. . . . . . . . . . . . . . . "Teorema de Cayley-Hamilton"@ca . "1120554488"^^ . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0430\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u0435\u043B\u0456"@uk . . "Twierdzenie Cayleya-Hamiltona m\u00F3wi, \u017Ce ka\u017Cda macierz kwadratowa nad cia\u0142em liczb rzeczywistych lub zespolonych jest pierwiastkiem swojego wielomianu charakterystycznego. Nazwa upami\u0119tnia matematyk\u00F3w: Arthura Cayleya i Williama Hamiltona Dok\u0142adniej; je\u017Celi jest macierz\u0105 oraz jest macierz\u0105 identyczno\u015Bciow\u0105 to wielomian charakterystyczny jest zdefiniowany jako: gdzie oznacza wyznacznik. Twierdzenie Cayleya-Hamiltona m\u00F3wi, \u017Ce podstawienie do wielomianu charakterystycznego daje w rezultacie macierz z\u0142o\u017Con\u0105 z samych zer: Wa\u017Cnym wnioskiem z teorii Cayleya-Hamiltona jest fakt, \u017Ce wielomian minimalny danej macierzy jest dzielnikiem wielomianu charakterystycznego. Jest to bardzo przydatne podczas znajdowania postaci Jordana danej macierzy."@pl . . . . "#0070BF"@en . . . "Em \u00E1lgebra linear, o teorema de Cayley-Hamilton (cujo nome faz refer\u00EAncia aos matem\u00E1ticos Arthur Cayley e William Hamilton) diz que o polin\u00F4mio m\u00EDnimo de uma matriz divide o seu polin\u00F4mio caracter\u00EDstico. Em outras palavras, seja uma matriz e o seu polin\u00F4mio caracter\u00EDstico, definido por: em que \u00E9 a fun\u00E7\u00E3o determinante e \u00E9 a matriz identidade de ordem Ent\u00E3o O teorema Cayley\u2013Hamilton \u00E9 equivalente \u00E0 afirma\u00E7\u00E3o de que o polin\u00F4mio m\u00EDnimo de uma matriz quadrada divide seu polin\u00F4mio caracter\u00EDstico."@pt . . "62655"^^ . "Cayleyho\u2013Hamiltonova v\u011Bta"@cs . . "Cayley\u2013Hamilton theorem"@en . . . "De stelling van Cayley-Hamilton is een stelling in de lineaire algebra die stelt dat elke vierkante re\u00EBle of complexe matrix voldoet aan zijn eigen karakteristieke vergelijking. De stelling is genoemd naar de wiskundigen Arthur Cayley en William Hamilton."@nl . . . . . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u0413\u0430\u0301\u043C\u0438\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u044D\u0301\u043B\u0438 \u2014 \u043A\u043B\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u044B,\u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0430\u0435\u0442, \u0447\u0442\u043E \u043B\u044E\u0431\u0430\u044F \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u0430\u044F \u043C\u0430\u0442\u0440\u0438\u0446\u0430 \u0443\u0434\u043E\u0432\u043B\u0435\u0442\u0432\u043E\u0440\u044F\u0435\u0442 \u0441\u0432\u043E\u0435\u043C\u0443 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u043C\u0443 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u044E.\u041D\u0430\u0437\u0432\u0430\u043D\u043D\u0430\u044F \u0432 \u0447\u0435\u0441\u0442\u044C \u0423\u0438\u043B\u044C\u044F\u043C\u0430 \u0413\u0430\u043C\u0438\u043B\u044C\u0442\u043E\u043D\u0430 \u0438 \u0410\u0440\u0442\u0443\u0440\u0430 \u041A\u044D\u043B\u0438."@ru . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0430\u043C\u0438\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u044D\u043B\u0438"@ru . "\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\uFF08\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u3066\u3044\u308A\u3001\u82F1: Cayley\u2013Hamilton theorem\uFF09\u3001\u307E\u305F\u306F\u30CF\u30DF\u30EB\u30C8\u30F3\u30FB\u30B1\u30A4\u30EA\u30FC\u306E\u5B9A\u7406\u3068\u306F\u3001\uFF08\u5B9F\u6570\u4F53\u3084\u8907\u7D20\u6570\u4F53\u306A\u3069\u306E\uFF09\u53EF\u63DB\u74B0\u4E0A\u306E\u6B63\u65B9\u884C\u5217\u306F\u56FA\u6709\u65B9\u7A0B\u5F0F\u3092\u6E80\u305F\u3059\u3068\u3044\u3046\u5B9A\u7406\u3067\u3042\u308B\u3002\u30A2\u30FC\u30B5\u30FC\u30FB\u30B1\u30A4\u30EA\u30FC\u3068\u30A6\u30A3\u30EA\u30A2\u30E0\u30FB\u30ED\u30FC\u30EF\u30F3\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306B\u56E0\u3080\u3002 n\u6B21\u6B63\u65B9\u884C\u5217 A \u306B\u5BFE\u3057\u3066\u3001In \u3092 n\u6B21\u5358\u4F4D\u884C\u5217\u3068\u3059\u308B\u3068\u3001A \u306E\u56FA\u6709\u591A\u9805\u5F0F\u306F \u3067\u5B9A\u7FA9\u3055\u308C\u308B\u3002\u3053\u3053\u3067 det \u306F\u884C\u5217\u5F0F\u3092\u8868\u3057\u3001\u03BB \u306F\u4FC2\u6570\u74B0\u306E\u5143\uFF08\u30B9\u30AB\u30E9\u30FC\uFF09\u3067\u3042\u308B\u3002\u5F15\u6570\u306E\u884C\u5217\u306F\u5404\u6210\u5206\u304C \u03BB \u306E 1\u6B21\u5F0F\u4EE5\u4E0B\u306E\u591A\u9805\u5F0F\uFF081\u6B21\u5F0F\u307E\u305F\u306F\u5B9A\u6570\uFF09\u3060\u304B\u3089\u3001\u305D\u306E\u884C\u5217\u5F0F\u3082 \u03BB \u306E