"In der Mathematik ist das Matrixexponential, auch als Matrixexponentialfunktion bezeichnet, eine Funktion auf der Menge der quadratischen Matrizen, welche analog zur gew\u00F6hnlichen (skalaren) Exponentialfunktion definiert ist. Das Matrixexponential stellt die Verbindung zwischen Lie-Algebra und der zugeh\u00F6rigen Lie-Gruppe her."@de . . "\u0415\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u0456"@uk . . . . . . . "\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u884C\u5217\u306E\u6307\u6570\u95A2\u6570\uFF08\u304E\u3087\u3046\u308C\u3064\u306E\u3057\u3059\u3046\u304B\u3093\u3059\u3046\u3001\u82F1\u8A9E: matrix exponential; \u884C\u5217\u4E57\uFF09\u306F\u3001\u6B63\u65B9\u884C\u5217\u306B\u5BFE\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u884C\u5217\u5024\u95A2\u6570\u3067\u3001\u901A\u5E38\u306E\uFF08\u5B9F\u307E\u305F\u306F\u8907\u7D20\u5909\u6570\u306E\uFF09\u6307\u6570\u95A2\u6570\u306B\u5BFE\u5FDC\u3059\u308B\u3082\u306E\u3067\u3042\u308B\u3002\u3088\u308A\u62BD\u8C61\u7684\u306B\u306F\u3001\u884C\u5217\u30EA\u30FC\u7FA4\u3068\u305D\u306E\u884C\u5217\u30EA\u30FC\u4EE3\u6570\u306E\u9593\u306E\u5BFE\u5FDC\u95A2\u4FC2\uFF08\u6307\u6570\u5199\u50CF\uFF09\u3092\u884C\u5217\u306E\u6307\u6570\u51FD\u6570\u304C\u8A18\u8FF0\u3059\u308B\u3002 n\u2009\u00D7\u2009n \u5B9F\u307E\u305F\u306F\u8907\u7D20\u884C\u5217 X \u306E\u6307\u6570\u95A2\u6570 eX \u307E\u305F\u306F exp(X) \u306F\u3001\u51AA\u7D1A\u6570 \u3067\u5B9A\u7FA9\u3055\u308C\u308B n-\u6B21\u6B63\u65B9\u884C\u5217\u3067\u3042\u308B\u3002\u3053\u306E\u7D1A\u6570\u306F\u4EFB\u610F\u306E X \u306B\u5BFE\u3057\u3066\u53CE\u675F\u3059\u308B\u304B\u3089\u3001\u884C\u5217 X \u306E\u6307\u6570\u95A2\u6570\u306F well-defined \u3067\u3042\u308B\u3002 X \u304C 1\u2009\u00D7\u20091 \u884C\u5217\u306E\u3068\u304D\u3001X-\u4E57 eX \u306F 1\u2009\u00D7\u20091 \u884C\u5217\u3067\u3042\u308A\u3001\u305D\u306E\u552F\u4E00\u306E\u6210\u5206\u306F X \u306E\u552F\u4E00\u306E\u6210\u5206\u306B\u5BFE\u3059\u308B\u901A\u5E38\u306E\u6307\u6570\u95A2\u6570\u306B\u4E00\u81F4\u3059\u308B\u3002\u3053\u308C\u3089\u306F\u3057\u3070\u3057\u3070\u540C\u4E00\u8996\u3055\u308C\u308B\u3002\u3053\u306E\u610F\u5473\u306B\u304A\u3044\u3066\u884C\u5217\u306E\u6307\u6570\u51FD\u6570\u306F\u3001\u901A\u5E38\u306E\u6307\u6570\u51FD\u6570\u306E\u4E00\u822C\u5316\u3067\u3042\u308B\u3002"@ja . "Matrisexponentialfunktion"@sv . . . . . . "672731"^^ . . . "L'exponencial d'una matriu \u00E9s una funci\u00F3 definida sobre les matrius quadrades, similar a la funci\u00F3 exponencial. Sigui una matriu de nombres reals o complexos. L'exponencial de , denotada per o \u00E9s la matriu definida per la s\u00E8rie de pot\u00E8ncies: Aquesta s\u00E8rie convergeix per a tota matriu . Observem que, si la matriu \u00E9s una matriu 1\u00D71, l'exponencial de correspon amb l'exponencial ordin\u00E0ria."