영어: trace 트레이스[*], 독일어: Spur 슈푸어[*])은 정사각 행렬의 주대각선 성분들의 합이다. 기호는 tr 또는 Sp. (ko)</li> <li style="display:none;">In de lineaire algebra, een deelgebied van de wiskunde, is het spoor (naar het Duitse Spur, in het Engels later vertaald door trace), aangeduid door sp of tr, van de vierkante matrix de som van de elementen van de hoofddiagonaal van , waarin het element in de -de rij en -de kolom van is. (nl)</li> <li style="display:none;">In algebra lineare, si definisce traccia di una matrice quadrata la somma di tutti gli elementi della sua diagonale principale. Nel caso di endomorfismi di uno spazio vettoriale, è possibile definire la traccia di un endomorfismo considerando la traccia della sua matrice associata rispetto ad una qualsiasi base dello spazio. Poiché la traccia è invariante per similitudine, questo valore non dipende dalla base scelta. (it)</li> <li style="display:none;">Na álgebra linear, o traço de uma matriz quadrada é a função matricial que associa a matriz à soma dos elementos da sua diagonal principal. Se A=[aij], então . O traço de uma aplicação linear num espaço vectorial de dimensão finita é o traço da matriz que representa essa aplicação em relação a uma dada base. Este traço está bem definido porque o traço de uma matriz é invariante por semelhanças (o que é uma consequência do facto de que tr(AB)=tr(BA), para quaisquer matrizes quadradas A e B da mesma ordem). (pt)</li> <li style="display:none;">Ślad macierzy – suma elementów na głównej przekątnej macierzy kwadratowej. (pl)</li> <li style="display:none;">Spåret är inom matematiken summan av diagonalelementen i en kvadratisk matris. Dvs, givet matrisen av storlek med elementen , är spåret: Beteckningen kommer från engelskans trace. Även beteckningen efter tyskans spur förekommer. (sv)</li> <li style="display:none;">След матрицы — операция, отображающая пространство квадратных матриц в поле, над которым определена матрица (для действительных матриц — в поле действительных чисел, для комплексных матриц — в поле комплексных чисел). След матрицы — это сумма элементов главной диагонали матрицы, то есть если элементы матрицы , то её след . Матрицы с нулевым следом называют бесследовыми (от англ. traceless или tracefree). В математических текстах встречается два обозначения операции взятия следа: (от англ. trace — след), и (от нем. Spur — след). В тензорном исчислении следом тензора второго ранга (один раз ковариантного и один раз контравариантного) называется сумма его диагональных элементов. Независимо от ковариантности и контравариантности, след тензора второго ранга вычисляется как двойное скалярное произведение тензора с метрическим тензором и является :. (ru)</li> <li style="display:none;">Слід матриці — операція, що відображає простір квадратних матриць у поле, над яким визначена матриця (див. функціонал). Слід матриці — це сума усіх її діагональних елементів, тобто якщо елементи матриці , її слід дорівнює: В математичних текстах зустрічається два позначення операції взяття сліду: (трейс, від англ. Trace — слід), і (шпур, від нім. Spur — слід). (uk)</li> <li style="display:none;">在线性代数中，一個的矩陣的跡（或跡數），是指的主對角線（從左上方至右下方的對角線）上各個元素的總和，一般記作或：其中代表矩陣的第i行j列上的元素的值。一個矩陣的跡是其特徵值的總和（按代數重數計算）。跡的英文為trace，是來自德文中的Spur這個單字（與英文中的Spoor是同源詞），在數學中，通常簡寫為「Sp」或「tr」。 (zh)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID">dbo:wikiPageID</a> </td><td class="col-10 text-break"><ul> <li>43270 (xsd:integer)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageLength">dbo:wikiPageLength</a> </td><td class="col-10 text-break"><ul> <li>36078 (xsd:nonNegativeInteger)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageRevisionID">dbo:wikiPageRevisionID</a> </td><td class="col-10 text-break"><ul> <li>1121214869 (xsd:integer)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageWikiLink">dbo:wikiPageWikiLink</a> </td><td 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من العنصر الأعلى يسارا إلى العنصر الأسفل يمينا). حيث aii تمثل المدخل على الصف i والعمود i للمصفوفة A.