n\u6B21\u30E2\u30CB\u30C3\u30AF\u591A\u9805\u5F0F\u306B\u306A\u308B\u3002\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\u306E\u4E3B\u5F35\u306F\u3001\u56FA\u6709\u591A\u9805\u5F0F\u3092\u884C\u5217\u591A\u9805\u5F0F\u3068\u898B\u308C\u3070 A \u304C\u96F6\u70B9\u3067\u3042\u308B\u3053\u3068\u3001\u3059\u306A\u308F\u3061\u4E0A\u8A18\u306E \u03BB \u3092\u884C\u5217 A \u3067\u7F6E\u304D\u63DB\u3048\u305F\u8A08\u7B97\u7D50\u679C\u304C\u96F6\u884C\u5217\u3067\u3042\u308B\u3053\u3068\u3001\u3059\u306A\u308F\u3061 \u306E\u6210\u7ACB\u3092\u8FF0\u3079\u308B\u3082\u306E\u3067\u3042\u308B\u3002 \u6CE8\u7F6E\u304D\u63DB\u3048\u306B\u304A\u3044\u3066\u3001\u03BB \u306E\u51AA\u306F\u3001A \u306E\u3001\u884C\u5217\u306E\u7A4D\u306B\u3088\u308B\u51AA\u306B\u7F6E\u304D\u63DB\u308F\u308B\u304B\u3089\u3001\u7279\u306B p(\u03BB) \u306E\u5B9A\u6570\u9805\u306F A0 \u3059\u306A\u308F\u3061\u5358\u4F4D\u884C\u5217\u306E\u5B9A\u6570\u500D\u306B\u7F6E\u304D\u63DB\u308F\u308B\u3002 \u5B9A\u7406\u306B\u3088\u308A\u3001\u7279\u306B An \u306F\u3001\u3088\u308A\u4F4E\u6B21\u306E A \u306E\u591A\u9805\u5F0F\u3067\u8868\u3055\u308C\u308B\u3053\u3068\u304C\u5206\u304B\u308B\u3002\u4FC2\u6570\u74B0\u304C\u306E\u3068\u304D\u3001\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\u306F\u300C\u4EFB\u610F\u306E\u6B63\u65B9\u884C\u5217 A \u306E\u6700\u5C0F\u591A\u9805\u5F0F\u306F A \u306E\u56FA\u6709\u591A\u9805\u5F0F\u3092\u6574\u9664\u3059\u308B\uFF08\u5272\u308A\u5207\u308B\uFF09\u300D\u3068\u3044\u3046\u4E3B\u5F35\u306B\u540C\u5024\u3067\u3042\u308B\u3002 \u3053\u306E\u5B9A\u7406\u306F1853\u5E74\u306B\u30CF\u30DF\u30EB\u30C8\u30F3\u304C\u521D\u3081\u3066\u8A3C\u660E\u3057\u305F\uFF08\u305D\u308C\u306F\u300C\u975E\u53EF\u63DB\u300D\u74B0\u3067\u3042\u308B\u56DB\u5143\u6570\u3092\u5909\u6570\u3068\u3059\u308B\u4E00\u6B21\u51FD\u6570\u306E\u9006\u3092\u7528\u3044\u305F\u3082\u306E\u3067\u3042\u3063\u305F\uFF09\u3002\u3053\u308C\u306F\u4E00\u822C\u306E\u5B9A\u7406\u306B\u304A\u3044\u3066\u3001\u5B9F4\u6B21\u307E\u305F\u306F\u8907\u7D202\u6B21\u3068\u3044\u3046\u7279\u5225\u306E\u5834\u5408\u306B\u5F53\u305F\u308B\u3082\u306E\u3067\u3042\u308B\u3002 \u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\u306F\u3001\u56DB\u5143\u6570\u4FC2\u6570\u306E\u884C\u5217\u306B\u5BFE\u3057\u3066\u3082\u6210\u7ACB\u3059\u308B\u3002 1858\u5E74\u306B\u30B1\u30A4\u30EA\u30FC\u306F 3\u6B21\u304A\u3088\u3073\u305D\u308C\u3088\u308A\u5C0F\u3055\u3044\u884C\u5217\u306B\u95A2\u3057\u3066\u5B9A\u7406\u3092\u8FF0\u3079\u3066\u3044\u308B\u304C\u3001\u8A3C\u660E\u306F 2\u6B21\u306E\u5834\u5408\u306E\u307F\u3092\u8457\u3057\u3066\u3044\u308B\u3002\u4E00\u822C\u306E\u5834\u5408\u304C\u521D\u3081\u3066\u8A3C\u660E\u3055\u308C\u305F\u306E\u306F1878\u5E74\u3067\u30D5\u30ED\u30D9\u30CB\u30A6\u30B9\u306B\u3088\u308B\u3002"@ja . "\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\uFF08\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u3066\u3044\u308A\u3001\u82F1: Cayley\u2013Hamilton theorem\uFF09\u3001\u307E\u305F\u306F\u30CF\u30DF\u30EB\u30C8\u30F3\u30FB\u30B1\u30A4\u30EA\u30FC\u306E\u5B9A\u7406\u3068\u306F\u3001\uFF08\u5B9F\u6570\u4F53\u3084\u8907\u7D20\u6570\u4F53\u306A\u3069\u306E\uFF09\u53EF\u63DB\u74B0\u4E0A\u306E\u6B63\u65B9\u884C\u5217\u306F\u56FA\u6709\u65B9\u7A0B\u5F0F\u3092\u6E80\u305F\u3059\u3068\u3044\u3046\u5B9A\u7406\u3067\u3042\u308B\u3002\u30A2\u30FC\u30B5\u30FC\u30FB\u30B1\u30A4\u30EA\u30FC\u3068\u30A6\u30A3\u30EA\u30A2\u30E0\u30FB\u30ED\u30FC\u30EF\u30F3\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306B\u56E0\u3080\u3002 n\u6B21\u6B63\u65B9\u884C\u5217 A \u306B\u5BFE\u3057\u3066\u3001In \u3092 n\u6B21\u5358\u4F4D\u884C\u5217\u3068\u3059\u308B\u3068\u3001A \u306E\u56FA\u6709\u591A\u9805\u5F0F\u306F \u3067\u5B9A\u7FA9\u3055\u308C\u308B\u3002\u3053\u3053\u3067 det \u306F\u884C\u5217\u5F0F\u3092\u8868\u3057\u3001\u03BB \u306F\u4FC2\u6570\u74B0\u306E\u5143\uFF08\u30B9\u30AB\u30E9\u30FC\uFF09\u3067\u3042\u308B\u3002\u5F15\u6570\u306E\u884C\u5217\u306F\u5404\u6210\u5206\u304C \u03BB \u306E 1\u6B21\u5F0F\u4EE5\u4E0B\u306E\u591A\u9805\u5F0F\uFF081\u6B21\u5F0F\u307E\u305F\u306F\u5B9A\u6570\uFF09\u3060\u304B\u3089\u3001\u305D\u306E\u884C\u5217\u5F0F\u3082 \u03BB \u306E n\u6B21\u30E2\u30CB\u30C3\u30AF\u591A\u9805\u5F0F\u306B\u306A\u308B\u3002\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\u306E\u4E3B\u5F35\u306F\u3001\u56FA\u6709\u591A\u9805\u5F0F\u3092\u884C\u5217\u591A\u9805\u5F0F\u3068\u898B\u308C\u3070 A \u304C\u96F6\u70B9\u3067\u3042\u308B\u3053\u3068\u3001\u3059\u306A\u308F\u3061\u4E0A\u8A18\u306E \u03BB \u3092\u884C\u5217 A \u3067\u7F6E\u304D\u63DB\u3048\u305F\u8A08\u7B97\u7D50\u679C\u304C\u96F6\u884C\u5217\u3067\u3042\u308B\u3053\u3068\u3001\u3059\u306A\u308F\u3061 \u306E\u6210\u7ACB\u3092\u8FF0\u3079\u308B\u3082\u306E\u3067\u3042\u308B\u3002 \u6CE8\u7F6E\u304D\u63DB\u3048\u306B\u304A\u3044\u3066\u3001\u03BB \u306E\u51AA\u306F\u3001A \u306E\u3001\u884C\u5217\u306E\u7A4D\u306B\u3088\u308B\u51AA\u306B\u7F6E\u304D\u63DB\u308F\u308B\u304B\u3089\u3001\u7279\u306B p(\u03BB) \u306E\u5B9A\u6570\u9805\u306F A0 \u3059\u306A\u308F\u3061\u5358\u4F4D\u884C\u5217\u306E\u5B9A\u6570\u500D\u306B\u7F6E\u304D\u63DB\u308F\u308B\u3002 \u5B9A\u7406\u306B\u3088\u308A\u3001\u7279\u306B An \u306F\u3001\u3088\u308A\u4F4E\u6B21\u306E A \u306E\u591A\u9805\u5F0F\u3067\u8868\u3055\u308C\u308B\u3053\u3068\u304C\u5206\u304B\u308B\u3002\u4FC2\u6570\u74B0\u304C\u306E\u3068\u304D\u3001\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406\u306F\u300C\u4EFB\u610F\u306E\u6B63\u65B9\u884C\u5217 A \u306E\u6700\u5C0F\u591A\u9805\u5F0F\u306F A \u306E\u56FA\u6709\u591A\u9805\u5F0F\u3092\u6574\u9664\u3059\u308B\uFF08\u5272\u308A\u5207\u308B\uFF09\u300D\u3068\u3044\u3046\u4E3B\u5F35\u306B\u540C\u5024\u3067\u3042\u308B\u3002"@ja . . "Der Satz von Cayley-Hamilton (nach Arthur Cayley und William Rowan Hamilton) ist ein Satz aus der linearen Algebra. Er besagt, dass jede quadratische Matrix Nullstelle ihres charakteristischen Polynoms ist."@de . . . . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u0413\u0430\u0301\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u0435\u0301\u043B\u0456 (\u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0412\u0456\u043B\u044C\u044F\u043C\u0430 \u0413\u0430\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430 \u0442\u0430 \u0410\u0440\u0442\u0443\u0440\u0430 \u041A\u0435\u043B\u0456) \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u043F\u0456\u0434\u0441\u0442\u0430\u043D\u043E\u0432\u043A\u0438 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u0434\u043E \u0457\u0457 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u0447\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0430 \u0442\u043E\u0442\u043E\u0436\u043D\u043E \u0434\u043E\u0440\u0456\u0432\u043D\u044E\u0454 \u043D\u0443\u043B\u044E: \u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0430\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430-\u041A\u0435\u043B\u0456 \u0434\u043E\u0437\u0432\u043E\u043B\u044F\u0454 \u0432\u0438\u0440\u0430\u0437\u0438\u0442\u0438 \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0438 \u0432\u0438\u0441\u043E\u043A\u043E\u0433\u043E \u0441\u0442\u0435\u043F\u0435\u043D\u044F \u0432\u0456\u0434 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u044F\u043A \u043B\u0456\u043D\u0456\u0439\u043D\u0456 \u043A\u043E\u043C\u0431\u0456\u043D\u0430\u0446\u0456\u0457 \u0422\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0454 \u0441\u043F\u0440\u0430\u0432\u0435\u0434\u043B\u0438\u0432\u0438\u043C \u0434\u043B\u044F \u043C\u0430\u0442\u0440\u0438\u0446\u044C \u0456\u0437 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0430\u043C\u0438 \u0456\u0437 \u0431\u0443\u0434\u044C-\u044F\u043A\u043E\u0433\u043E \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F \u0437 \u043E\u0434\u0438\u043D\u0438\u0446\u0435\u044E \u0437\u043E\u043A\u0440\u0435\u043C\u0430 \u0431\u0443\u0434\u044C-\u044F\u043A\u043E\u0433\u043E \u043F\u043E\u043B\u044F."@uk . . . . . . . . . . "#F9FFF7"@en . "Teorema de Cayley-Hamilton"@pt . . . . "In algebra lineare, il teorema di Hamilton-Cayley, il cui nome \u00E8 dovuto a William Rowan Hamilton e Arthur Cayley, asserisce che ogni trasformazione lineare di uno spazio vettoriale (o equivalentemente ogni matrice quadrata) \u00E8 una radice del suo polinomio caratteristico, visto come polinomio a coefficienti numerici nell'anello delle trasformazioni lineari (o delle matrici quadrate). Pi\u00F9 precisamente, se \u00E8 la trasformazione lineare nello spazio -dimensionale (o, equivalentemente, una matrice ) e \u00E8 l'operatore identit\u00E0 (o, equivalentemente, la matrice identit\u00E0), allora vale: Questo risultato implica che il polinomio minimo divide il polinomio caratteristico, ed \u00E8 quindi utile per trovare la forma canonica di Jordan di una applicazione o matrice. Inoltre, rende effettuabile analiticamente il calcolo di qualsiasi funzione di matrice. Il teorema di Hamilton\u2013Cayley vale anche per matrici quadrate su anelli commutativi."