@ca . . . . . . . . . . . . "6"^^ . . . . . . . . . . "Matrisexponentialfunktionen \u00E4r inom matematiken en ut\u00F6kning av exponentialfunktionen fr\u00E5n komplexa tal till att g\u00E4lla \u00E4ven kvadratiska matriser, s\u00E5 att man f\u00E5r en matrisfunktion."@sv . . . . . "Em matem\u00E1tica, a exponencial matricial \u00E9 uma fun\u00E7\u00E3o matricial definida no conjunto das matrizes quadradas e possui propriedades semelhantes \u00E0 fun\u00E7\u00E3o exponencial definida nos n\u00FAmeros reais (ou complexos). Mais abstratamente falando a exponencial matricial estabelece uma conex\u00E3o entre a \u00E1lgebra de Lie das matrizes e o seu correspondente grupo de Lie. Seja uma matriz real ou complexa , define-se pela seguinte s\u00E9rie de pot\u00EAncias: , onde \u00E9 a matriz identidade A converg\u00EAncia desta s\u00E9rie \u00E9 garantida pelo teste M de Weierstrass."@pt . "Eksponenta macierzy"@pl . "In der Mathematik ist das Matrixexponential, auch als Matrixexponentialfunktion bezeichnet, eine Funktion auf der Menge der quadratischen Matrizen, welche analog zur gew\u00F6hnlichen (skalaren) Exponentialfunktion definiert ist. Das Matrixexponential stellt die Verbindung zwischen Lie-Algebra und der zugeh\u00F6rigen Lie-Gruppe her."@de . . . . . . "1.2"^^ . . . . . . . . "\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u884C\u5217\u306E\u6307\u6570\u95A2\u6570\uFF08\u304E\u3087\u3046\u308C\u3064\u306E\u3057\u3059\u3046\u304B\u3093\u3059\u3046\u3001\u82F1\u8A9E: matrix exponential; \u884C\u5217\u4E57\uFF09\u306F\u3001\u6B63\u65B9\u884C\u5217\u306B\u5BFE\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u884C\u5217\u5024\u95A2\u6570\u3067\u3001\u901A\u5E38\u306E\uFF08\u5B9F\u307E\u305F\u306F\u8907\u7D20\u5909\u6570\u306E\uFF09\u6307\u6570\u95A2\u6570\u306B\u5BFE\u5FDC\u3059\u308B\u3082\u306E\u3067\u3042\u308B\u3002\u3088\u308A\u62BD\u8C61\u7684\u306B\u306F\u3001\u884C\u5217\u30EA\u30FC\u7FA4\u3068\u305D\u306E\u884C\u5217\u30EA\u30FC\u4EE3\u6570\u306E\u9593\u306E\u5BFE\u5FDC\u95A2\u4FC2\uFF08\u6307\u6570\u5199\u50CF\uFF09\u3092\u884C\u5217\u306E\u6307\u6570\u51FD\u6570\u304C\u8A18\u8FF0\u3059\u308B\u3002 n\u2009\u00D7\u2009n \u5B9F\u307E\u305F\u306F\u8907\u7D20\u884C\u5217 X \u306E\u6307\u6570\u95A2\u6570 eX \u307E\u305F\u306F exp(X) \u306F\u3001\u51AA\u7D1A\u6570 \u3067\u5B9A\u7FA9\u3055\u308C\u308B n-\u6B21\u6B63\u65B9\u884C\u5217\u3067\u3042\u308B\u3002\u3053\u306E\u7D1A\u6570\u306F\u4EFB\u610F\u306E X \u306B\u5BFE\u3057\u3066\u53CE\u675F\u3059\u308B\u304B\u3089\u3001\u884C\u5217 X \u306E\u6307\u6570\u95A2\u6570\u306F well-defined \u3067\u3042\u308B\u3002 X \u304C 1\u2009\u00D7\u20091 \u884C\u5217\u306E\u3068\u304D\u3001X-\u4E57 eX \u306F 1\u2009\u00D7\u20091 \u884C\u5217\u3067\u3042\u308A\u3001\u305D\u306E\u552F\u4E00\u306E\u6210\u5206\u306F X \u306E\u552F\u4E00\u306E\u6210\u5206\u306B\u5BFE\u3059\u308B\u901A\u5E38\u306E\u6307\u6570\u95A2\u6570\u306B\u4E00\u81F4\u3059\u308B\u3002\u3053\u308C\u3089\u306F\u3057\u3070\u3057\u3070\u540C\u4E00\u8996\u3055\u308C\u308B\u3002\u3053\u306E\u610F\u5473\u306B\u304A\u3044\u3066\u884C\u5217\u306E\u6307\u6570\u51FD\u6570\u306F\u3001\u901A\u5E38\u306E\u6307\u6570\u51FD\u6570\u306E\u4E00\u822C\u5316\u3067\u3042\u308B\u3002"@ja . . "\uD589\uB82C \uC9C0\uC218 \uD568\uC218(\u884C\u5217\u6307\u6578\u51FD\u6578, matrix exponential)\uB780 \uC815\uC0AC\uAC01\uD589\uB82C\uC5D0 \uB300\uD55C \uC77C\uC885\uC758 \uC9C0\uC218 \uD568\uC218\uB2E4."