Matrix calculus (original) (raw)
في الرياضيات, يكون حسبان المصفوفات (بالإنجليزية: matrix calculus) عبارة عن ترميز متخصص للقيام بحسبان متعدد المتغيرات, وخصوصاً على فراغات المصفوفات, حيث تعرف أيضاً باسم تفاضل المصفوفات (بالإنجليزية: matrix derivative). هذا النوع من الترميز مناسب تماماً لوصف أنظمة المعادلات التفاضلية, وأيضاً لأخذ تفاضلات الدوال ذو القيم المصفوفية وذلك بالنسبة إلى المتغيرات المصفوفية. يُستعمل ذا الترميز عادةً في الإحصاء وفي الهندسة, بينما يُفضل استعمال Tensor index notation في الفيزياء.
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| dbo:abstract | في الرياضيات, يكون حسبان المصفوفات (بالإنجليزية: matrix calculus) عبارة عن ترميز متخصص للقيام بحسبان متعدد المتغيرات, وخصوصاً على فراغات المصفوفات, حيث تعرف أيضاً باسم تفاضل المصفوفات (بالإنجليزية: matrix derivative). هذا النوع من الترميز مناسب تماماً لوصف أنظمة المعادلات التفاضلية, وأيضاً لأخذ تفاضلات الدوال ذو القيم المصفوفية وذلك بالنسبة إلى المتغيرات المصفوفية. يُستعمل ذا الترميز عادةً في الإحصاء وفي الهندسة, بينما يُفضل استعمال Tensor index notation في الفيزياء. (ar) V matematice je maticový počet speciální zápis pro realizaci matematického počtu více proměnných, zvláště v maticových prostorech. Shromažďuje různé parciální derivace jedné funkce s ohledem na více proměnných, a/nebo parciální derivace funkce více proměnných s ohledem na jednu proměnnou, do vektorů a matic, které mohou být považovány za jednu entitu. To značně zjednodušuje operace jako hledání maxima nebo minima funkce více proměnných a řešení systému diferenciálních rovnic. Notace (zápis) použitý zde se obvykle používá v statistice a inženýrství, zatímco tenzorová indexová notace se upřednostňuje ve fyzice. Dvě notační (zápisové) konvence rozdělily obor maticového počtu do dvou separátních skupin. Tyto dvě skupiny možno rozeznat podle toho, jak zapisují derivaci skaláru s ohledem na vektor jako sloupcový vektor nebo jako řádkový vektor. Obě tyto konvence jsou možné i když se udělá obecný předpoklad, že vektory nutno považovat za sloupcové vektory, když se kombinují s maticemi (dříve než ). Jediná konvence může být poněkud standardní přes jeden obor, který obvykle používá maticový počet (např. ekonometrie, statistika, a strojové učení. Ale i v daném oboru různí autoři používají odlišné konvence. Autoři obou skupin často píšou jakoby jejich specifická konvence byla standard. Vážné chyby mohou rezultovat při kombinaci výsledků od různých autorů bez pečlivého ověření, že jsou použity kompatibilní notace. Proto je nutno věnovat velkou pozornost zajištění zápisové jednoty. Definice těchto dvou konvencí a porovnání mezi nimi jsou dále v článku. (cs) In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific convention were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the section. (en) Dalam matematika kalkulus matriks adalah notasi khusus untuk menghitung kalkulus multivariabel (kalkulus peubah banyak), terutama pada ruang matriks. Pada ruang matriks notasi ini mendefinisikan turunan matriks. Notasi ini cocok untuk memerikan sistem persamaan diferensial, dan mengambil turunan dari fungsi matriks terhadap variabel berbentuk matriks pula. Kalkulus matriks umum digunakan dalam statistika dan rekayasa, sedangkan lebih disukai dalam fisika. (in) Na