@it . . . "Cayleyho-Hamiltonova v\u011Bta je matematick\u00E9 tvrzen\u00ED z oboru line\u00E1rn\u00ED algebry pojmenovan\u00E9 po Arthurovi Cayleym a Williamu Hamiltonovi, kter\u00E9 \u0159\u00EDk\u00E1, \u017Ee ka\u017Ed\u00E1 \u010Dtvercov\u00E1 matice nad komutativn\u00EDm okruhem (tedy speci\u00E1ln\u011B nap\u0159\u00EDklad nad t\u011Blesem re\u00E1ln\u00FDch \u010D\u00EDsel nebo t\u011Blesemkomplexn\u00EDm \u010D\u00EDsel) je ko\u0159enem sv\u00E9ho charakteristick\u00E9ho polynomu. V p\u0159\u00EDpad\u011B t\u011Bles to znamen\u00E1, \u017Ee charakteristick\u00FD polynom je d\u011Bliteln\u00FD minim\u00E1ln\u00EDm polynomem. Podrobn\u011Bji \u0159e\u010Deno znamen\u00E1 Cayleyho-Hamiltonova v\u011Bta, \u017Ee pro danou \u010Dtvercovou matici plat\u00ED, \u017Ee je ko\u0159enem polynomu , tedy polynomu, kter\u00FD vznikne v\u00FDpo\u010Dtem determinantu matice vznikl\u00E9 rozd\u00EDlem jednotkov\u00E9 matice \u0159\u00E1du pron\u00E1soben\u00E9 skal\u00E1rn\u00ED nezn\u00E1mou a matice . Za jej\u00ED zobecn\u011Bn\u00ED lze pokl\u00E1dat ."@cs . . . "MacDonald"@en . . . "In algebra lineare, il teorema di Hamilton-Cayley, il cui nome \u00E8 dovuto a William Rowan Hamilton e Arthur Cayley, asserisce che ogni trasformazione lineare di uno spazio vettoriale (o equivalentemente ogni matrice quadrata) \u00E8 una radice del suo polinomio caratteristico, visto come polinomio a coefficienti numerici nell'anello delle trasformazioni lineari (o delle matrici quadrate). Pi\u00F9 precisamente, se \u00E8 la trasformazione lineare nello spazio -dimensionale (o, equivalentemente, una matrice ) e \u00E8 l'operatore identit\u00E0 (o, equivalentemente, la matrice identit\u00E0), allora vale:"@it . . . . . . . . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u0413\u0430\u0301\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u0435\u0301\u043B\u0456 (\u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0412\u0456\u043B\u044C\u044F\u043C\u0430 \u0413\u0430\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430 \u0442\u0430 \u0410\u0440\u0442\u0443\u0440\u0430 \u041A\u0435\u043B\u0456) \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u043F\u0456\u0434\u0441\u0442\u0430\u043D\u043E\u0432\u043A\u0438 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u043E\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u0434\u043E \u0457\u0457 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u0447\u043D\u043E\u0433\u043E \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0430 \u0442\u043E\u0442\u043E\u0436\u043D\u043E \u0434\u043E\u0440\u0456\u0432\u043D\u044E\u0454 \u043D\u0443\u043B\u044E: \u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0430\u043C\u0456\u043B\u044C\u0442\u043E\u043D\u0430-\u041A\u0435\u043B\u0456 \u0434\u043E\u0437\u0432\u043E\u043B\u044F\u0454 \u0432\u0438\u0440\u0430\u0437\u0438\u0442\u0438 \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0438 \u0432\u0438\u0441\u043E\u043A\u043E\u0433\u043E \u0441\u0442\u0435\u043F\u0435\u043D\u044F \u0432\u0456\u0434 \u043C\u0430\u0442\u0440\u0438\u0446\u0456 \u044F\u043A \u043B\u0456\u043D\u0456\u0439\u043D\u0456 \u043A\u043E\u043C\u0431\u0456\u043D\u0430\u0446\u0456\u0457 \u0422\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0454 \u0441\u043F\u0440\u0430\u0432\u0435\u0434\u043B\u0438\u0432\u0438\u043C \u0434\u043B\u044F \u043C\u0430\u0442\u0440\u0438\u0446\u044C \u0456\u0437 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0430\u043C\u0438 \u0456\u0437 \u0431\u0443\u0434\u044C-\u044F\u043A\u043E\u0433\u043E \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F \u0437 \u043E\u0434\u0438\u043D\u0438\u0446\u0435\u044E \u0437\u043E\u043A\u0440\u0435\u043C\u0430 \u0431\u0443\u0434\u044C-\u044F\u043A\u043E\u0433\u043E \u043F\u043E\u043B\u044F."@uk . . . "\u5728\u7DDA\u6027\u4EE3\u6578\u4E2D\uFF0C\u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\uFF08\u82F1\u8A9E\uFF1ACayley\u2013Hamilton theorem\uFF09\uFF08\u4EE5\u6578\u5B78\u5BB6\u963F\u745F\u00B7\u51F1\u840A\u8207\u5A01\u5EC9\u00B7\u5362\u4E91\u00B7\u54C8\u5BC6\u987F\u547D\u540D\uFF09\u8868\u660E\u6BCF\u500B\u4F48\u65BC\u4EFB\u4F55\u4EA4\u63DB\u74B0\u4E0A\u7684\u5BE6\u6216\u8907\u65B9\u9663\u90FD\u6EFF\u8DB3\u5176\u7279\u5FB5\u65B9\u7A0B\u5F0F\u3002 \u660E\u78BA\u5730\u8AAA\uFF1A\u8A2D\u70BA\u7D66\u5B9A\u7684\u77E9\u9663\uFF0C\u4E26\u8A2D\u70BA\u55AE\u4F4D\u77E9\u9663\uFF0C\u5247\u7684\u7279\u5FB5\u591A\u9805\u5F0F\u5B9A\u7FA9\u70BA\uFF1A \u5176\u4E2D\u8868\u884C\u5217\u5F0F\u51FD\u6578\u3002\u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\u65B7\u8A00\uFF1A \u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\u7B49\u50F9\u65BC\u65B9\u9663\u7684\u7279\u5FB5\u591A\u9805\u5F0F\u6703\u88AB\u5176\u6975\u5C0F\u591A\u9805\u5F0F\u6574\u9664\uFF0C\u9019\u5728\u5C0B\u627E\u82E5\u5C14\u5F53\u6807\u51C6\u5F62\u6642\u7279\u5225\u6709\u7528\u3002"@zh . . . "173547"^^ . . "Stelling van Cayley-Hamilton"@nl . . . . . . . "\u5728\u7DDA\u6027\u4EE3\u6578\u4E2D\uFF0C\u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\uFF08\u82F1\u8A9E\uFF1ACayley\u2013Hamilton