@ko . "\u042D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u044B"@ru . . . . . . "En math\u00E9matiques, et plus particuli\u00E8rement en analyse, l'exponentielle d'une matrice est une fonction g\u00E9n\u00E9ralisant la fonction exponentielle aux matrices et aux endomorphismes par le calcul fonctionnel. Elle fait en particulier le pont entre un groupe de Lie et son alg\u00E8bre de Lie."@fr . . "\u042D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u044B \u2014 \u043C\u0430\u0442\u0440\u0438\u0447\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u0442 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0439 \u043C\u0430\u0442\u0440\u0438\u0446\u044B, \u0430\u043D\u0430\u043B\u043E\u0433\u0438\u0447\u043D\u0430\u044F \u043E\u0431\u044B\u0447\u043D\u043E\u0439 \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438. \u041C\u0430\u0442\u0440\u0438\u0447\u043D\u0430\u044F \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u0443\u0441\u0442\u0430\u043D\u0430\u0432\u043B\u0438\u0432\u0430\u0435\u0442 \u0441\u0432\u044F\u0437\u044C \u043C\u0435\u0436\u0434\u0443 \u0430\u043B\u0433\u0435\u0431\u0440\u043E\u0439 \u041B\u0438 \u043C\u0430\u0442\u0440\u0438\u0446 \u0438 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0433\u0440\u0443\u043F\u043F\u043E\u0439 \u041B\u0438. \u0414\u043B\u044F \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043D\u043D\u043E\u0439 \u0438\u043B\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u043E\u0439 \u043C\u0430\u0442\u0440\u0438\u0446\u044B \u0440\u0430\u0437\u043C\u0435\u0440\u0430 \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043E\u0442 , \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u043C\u0430\u044F \u043A\u0430\u043A \u0438\u043B\u0438 , \u2014 \u044D\u0442\u043E \u043C\u0430\u0442\u0440\u0438\u0446\u0430 , \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u043C\u0430\u044F \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u043C \u0440\u044F\u0434\u043E\u043C: , \u0433\u0434\u0435 \u2014 k-\u044F \u0441\u0442\u0435\u043F\u0435\u043D\u044C \u043C\u0430\u0442\u0440\u0438\u0446\u044B .\u0414\u0430\u043D\u043D\u044B\u0439 \u0440\u044F\u0434 \u0432\u0441\u0435\u0433\u0434\u0430 \u0441\u0445\u043E\u0434\u0438\u0442\u0441\u044F, \u0442\u0430\u043A \u0447\u0442\u043E \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043E\u0442 \u0432\u0441\u0435\u0433\u0434\u0430 \u043A\u043E\u0440\u0440\u0435\u043A\u0442\u043D\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0430. \u0415\u0441\u043B\u0438 \u2014 \u043C\u0430\u0442\u0440\u0438\u0446\u0430 \u0440\u0430\u0437\u043C\u0435\u0440\u0430 , \u0442\u043E \u043C\u0430\u0442\u0440\u0438\u0447\u043D\u0430\u044F \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043E\u0442 \u0435\u0441\u0442\u044C \u043C\u0430\u0442\u0440\u0438\u0446\u0430 \u0440\u0430\u0437\u043C\u0435\u0440\u043D\u043E\u0441\u0442\u0438 , \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u043A\u043E\u0442\u043E\u0440\u043E\u0439 \u0440\u0430\u0432\u0435\u043D \u043E\u0431\u044B\u0447\u043D\u043E\u0439 \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0435 \u043E\u0442 \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u043E\u0433\u043E \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u0430 ."@ru . . . . . . . . . "En matem\u00E1ticas, la exponencial de matrices es una funci\u00F3n definida sobre las matrices cuadradas an\u00E1loga a la funci\u00F3n exponencial. Es usada para resolver sistemas de ecuaciones diferenciales lineales."@es . . . "Exponencial matricial"@pt . "#000000"@en . "\u042D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u044B \u2014 \u043C\u0430\u0442\u0440\u0438\u0447\u043D\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u043E\u0442 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0439 \u043C\u0430\u0442\u0440\u0438\u0446\u044B, \u0430\u043D\u0430\u043B\u043E\u0433\u0438\u0447\u043D\u0430\u044F \u043E\u0431\u044B\u0447\u043D\u043E\u0439 \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438. \u041C\u0430\u0442\u0440\u0438\u0447\u043D\u0430\u044F \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u0443\u0441\u0442\u0430\u043D\u0430\u0432\u043B\u0438\u0432\u0430\u0435\u0442 \u0441\u0432\u044F\u0437\u044C \u043C\u0435\u0436\u0434\u0443 \u0430\u043B\u0433\u0435\u0431\u0440\u043E\u0439 \u041B\u0438 \u043C\u0430\u0442\u0440\u0438\u0446 \u0438 \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0439 \u0433\u0440\u0443\u043F\u043F\u043E\u0439 \u041B\u0438. \u0414\u043B\u044F \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043D\u043D\u043E\u0439 \u0438\u043B\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u043E\u0439 \u043C\u0430\u0442\u0440\u0438\u0446\u044B \u0440\u0430\u0437\u043C\u0435\u0440\u0430 \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043E\u0442 , \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u043C\u0430\u044F \u043A\u0430\u043A \u0438\u043B\u0438 , \u2014 \u044D\u0442\u043E \u043C\u0430\u0442\u0440\u0438\u0446\u0430 , \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u043C\u0430\u044F \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u043C \u0440\u044F\u0434\u043E\u043C: , \u0433\u0434\u0435 \u2014 k-\u044F \u0441\u0442\u0435\u043F\u0435\u043D\u044C \u043C\u0430\u0442\u0440\u0438\u0446\u044B .\u0414\u0430\u043D\u043D\u044B\u0439 \u0440\u044F\u0434 \u0432\u0441\u0435\u0433\u0434\u0430 \u0441\u0445\u043E\u0434\u0438\u0442\u0441\u044F, \u0442\u0430\u043A \u0447\u0442\u043E \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043E\u0442 \u0432\u0441\u0435\u0433\u0434\u0430 \u043A\u043E\u0440\u0440\u0435\u043A\u0442\u043D\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0430."