أثر المصفوفة هو مجموع قيمها الذاتية، حقيقية كانت أم عقدية. أثر مصفوفة لا يتغير بتغيير القاعدة، صانعًا منها . (ar)</li> <li style="display:none;">En àlgebra lineal, la traça d'una matriu quadrada A d'nxn es defineix com la suma dels elements de la diagonal principal d'A, és a dir on aij representa l'element que és a la fila i-èsima i a la columna j-èsima d'A. (ca)</li> <li style="display:none;">Stopa je v oboru lineární algebry označení pro součet prvků na hlavní diagonále čtvercové matice. Pro matici: má tedy stopa podobu (cs)</li> <li style="display:none;">Die Spur (Spurfunktion, Spurabbildung) ist ein Konzept in den mathematischen Teilgebieten der Linearen Algebra sowie der Funktionalanalysis und wird auch in der Theorie der Körper und Körpererweiterungen verwendet. (de)</li> <li style="display:none;">En lineara algebro, la spuro de n-per-n kvadrata matrico A estas sumo de la eroj sur la ĉefdiagonalo de A, kio estas kie . (eo)</li> <li style="display:none;">Aljebra linealean, matrize karratu baten aztarna bere diagonal nagusiko elementuen batura da. Hau da, non aij i-garren errenkadan eta j-garren zutabean dagoen elementua den. (eu)</li> <li style="display:none;">数学の線型代数学において、正方行列の跡（せき、英: trace; トレース、独: Spur; シュプール）あるいは対角和（たいかくわ）とは、主対角成分の総和である。つまりを指す。それは基底変換に関して不変であり、また固有値の総和（固有値和）に等しい。即ち、行列の跡は行列の相似を除いて定まり、したがって一般に行列に対応する線型写像の跡として定義することができる。行列の跡は、正方行列に対してのみ定義されることに注意せよ。この語は（この同じ数学的対象を意味する）ドイツ語のSpurからの翻訳借用である。 (ja)</li> <li style="display:none;">선형대수학에서 대각합(對角合, 영어: trace 트레이스[*], 독일어: Spur 슈푸어[*])은 정사각 행렬의 주대각선 성분들의 합이다. 기호는 tr 또는 Sp. (ko)</li> <li style="display:none;">In de lineaire algebra, een deelgebied van de wiskunde, is het spoor (naar het Duitse Spur, in het Engels later vertaald door trace), aangeduid door sp of tr, van de vierkante matrix de som van de elementen van de hoofddiagonaal van , waarin het element in de -de rij en -de kolom van is. (nl)</li> <li style="display:none;">In algebra lineare, si definisce traccia di una matrice quadrata la somma di tutti gli elementi della sua diagonale principale. Nel caso di endomorfismi di uno spazio vettoriale, è possibile definire la traccia di un endomorfismo considerando la traccia della sua matrice associata rispetto ad una qualsiasi base dello spazio. Poiché la traccia è invariante per similitudine, questo valore non dipende dalla base scelta. (it)</li> <li style="display:none;">Na álgebra linear, o traço de uma matriz quadrada é a função matricial que associa a matriz à soma dos elementos da sua diagonal principal. Se A=[aij], então . O traço de uma aplicação linear num espaço vectorial de dimensão finita é o traço da matriz que representa essa aplicação em relação a uma dada base. Este traço está bem definido porque o traço de uma matriz é invariante por semelhanças (o que é uma consequência do facto de que tr(AB)=tr(BA), para quaisquer matrizes quadradas A e B da mesma ordem). (pt)</li> <li style="display:none;">Ślad macierzy – suma elementów na głównej przekątnej macierzy kwadratowej. (pl)</li> <li style="display:none;">Spåret är inom matematiken summan av diagonalelementen i en kvadratisk matris. Dvs, givet matrisen av storlek med elementen , är spåret: Beteckningen kommer från engelskans trace. Även beteckningen efter tyskans spur förekommer. (sv)</li> <li style="display:none;">Слід матриці — операція, що відображає простір квадратних матриць у поле, над яким визначена матриця (див. функціонал). Слід матриці — це сума усіх її діагональних елементів, тобто якщо елементи матриці , її слід дорівнює: В математичних текстах зустрічається два позначення операції взяття сліду: (трейс, від англ. Trace — слід), і (шпур, від нім. Spur — слід). (uk)</li> <li style="display:none;">在线性代数中，一個的矩陣的跡（或跡數），是指的主對角線（從左上方至右下方的對角線）上各個元素的總和，一般記作或：其中代表矩陣的第i行j列上的元素的值。