theorem\uFF09\uFF08\u4EE5\u6578\u5B78\u5BB6\u963F\u745F\u00B7\u51F1\u840A\u8207\u5A01\u5EC9\u00B7\u5362\u4E91\u00B7\u54C8\u5BC6\u987F\u547D\u540D\uFF09\u8868\u660E\u6BCF\u500B\u4F48\u65BC\u4EFB\u4F55\u4EA4\u63DB\u74B0\u4E0A\u7684\u5BE6\u6216\u8907\u65B9\u9663\u90FD\u6EFF\u8DB3\u5176\u7279\u5FB5\u65B9\u7A0B\u5F0F\u3002 \u660E\u78BA\u5730\u8AAA\uFF1A\u8A2D\u70BA\u7D66\u5B9A\u7684\u77E9\u9663\uFF0C\u4E26\u8A2D\u70BA\u55AE\u4F4D\u77E9\u9663\uFF0C\u5247\u7684\u7279\u5FB5\u591A\u9805\u5F0F\u5B9A\u7FA9\u70BA\uFF1A \u5176\u4E2D\u8868\u884C\u5217\u5F0F\u51FD\u6578\u3002\u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\u65B7\u8A00\uFF1A \u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406\u7B49\u50F9\u65BC\u65B9\u9663\u7684\u7279\u5FB5\u591A\u9805\u5F0F\u6703\u88AB\u5176\u6975\u5C0F\u591A\u9805\u5F0F\u6574\u9664\uFF0C\u9019\u5728\u5C0B\u627E\u82E5\u5C14\u5F53\u6807\u51C6\u5F62\u6642\u7279\u5225\u6709\u7528\u3002"@zh . "En \u00E1lgebra lineal, el teorema de Cayley-Hamilton (que lleva los nombres de los matem\u00E1ticos Arthur Cayley y William Hamilton) asegura que todo endomorfismo de un espacio vectorial de dimensi\u00F3n finita sobre un cuerpo cualquiera anula su propio polinomio caracter\u00EDstico. En t\u00E9rminos matriciales, eso significa que : si A es una matriz cuadrada de orden n y si es su polinomio caracter\u00EDstico (polinomio de indeterminada \u03BB), entonces al sustituir formalmente \u03BB por la matriz A en el polinomio, el resultado es la matriz nula: El teorema de Cayley-Hamilton se aplica tambi\u00E9n a matrices cuadradas de coeficientes en un anillo conmutativo cualquiera. Un corolario importante del teorema de Cayley-Hamilton afirma que el polinomio m\u00EDnimo de una matriz dada es un divisor de su polinomio caracter\u00EDstico, y no solo eso, el polinomio m\u00EDnimo tiene los mismos factores irreducibles que el polinomio caracter\u00EDstico."@es . . . "Twierdzenie Cayleya-Hamiltona m\u00F3wi, \u017Ce ka\u017Cda macierz kwadratowa nad cia\u0142em liczb rzeczywistych lub zespolonych jest pierwiastkiem swojego wielomianu charakterystycznego. Nazwa upami\u0119tnia matematyk\u00F3w: Arthura Cayleya i Williama Hamiltona Dok\u0142adniej; je\u017Celi jest macierz\u0105 oraz jest macierz\u0105 identyczno\u015Bciow\u0105 to wielomian charakterystyczny jest zdefiniowany jako: gdzie oznacza wyznacznik. Twierdzenie Cayleya-Hamiltona m\u00F3wi, \u017Ce podstawienie do wielomianu charakterystycznego daje w rezultacie macierz z\u0142o\u017Con\u0105 z samych zer:"@pl . . . . . . "El teorema de Cayley-Hamilton \u00E9s un resultat fonamental en l'\u00E0lgebra lineal segons el qual, donada una matriu A i el seu polinomi caracter\u00EDstic Q(x), aquest s'anul\u00B7la en avaluar-lo en A.\u00C9s a dir, que Q(A)=0. \u00C9s immediat que dins el polinomi, la matriu A \u00E9s commutativa respecte a l'operaci\u00F3 producte. Una formulaci\u00F3 equivalent n'\u00E9s l'afirmaci\u00F3 que el polinomi caracter\u00EDstic de A \u00E9s un m\u00FAltiple del polinomi m\u00EDnim de A o, cosa que \u00E9s el mateix, que el polinomi caracter\u00EDstic de A \u00E9s un element de l'ideal principal de polinomis anul\u00B7ladors de A. Rep el seu nom dels matem\u00E0tics Arthur Cayley i William Rowan Hamilton."@ca . . . . "Satz von Cayley-Hamilton"@de . . "En alg\u00E8bre lin\u00E9aire, le th\u00E9or\u00E8me de Cayley-Hamilton affirme que tout endomorphisme d'un espace vectoriel de dimension finie sur un corps commutatif quelconque annule son propre polyn\u00F4me caract\u00E9ristique. En termes de matrice, cela signifie que si A est une matrice carr\u00E9e d'ordre n et si est son polyn\u00F4me caract\u00E9ristique (polyn\u00F4me d'ind\u00E9termin\u00E9e X), alors en rempla\u00E7ant formellement X par la matrice A dans le polyn\u00F4me, le r\u00E9sultat est la matrice nulle : Le th\u00E9or\u00E8me de Cayley-Hamilton s'applique aussi \u00E0 des matrices carr\u00E9es \u00E0 coefficients dans un anneau commutatif quelconque."@fr . . . . . "In linear algebra, the Cayley\u2013Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation. If A is a given n\u2009\u00D7\u2009n matrix and In is the n\u2009\u00D7\u2009n identity matrix, then the characteristic polynomial of A is defined as , where det is the determinant operation and \u03BB is a variable for a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in \u03BB, the determinant is also a degree-n monic polynomial in \u03BB, One can create an analogous polynomial in the matrix A instead of the scalar variable \u03BB, defined as The Cayley\u2013Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that . The theorem allows An to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley\u2013Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. The theorem was first proven in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This corresponds to the special case of certain 4\u2009\u00D7\u20094 real or 2\u2009\u00D7\u20092 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3\u2009\u00D7\u20093 and smaller matrices, but only published a proof for the 2\u2009\u00D7\u20092 case. The general case was first proved by Ferdinand Frobenius in 1878."