@ru . . "Matrix Exponential"@en . "#0073CF"@en . . . "Matrixexponential"@de . . . . "\u0415\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u2014 \u043C\u0430\u0442\u0440\u0438\u0447\u043D\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0432\u0456\u0434 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u0456, \u0449\u043E \u043C\u0430\u0454 \u0431\u0430\u0433\u0430\u0442\u043E \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0438\u0445 \u0437\u0432\u0438\u0447\u0430\u0439\u043D\u0456\u0439 \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0456\u0439\u043D\u0456\u0439 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0434\u0456\u0439\u0441\u043D\u0438\u0445 \u0447\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B. \u041C\u0430\u0442\u0440\u0438\u0447\u043D\u0430 \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u0432\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u044E\u0454 \u0437\u0432'\u044F\u0437\u043E\u043A \u043C\u0456\u0436 \u0430\u043B\u0433\u0435\u0431\u0440\u043E\u044E \u041B\u0456 \u043C\u0430\u0442\u0440\u0438\u0446\u044C \u0456 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E\u044E \u0433\u0440\u0443\u043F\u043E\u044E \u041B\u0456."@uk . . . "\u77E9\u9635\u6307\u6570"@zh . "MatrixExponential"@en . "\uD589\uB82C \uC9C0\uC218 \uD568\uC218(\u884C\u5217\u6307\u6578\u51FD\u6578, matrix exponential)\uB780 \uC815\uC0AC\uAC01\uD589\uB82C\uC5D0 \uB300\uD55C \uC77C\uC885\uC758 \uC9C0\uC218 \uD568\uC218\uB2E4."@ko . "Matrix exponential"@en . . . . "En matem\u00E1ticas, la exponencial de matrices es una funci\u00F3n definida sobre las matrices cuadradas an\u00E1loga a la funci\u00F3n exponencial. Es usada para resolver sistemas de ecuaciones diferenciales lineales."@es . "\u77E9\u9635\u6307\u6570\uFF08matrix exponential\uFF09\u662F\u65B9\u5757\u77E9\u9635\u7684\u4E00\u79CD\u77E9\u9635\u51FD\u6570\uFF0C\u4E0E\u6307\u6570\u51FD\u6570\u7C7B\u4F3C\u3002\u77E9\u9635\u6307\u6570\u7ED9\u51FA\u4E86\u77E9\u9635\u674E\u4EE3\u6570\u4E0E\u5BF9\u5E94\u7684\u674E\u7FA4\u4E4B\u95F4\u7684\u5173\u7CFB\u3002 \u8BBEX\u4E3An\u00D7n\u7684\u5B9E\u6570\u6216\u590D\u6570\u77E9\u9635\u3002X\u7684\u6307\u6570\uFF0C\u7528eX\u6216exp(X)\u6765\u8868\u793A\uFF0C\u662F\u7531\u4EE5\u4E0B\u5E42\u7EA7\u6570\u6240\u7ED9\u51FA\u7684n\u00D7n\u77E9\u9635\uFF1A \u4EE5\u4E0A\u7684\u7EA7\u6570\u603B\u662F\u6536\u655B\u7684\uFF0C\u56E0\u6B64X\u7684\u6307\u6570\u662F\u5B9A\u4E49\u826F\u597D\u7684\u3002\u6CE8\u610F\uFF0C\u5982\u679CX\u662F1\u00D71\u7684\u77E9\u9635\uFF0C\u5219X\u7684\u77E9\u9635\u6307\u6570\u5C31\u662F\u7531X\u7684\u5143\u7D20\u7684\u6307\u6570\u6240\u7EC4\u6210\u76841\u00D71\u77E9\u9635\u3002"@zh . . . . "L'exponencial d'una matriu \u00E9s una funci\u00F3 definida sobre les matrius quadrades, similar a la funci\u00F3 exponencial. Sigui una matriu de nombres reals o complexos. L'exponencial de , denotada per o \u00E9s la matriu definida per la s\u00E8rie de pot\u00E8ncies: Aquesta s\u00E8rie convergeix per a tota matriu . Observem que, si la matriu \u00E9s una matriu 1\u00D71, l'exponencial de correspon amb l'exponencial ordin\u00E0ria."