matemática, o Cálculo matricial é uma notação especial para tratar o cálculo multivariável, especialmente em espaços de matrizes, onde está definida a . Esta notação é conveniente para descrever sistemas de equações diferenciais e para calcular o diferencial de funções de matrizes. Esta notação é utilizada em Estatística e Engenharia; físicos preferem usar a notação de Einstein. O princípio básico desta notação é tratar cada vetor como uma , e identificar uma matriz 1x1 com o escalar. (pt) 在数学中, 矩阵微积分是多元微积分的一种特殊表达，尤其是在矩阵空间上进行讨论的时候。它把单个函数对多个变量或者多元函数对单个变量的偏导数写成向量和矩阵的形式，使其可以被当成一个整体被处理。這使得要在多元函數尋找最大或最小值，又或是要為微分方程系統尋解的過程大幅簡化。这里我们主要使用统计学和工程学中的惯用记法，而更常用于物理学中。 (zh) |
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| rdfs:comment | في الرياضيات, يكون حسبان المصفوفات (بالإنجليزية: matrix calculus) عبارة عن ترميز متخصص للقيام بحسبان متعدد المتغيرات, وخصوصاً على فراغات المصفوفات, حيث تعرف أيضاً باسم تفاضل المصفوفات (بالإنجليزية: matrix derivative). هذا النوع من الترميز مناسب تماماً لوصف أنظمة المعادلات التفاضلية, وأيضاً لأخذ تفاضلات الدوال ذو القيم المصفوفية وذلك بالنسبة إلى المتغيرات المصفوفية. يُستعمل ذا الترميز عادةً في الإحصاء وفي الهندسة, بينما يُفضل استعمال Tensor index notation في الفيزياء. (ar) Dalam matematika kalkulus matriks adalah notasi khusus untuk menghitung kalkulus multivariabel (kalkulus peubah banyak), terutama pada ruang matriks. Pada ruang matriks notasi ini mendefinisikan turunan matriks. Notasi ini cocok untuk memerikan sistem persamaan diferensial, dan mengambil turunan dari fungsi matriks terhadap variabel berbentuk matriks pula. Kalkulus matriks umum digunakan dalam statistika dan rekayasa, sedangkan lebih disukai dalam fisika. (in) Na matemática, o Cálculo matricial é uma notação especial para tratar o cálculo multivariável, especialmente em espaços de matrizes, onde está definida a . Esta notação é conveniente para descrever sistemas de equações diferenciais e para calcular o diferencial de funções de matrizes. Esta notação é utilizada em Estatística e Engenharia; físicos preferem usar a notação de Einstein. O princípio básico desta notação é tratar cada vetor como uma , e identificar uma matriz 1x1 com o escalar. (pt) 在数学中, 矩阵微积分是多元微积分的一种特殊表达，尤其是在矩阵空间上进行讨论的时候。它把单个函数对多个变量或者多元函数对单个变量的偏导数写成向量和矩阵的形式，使其可以被当成一个整体被处理。這使得要在多元函數尋找最大或最小值，又或是要為微分方程系統尋解的過程大幅簡化。这里我们主要使用统计学和工程学中的惯用记法，而更常用于物理学中。 (zh) V matematice je maticový počet speciální zápis pro realizaci matematického počtu více proměnných, zvláště v maticových prostorech. Shromažďuje různé parciální derivace jedné funkce s ohledem na více proměnných, a/nebo parciální derivace funkce více proměnných s ohledem na jednu proměnnou, do vektorů a matic, které mohou být považovány za jednu entitu. To značně zjednodušuje operace jako hledání maxima nebo minima funkce více proměnných a řešení systému diferenciálních rovnic. Notace (zápis) použitý zde se obvykle používá v statistice a inženýrství, zatímco tenzorová indexová notace se upřednostňuje ve fyzice. (cs) In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. (en) |
| rdfs:label | حساب المصفوفات (ar) Maticový počet (cs) Kalkulus matriks (in) Matrix calculus (en) Cálculo matricial (pt) 矩阵微积分 (zh) |
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