一個矩陣的跡是其特徵值的總和（按代數重數計算）。跡的英文為trace，是來自德文中的Spur這個單字（與英文中的Spoor是同源詞），在數學中，通常簡寫為「Sp」或「tr」。 (zh)</li> <li style="display:none;">En álgebra lineal, la traza de una matriz cuadrada A de nxn está definida como la suma de los elementos de la diagonal principal de A. Es decir, donde aij representa el elemento que está en la fila i-ésima y en la columna j-ésima de A.Para cualquier otra matriz, la traza es la suma de sus valores propios. Puede comprobarse que si Af es la matriz de dicha aplicación respecto a dicha base la cantidad anterior es igual a la traza de la matriz A. Y de hecho si Bf es la matriz de la misma aplicación respecto a cualquier otra base ortonormal se tiene: (es)</li> <li style="display:none;">En algèbre linéaire, la trace d'une matrice carrée A est définie comme la somme de ses coefficients diagonaux et souvent notée Tr(A). La trace peut être vue comme une forme linéaire sur l'espace vectoriel des matrices. Elle vérifie l'identité : Tr(AB) = Tr(BA), et est en conséquence invariante par similitude. De façon voisine, si u est un endomorphisme d'un espace vectoriel de dimension finie sur un corps commutatif K, on peut définir la trace de l'opérateur u, par exemple comme trace de sa matrice dans n'importe quelle base. (fr)</li> <li style="display:none;">Dalam aljabar linear, teras (juga disebut dengan trace), dari matriks persegi didefinisikan sebagai jumlah dari setiap elemen pada diagonal utama matriks tersebut. Notasi yang digunakan untuk mewakili teras dari matriks A adalah tr(A). Nilai teras juga sama dengan jumlah nilai eigen (kompleks) berserta kelipatannya dari matriks tersebut; dan tidak bergantung pada basis yang dipakai. Sifat ini dapat digunakan untuk mendefinisikan teras dari operator linear secara umum. Nilai teras hanya terdefinisi untuk matriks persegi (matriks berukuran n × n). (in)</li> <li>In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n). The trace is related to the derivative of the determinant (see Jacobi's formula). (en)</li> <li style="display:none;">След матрицы — операция, отображающая пространство квадратных матриц в поле, над которым определена матрица (для действительных матриц — в поле действительных чисел, для комплексных матриц — в поле комплексных чисел). След матрицы — это сумма элементов главной диагонали матрицы, то есть если элементы матрицы , то её след . Матрицы с нулевым следом называют бесследовыми (от англ. traceless или tracefree). В математических текстах встречается два обозначения операции взятия следа: (от англ. trace — след), и (от нем. Spur — след). (ru)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">أثر (جبر خطي) (ar)</li> <li style="display:none;">Traça (àlgebra lineal) (ca)</li> <li style="display:none;">Stopa (algebra) (cs)</li> <li style="display:none;">Spur (Mathematik) (de)</li> <li style="display:none;">Ίχνος πίνακα (el)</li> <li style="display:none;">Spuro (lineara algebro) (eo)</li> <li style="display:none;">Traza (álgebra lineal) (es)</li> <li style="display:none;">Aztarna (aljebra) (eu)</li> <li style="display:none;">Teras (aljabar linear) (in)</li> <li style="display:none;">Trace (algèbre) (fr)</li> <li style="display:none;">Traccia (matrice) (it)</li> <li style="display:none;">跡 (線型代数学) (ja)</li> <li style="display:none;">대각합 (ko)</li> <li style="display:none;">Spoor (lineaire algebra) (nl)</li> <li style="display:none;">Ślad (algebra liniowa) (pl)</li> <li 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