@ca . "\u0415\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u2014 \u043C\u0430\u0442\u0440\u0438\u0447\u043D\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0432\u0456\u0434 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u0456, \u0449\u043E \u043C\u0430\u0454 \u0431\u0430\u0433\u0430\u0442\u043E \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0438\u0445 \u0437\u0432\u0438\u0447\u0430\u0439\u043D\u0456\u0439 \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0456\u0439\u043D\u0456\u0439 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0434\u0456\u0439\u0441\u043D\u0438\u0445 \u0447\u0438 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B. \u041C\u0430\u0442\u0440\u0438\u0447\u043D\u0430 \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0442\u0430 \u0432\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u044E\u0454 \u0437\u0432'\u044F\u0437\u043E\u043A \u043C\u0456\u0436 \u0430\u043B\u0433\u0435\u0431\u0440\u043E\u044E \u041B\u0456 \u043C\u0430\u0442\u0440\u0438\u0446\u044C \u0456 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E\u044E \u0433\u0440\u0443\u043F\u043E\u044E \u041B\u0456."@uk . "In algebra lineare, l'esponenziale di matrice \u00E8 la funzione di matrice corrispondente alla funzione esponenziale di una matrice quadrata. La matrice esponenziale compare ad esempio nella risoluzione dei sistemi lineari di equazioni differenziali. Ha quindi un'importante applicazione nella teoria dei sistemi e nella teoria dei controlli automatici."@it . . . . . . . . "En math\u00E9matiques, et plus particuli\u00E8rement en analyse, l'exponentielle d'une matrice est une fonction g\u00E9n\u00E9ralisant la fonction exponentielle aux matrices et aux endomorphismes par le calcul fonctionnel. Elle fait en particulier le pont entre un groupe de Lie et son alg\u00E8bre de Lie."@fr . . . . . . . . . "Eksponenta macierzy \u2013 funkcja macierzowa zdefiniowana dla macierzy kwadratowych analogicznie jak klasyczna funkcja wyk\u0142adnicza. Eksponent\u0105 macierzy rzeczywistej lub zespolonej wymiaru jest macierz wymiaru oznaczana jako albo zadana przez szereg pot\u0119gowy: przy czym przyjmuje si\u0119: \n* \n* w szczeg\u00F3lno\u015Bci gdzie \u2013 macierz jednostkowa \u2013 macierz zerowa"@pl . . . . . "In algebra lineare, l'esponenziale di matrice \u00E8 la funzione di matrice corrispondente alla funzione esponenziale di una matrice quadrata. La matrice esponenziale compare ad esempio nella risoluzione dei sistemi lineari di equazioni differenziali. Ha quindi un'importante applicazione nella teoria dei sistemi e nella teoria dei controlli automatici."@it . . "55339"^^ . . "Em matem\u00E1tica, a exponencial matricial \u00E9 uma fun\u00E7\u00E3o matricial definida no conjunto das matrizes quadradas e possui propriedades semelhantes \u00E0 fun\u00E7\u00E3o exponencial definida nos n\u00FAmeros reais (ou complexos). Mais abstratamente falando a exponencial matricial estabelece uma conex\u00E3o entre a \u00E1lgebra de Lie das matrizes e o seu correspondente grupo de Lie. Seja uma matriz real ou complexa , define-se pela seguinte s\u00E9rie de pot\u00EAncias: , onde \u00E9 a matriz identidade A converg\u00EAncia desta s\u00E9rie \u00E9 garantida pelo teste M de Weierstrass."@pt . . . . . . . . . . . . . . . . . "\uD589\uB82C \uC9C0\uC218 \uD568\uC218"@ko . . "1122134034"^^ . . . "\u884C\u5217\u6307\u6570\u95A2\u6570"@ja . . . . . . "Exponencial de una matriz"@es . . . . . . . . . "."@en . . . "Matrice esponenziale"@it . . . . . . . . . . . "Exponencial d'una matriu"@ca . . . . . . . . "\u77E9\u9635\u6307\u6570\uFF08matrix exponential\uFF09\u662F\u65B9\u5757\u77E9\u9635\u7684\u4E00\u79CD\u77E9\u9635\u51FD\u6570\uFF0C\u4E0E\u6307\u6570\u51FD\u6570\u7C7B\u4F3C\u3002\u77E9\u9635\u6307\u6570\u7ED9\u51FA\u4E86\u77E9\u9635\u674E\u4EE3\u6570\u4E0E\u5BF9\u5E94\u7684\u674E\u7FA4\u4E4B\u95F4\u7684\u5173\u7CFB\u3002 \u8BBEX\u4E3An\u00D7n\u7684\u5B9E\u6570\u6216\u590D\u6570\u77E9\u9635\u3002X\u7684\u6307\u6570\uFF0C\u7528eX\u6216exp(X)\u6765\u8868\u793A\uFF0C\u662F\u7531\u4EE5\u4E0B\u5E42\u7EA7\u6570\u6240\u7ED9\u51FA\u7684n\u00D7n\u77E9\u9635\uFF1A \u4EE5\u4E0A\u7684\u7EA7\u6570\u603B\u662F\u6536\u655B\u7684\uFF0C\u56E0\u6B64X\u7684\u6307\u6570\u662F\u5B9A\u4E49\u826F\u597D\u7684\u3002\u6CE8\u610F\uFF0C\u5982\u679CX\u662F1\u00D71\u7684\u77E9\u9635\uFF0C\u5219X\u7684\u77E9\u9635\u6307\u6570\u5C31\u662F\u7531X\u7684\u5143\u7D20\u7684\u6307\u6570\u6240\u7EC4\u6210\u76841\u00D71\u77E9\u9635\u3002"@zh . . "In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n\u00D7n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n\u00D7n matrix given by the power series where is defined to be the identity matrix with the same dimensions as ."@en . . . . . . "In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n\u00D7n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n\u00D7n matrix given by the power series where is defined to be the identity matrix with the same dimensions as . The above series always converges, so the exponential of X is well-defined. If X is a 1\u00D71 matrix the matrix exponential of X is a 1\u00D71 matrix whose single element is the ordinary exponential of the single element of X."@en . "Matrisexponentialfunktionen \u00E4r inom matematiken en ut\u00F6kning av exponentialfunktionen fr\u00E5n komplexa tal till att g\u00E4lla \u00E4ven kvadratiska matriser, s\u00E5 att man f\u00E5r en matrisfunktion."@sv . "Eksponenta macierzy \u2013 funkcja macierzowa zdefiniowana dla macierzy kwadratowych analogicznie jak klasyczna funkcja wyk\u0142adnicza. Eksponent\u0105 macierzy rzeczywistej lub zespolonej wymiaru jest macierz wymiaru oznaczana jako albo zadana przez szereg pot\u0119gowy: przy czym przyjmuje si\u0119: \n* \n* w szczeg\u00F3lno\u015Bci gdzie \u2013 macierz jednostkowa \u2013 macierz zerowa"@pl . "Exponentielle d'une matrice"@fr . . .