@en . . . . "Cayleyho-Hamiltonova v\u011Bta je matematick\u00E9 tvrzen\u00ED z oboru line\u00E1rn\u00ED algebry pojmenovan\u00E9 po Arthurovi Cayleym a Williamu Hamiltonovi, kter\u00E9 \u0159\u00EDk\u00E1, \u017Ee ka\u017Ed\u00E1 \u010Dtvercov\u00E1 matice nad komutativn\u00EDm okruhem (tedy speci\u00E1ln\u011B nap\u0159\u00EDklad nad t\u011Blesem re\u00E1ln\u00FDch \u010D\u00EDsel nebo t\u011Blesemkomplexn\u00EDm \u010D\u00EDsel) je ko\u0159enem sv\u00E9ho charakteristick\u00E9ho polynomu. V p\u0159\u00EDpad\u011B t\u011Bles to znamen\u00E1, \u017Ee charakteristick\u00FD polynom je d\u011Bliteln\u00FD minim\u00E1ln\u00EDm polynomem. Za jej\u00ED zobecn\u011Bn\u00ED lze pokl\u00E1dat ."@cs . . . "Atiyah"@en . . . . . . "\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uCF00\uC77C\uB9AC-\uD574\uBC00\uD134 \uC815\uB9AC(\uC601\uC5B4: Cayley\u2013Hamilton theorem)\uB294 \uC815\uC0AC\uAC01 \uD589\uB82C\uC774 \uC790\uAE30 \uC790\uC2E0\uC758 \uD2B9\uC131 \uBC29\uC815\uC2DD\uC744 \uB9CC\uC871\uC2DC\uD0A8\uB2E4\uB294 \uC815\uB9AC\uC774\uB2E4. \uC544\uC11C \uCF00\uC77C\uB9AC\uC640 \uC70C\uB9AC\uC5C4 \uB85C\uC5B8 \uD574\uBC00\uD134\uC758 \uC774\uB984\uC5D0\uC11C \uB530\uC654\uB2E4."@ko . . "#0073CF"@en . . "\u30B1\u30A4\u30EA\u30FC\u30FB\u30CF\u30DF\u30EB\u30C8\u30F3\u306E\u5B9A\u7406"@ja . "Der Satz von Cayley-Hamilton (nach Arthur Cayley und William Rowan Hamilton) ist ein Satz aus der linearen Algebra. Er besagt, dass jede quadratische Matrix Nullstelle ihres charakteristischen Polynoms ist."@de . . . . . "En \u00E1lgebra lineal, el teorema de Cayley-Hamilton (que lleva los nombres de los matem\u00E1ticos Arthur Cayley y William Hamilton) asegura que todo endomorfismo de un espacio vectorial de dimensi\u00F3n finita sobre un cuerpo cualquiera anula su propio polinomio caracter\u00EDstico. En t\u00E9rminos matriciales, eso significa que : si A es una matriz cuadrada de orden n y si es su polinomio caracter\u00EDstico (polinomio de indeterminada \u03BB), entonces al sustituir formalmente \u03BB por la matriz A en el polinomio, el resultado es la matriz nula:"@es . "Prop. 2.4"@en . "\u03A3\u03C4\u03B7 \u03B3\u03C1\u03B1\u03BC\u03BC\u03B9\u03BA\u03AE \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B1, \u03C4\u03BF \u03B8\u03B5\u03CE\u03C1\u03B7\u03BC\u03B1 \u039A\u03AD\u03B9\u03BB\u03B5\u03CA \u2013 \u03A7\u03AC\u03BC\u03B9\u03BB\u03C4\u03BF\u03BD (\u03C0\u03AE\u03C1\u03B5 \u03C4\u03BF \u03CC\u03BD\u03BF\u03BC\u03AC \u03C4\u03BF\u03C5 \u03B1\u03C0\u03CC \u03C4\u03BF\u03C5\u03C2 \u03BC\u03B1\u03B8\u03B7\u03BC\u03B1\u03C4\u03B9\u03BA\u03BF\u03CD\u03C2 \u03BA\u03B1\u03B9 \u0393\u03BF\u03C5\u03AF\u03BB\u03B9\u03B1\u03BC \u03A1\u03CC\u03BF\u03C5\u03B1\u03BD \u03A7\u03AC\u03BC\u03B9\u03BB\u03C4\u03BF\u03BD) \u03B1\u03BD\u03B1\u03C6\u03AD\u03C1\u03B5\u03B9 \u03CC\u03C4\u03B9 \u03BA\u03AC\u03B8\u03B5 \u03C4\u03B5\u03C4\u03C1\u03B1\u03B3\u03C9\u03BD\u03B9\u03BA\u03CC\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2 \u03C3\u03B5 \u03AD\u03BD\u03B1\u03BD \u03B1\u03BD\u03C4\u03B9\u03BC\u03B5\u03C4\u03B1\u03B8\u03B5\u03C4\u03B9\u03BA\u03CC \u03B4\u03B1\u03BA\u03C4\u03CD\u03BB\u03B9\u03BF (\u03CC\u03C0\u03C9\u03C2 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03C4\u03B1 \u03C3\u03CE\u03BC\u03B1\u03C4\u03B1 \u03C4\u03C9\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03AE \u03C4\u03C9\u03BD \u03BC\u03B9\u03B3\u03B1\u03B4\u03B9\u03BA\u03CE\u03BD \u03B1\u03C1\u03B9\u03B8\u03BC\u03CE\u03BD) \u03B9\u03BA\u03B1\u03BD\u03BF\u03C0\u03BF\u03B9\u03B5\u03AF \u03C4\u03B7 \u03B4\u03B9\u03BA\u03AE \u03C4\u03BF\u03C5 \u03C7\u03B1\u03C1\u03B1\u03BA\u03C4\u03B7\u03C1\u03B9\u03C3\u03C4\u03B9\u03BA\u03AE \u03B5\u03BE\u03AF\u03C3\u03C9\u03C3\u03B7. \u0391\u03BD \u0391 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03B4\u03BF\u03B8\u03B5\u03AF\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2 \u03BA\u03B1\u03B9 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03BF \u03C4\u03B1\u03C5\u03C4\u03BF\u03C4\u03B9\u03BA\u03CC\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2, \u03C4\u03CC\u03C4\u03B5 \u03C4\u03BF \u03C7\u03B1\u03C1\u03B1\u03BA\u03C4\u03B7\u03C1\u03B9\u03C3\u03C4\u03B9\u03BA\u03CC \u03C0\u03BF\u03BB\u03C5\u03CE\u03BD\u03C5\u03BC\u03BF \u03C4\u03BF\u03C5 A \u03BF\u03C1\u03AF\u03B6\u03B5\u03C4\u03B1\u03B9 \u03C9\u03C2"@el . . . . "\u0641\u064A \u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062E\u0637\u064A\u060C \u0645\u0628\u0631\u0647\u0646\u0629 \u0643\u0627\u064A\u0644\u064A-\u0647\u0627\u0645\u064A\u0644\u062A\u0648\u0646 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cayley\u2013Hamilton theorem)\u200F \u062A\u0646\u0635 \u0639\u0644\u0649 \u0623\u0646 \u0643\u0644 \u0645\u0635\u0641\u0648\u0641\u0629 \u0645\u0631\u0628\u0639\u0629 \u0645\u0639\u0631\u0641\u0629 \u0639\u0644\u0649 \u062D\u0644\u0642\u0629 \u062A\u0628\u0627\u062F\u0644\u064A\u0629 (\u0645\u062C\u0645\u0648\u0639\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u062D\u0642\u064A\u0642\u064A\u0629 \u0623\u0648 \u0627\u0644\u0639\u0642\u062F\u064A\u0629 \u0645\u062B\u0627\u0644\u064A\u0646) \u062A\u062D\u0642\u0642 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u0645\u064A\u0632\u0629 \u0627\u0644\u062E\u0627\u0635\u0629 \u0628\u0647\u0627. \u0633\u0645\u064A\u062A \u0647\u0627\u062A\u0647 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0629 \u0647\u0643\u0630\u0627 \u0646\u0633\u067E\u0629 \u0644\u0639\u0627\u0644\u0645\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0627\u0644\u0644\u0630\u064A\u0646 \u0648\u0636\u0639\u0627\u0647\u0627 \u0648\u0647\u0645\u0627 \u0623\u0631\u062B\u0648\u0631 \u0643\u0627\u064A\u0644\u064A \u0648\u0648\u064A\u0644\u064A\u0627\u0645 \u0647\u0627\u0645\u064A\u0644\u062A\u0648\u0646."@ar . . . . "Teorema di Hamilton-Cayley"@it . . . "\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uCF00\uC77C\uB9AC-\uD574\uBC00\uD134 \uC815\uB9AC(\uC601\uC5B4: Cayley\u2013Hamilton theorem)\uB294 \uC815\uC0AC\uAC01 \uD589\uB82C\uC774 \uC790\uAE30 \uC790\uC2E0\uC758 \uD2B9\uC131 \uBC29\uC815\uC2DD\uC744 \uB9CC\uC871\uC2DC\uD0A8\uB2E4\uB294 \uC815\uB9AC\uC774\uB2E4. \uC544\uC11C \uCF00\uC77C\uB9AC\uC640 \uC70C\uB9AC\uC5C4 \uB85C\uC5B8 \uD574\uBC00\uD134\uC758 \uC774\uB984\uC5D0\uC11C \uB530\uC654\uB2E4."@ko . . . . . . . "\u03A3\u03C4\u03B7 \u03B3\u03C1\u03B1\u03BC\u03BC\u03B9\u03BA\u03AE \u03AC\u03BB\u03B3\u03B5\u03B2\u03C1\u03B1, \u03C4\u03BF \u03B8\u03B5\u03CE\u03C1\u03B7\u03BC\u03B1 \u039A\u03AD\u03B9\u03BB\u03B5\u03CA \u2013 \u03A7\u03AC\u03BC\u03B9\u03BB\u03C4\u03BF\u03BD (\u03C0\u03AE\u03C1\u03B5 \u03C4\u03BF \u03CC\u03BD\u03BF\u03BC\u03AC \u03C4\u03BF\u03C5 \u03B1\u03C0\u03CC \u03C4\u03BF\u03C5\u03C2 \u03BC\u03B1\u03B8\u03B7\u03BC\u03B1\u03C4\u03B9\u03BA\u03BF\u03CD\u03C2 \u03BA\u03B1\u03B9 \u0393\u03BF\u03C5\u03AF\u03BB\u03B9\u03B1\u03BC \u03A1\u03CC\u03BF\u03C5\u03B1\u03BD \u03A7\u03AC\u03BC\u03B9\u03BB\u03C4\u03BF\u03BD) \u03B1\u03BD\u03B1\u03C6\u03AD\u03C1\u03B5\u03B9 \u03CC\u03C4\u03B9 \u03BA\u03AC\u03B8\u03B5 \u03C4\u03B5\u03C4\u03C1\u03B1\u03B3\u03C9\u03BD\u03B9\u03BA\u03CC\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2 \u03C3\u03B5 \u03AD\u03BD\u03B1\u03BD \u03B1\u03BD\u03C4\u03B9\u03BC\u03B5\u03C4\u03B1\u03B8\u03B5\u03C4\u03B9\u03BA\u03CC \u03B4\u03B1\u03BA\u03C4\u03CD\u03BB\u03B9\u03BF (\u03CC\u03C0\u03C9\u03C2 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03C4\u03B1 \u03C3\u03CE\u03BC\u03B1\u03C4\u03B1 \u03C4\u03C9\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03AE \u03C4\u03C9\u03BD \u03BC\u03B9\u03B3\u03B1\u03B4\u03B9\u03BA\u03CE\u03BD \u03B1\u03C1\u03B9\u03B8\u03BC\u03CE\u03BD) \u03B9\u03BA\u03B1\u03BD\u03BF\u03C0\u03BF\u03B9\u03B5\u03AF \u03C4\u03B7 \u03B4\u03B9\u03BA\u03AE \u03C4\u03BF\u03C5 \u03C7\u03B1\u03C1\u03B1\u03BA\u03C4\u03B7\u03C1\u03B9\u03C3\u03C4\u03B9\u03BA\u03AE \u03B5\u03BE\u03AF\u03C3\u03C9\u03C3\u03B7. \u0391\u03BD \u0391 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03B4\u03BF\u03B8\u03B5\u03AF\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2 \u03BA\u03B1\u03B9 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03BF \u03C4\u03B1\u03C5\u03C4\u03BF\u03C4\u03B9\u03BA\u03CC\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1\u03C2, \u03C4\u03CC\u03C4\u03B5 \u03C4\u03BF 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\u03C3\u03C4\u03BF \u03BB, \u03B7 \u03BF\u03C1\u03AF\u03B6\u03BF\u03C5\u03C3\u03B1 \u03B5\u03AF\u03BD\u03B1\u03B9 \u03B5\u03C0\u03AF\u03C3\u03B7\u03C2 \u03BC\u03B9\u03B1 n-\u03BF\u03C3\u03C4\u03AE\u03C2 \u03C4\u03AC\u03BE\u03B7\u03C2 \u03BA\u03B1\u03BD\u03BF\u03BD\u03B9\u03BA\u03BF\u03CD \u03C0\u03BF\u03BB\u03C5\u03C9\u03BD\u03CD\u03BC\u03BF\u03C5 \u03C3\u03C4\u03BF \u03BB. \u03A4\u03BF \u03B8\u03B5\u03CE\u03C1\u03B7\u03BC\u03B1 Cayley\u2013Hamilton \u03B4\u03B7\u03BB\u03CE\u03BD\u03B5\u03B9 \u03CC\u03C4\u03B9 \u03B1\u03BD\u03C4\u03B9\u03BA\u03B1\u03B8\u03B9\u03C3\u03C4\u03CE\u03BD\u03C4\u03B1\u03C2 \u03C3\u03C4\u03BF\u03BD \u03C0\u03AF\u03BD\u03B1\u03BA\u03B1 A \u03B3\u03B9\u03B1 \u03BB \u03C3\u03B5 \u03B1\u03C5\u03C4\u03CC \u03C4\u03BF \u03C0\u03BF\u03BB\u03C5\u03CE\u03BD\u03C5\u03BC\u03BF \u03AD\u03C7\u03B5\u03B9 \u03C9\u03C2 \u03B1\u03C0\u03BF\u03C4\u03AD\u03BB\u03B5\u03C3\u03BC\u03B1 \u03C4\u03BF\u03BD \u03BC\u03B7\u03B4\u03B5\u03BD\u03B9\u03BA\u03CC 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\u03B1\u03BD\u03B1\u03C6\u03BF\u03C1\u03B9\u03BA\u03AC \u03BC\u03B5 \u03C4\u03BF\u03C5\u03C2 \u03B1\u03BD\u03C4\u03AF\u03C3\u03C4\u03C1\u03BF\u03C6\u03BF\u03C5\u03C2 \u03CC\u03C1\u03BF\u03C5\u03C2 \u03C4\u03C9\u03BD \u03B3\u03C1\u03B1\u03BC\u03BC\u03B9\u03BA\u03CE\u03BD \u03C3\u03C5\u03BD\u03B1\u03C1\u03C4\u03AE\u03C3\u03B5\u03C9\u03BD \u03C4\u03C9\u03BD \u03C4\u03B5\u03C4\u03C1\u03B1\u03B4\u03B9\u03BA\u03CE\u03BD \u03B1\u03C1\u03B9\u03B8\u03BC\u03CE\u03BD, \u03AD\u03BD\u03B1\u03C2 \u03BC\u03B7-\u03B1\u03BD\u03C4\u03B9\u03BC\u03B5\u03C4\u03B1\u03B8\u03B5\u03C4\u03B9\u03BA\u03CC\u03C2 \u03B4\u03B1\u03BA\u03C4\u03CD\u03BB\u03B9\u03BF\u03C2, \u03B1\u03C0\u03CC \u03C4\u03BF\u03BD Hamilton. \u0391\u03C5\u03C4\u03CC \u03B1\u03BD\u03C4\u03B9\u03C3\u03C4\u03BF\u03B9\u03C7\u03B5\u03AF \u03C3\u03C4\u03B7\u03BD \u03B5\u03B9\u03B4\u03B9\u03BA\u03AE \u03C0\u03B5\u03C1\u03AF\u03C0\u03C4\u03C9\u03C3\u03B7 \u03BF\u03C1\u03B9\u03C3\u03BC\u03AD\u03BD\u03C9\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03C0\u03C1\u03B1\u03B3\u03BC\u03B1\u03C4\u03B9\u03BA\u03CE\u03BD \u03AE \u03BC\u03B9\u03B3\u03B1\u03B4\u03B9\u03BA\u03CE\u03BD \u03C0\u03B9\u03BD\u03AC\u03BA\u03C9\u03BD. \u03A4\u03BF \u03B8\u03B5\u03CE\u03C1\u03B7\u03BC\u03B1 \u03B9\u03C3\u03C7\u03CD\u03B5\u03B9 \u03B3\u03B5\u03BD\u03B9\u03BA\u03AC \u03B3\u03B9\u03B1 \u03C4\u03B5\u03C4\u03C1\u03B1\u03B4\u03B9\u03BA\u03BF\u03CD\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B5\u03C2. \u039F Cayley \u03C4\u03BF 1858 \u03C4\u03BF \u03B4\u03AE\u03BB\u03C9\u03C3\u03B5 \u03B3\u03B9\u03B1 \u03BA\u03B1\u03B9 \u03BC\u03B9\u03BA\u03C1\u03CC\u03C4\u03B5\u03C1\u03BF\u03C5\u03C2 \u03C0\u03AF\u03BD\u03B1\u03BA\u03B5\u03C2, \u03B1\u03BB\u03BB\u03AC \u03B4\u03B7\u03BC\u03BF\u03C3\u03AF\u03B5\u03C5\u03C3\u03B5 \u03BC\u03B9\u03B1 \u03B1\u03C0\u03CC\u03B4\u03B5\u03B9\u03BE\u03B7 \u03BC\u03CC\u03BD\u03BF \u03B3\u03B9\u03B1 \u03C4\u03B7 \u03C0\u03B5\u03C1\u03AF\u03C0\u03C4\u03C9\u03C3\u03B7. \u0397 \u03B3\u03B5\u03BD\u03B9\u03BA\u03AE \u03C0\u03B5\u03C1\u03AF\u03C0\u03C4\u03C9\u03C3\u03B7 \u03B1\u03C0\u03BF\u03B4\u03B5\u03AF\u03C7\u03C4\u03B7\u03BA\u03B5 \u03B3\u03B9\u03B1 \u03C0\u03C1\u03CE\u03C4\u03B7 \u03C6\u03BF\u03C1\u03AC \u03B1\u03C0\u03CC \u03C4\u03BF\u03BD Frobenius \u03C4\u03BF 1878."@el . . . . . "De stelling van Cayley-Hamilton is een stelling in de lineaire algebra die stelt dat elke vierkante re\u00EBle of complexe matrix voldoet aan zijn eigen karakteristieke vergelijking. De stelling is genoemd naar de wiskundigen Arthur Cayley en William Hamilton."@nl . . . . "\u0641\u064A \u0627\u0644\u062C\u0628\u0631 \u0627\u0644\u062E\u0637\u064A\u060C \u0645\u0628\u0631\u0647\u0646\u0629 \u0643\u0627\u064A\u0644\u064A-\u0647\u0627\u0645\u064A\u0644\u062A\u0648\u0646 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Cayley\u2013Hamilton theorem)\u200F \u062A\u0646\u0635 \u0639\u0644\u0649 \u0623\u0646 \u0643\u0644 \u0645\u0635\u0641\u0648\u0641\u0629 \u0645\u0631\u0628\u0639\u0629 \u0645\u0639\u0631\u0641\u0629 \u0639\u0644\u0649 \u062D\u0644\u0642\u0629 \u062A\u0628\u0627\u062F\u0644\u064A\u0629 (\u0645\u062C\u0645\u0648\u0639\u0629 \u0627\u0644\u0623\u0639\u062F\u0627\u062F \u0627\u0644\u062D\u0642\u064A\u0642\u064A\u0629 \u0623\u0648 \u0627\u0644\u0639\u0642\u062F\u064A\u0629 \u0645\u062B\u0627\u0644\u064A\u0646) \u062A\u062D\u0642\u0642 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u0645\u0645\u064A\u0632\u0629 \u0627\u0644\u062E\u0627\u0635\u0629 \u0628\u0647\u0627. \u0633\u0645\u064A\u062A \u0647\u0627\u062A\u0647 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0629 \u0647\u0643\u0630\u0627 \u0646\u0633\u067E\u0629 \u0644\u0639\u0627\u0644\u0645\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0627\u0644\u0644\u0630\u064A\u0646 \u0648\u0636\u0639\u0627\u0647\u0627 \u0648\u0647\u0645\u0627 \u0623\u0631\u062B\u0648\u0631 \u0643\u0627\u064A\u0644\u064A \u0648\u0648\u064A\u0644\u064A\u0627\u0645 \u0647\u0627\u0645\u064A\u0644\u062A\u0648\u0646."@ar . . . . . "Cayley\u2013Hamilton theorem"@en . "In linear algebra, the Cayley\u2013Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation."@en . "::"@en . . . . . "Th\u00E9or\u00E8me de Cayley-Hamilton"@fr . . . "\u0398\u03B5\u03CE\u03C1\u03B7\u03BC\u03B1 \u039A\u03AD\u03B9\u03BB\u03B5\u03CA-\u03A7\u03AC\u03BC\u03B9\u03BB\u03C4\u03BF\u03BD"@el . . . . "El teorema de Cayley-Hamilton \u00E9s un resultat fonamental en l'\u00E0lgebra lineal segons el qual, donada una matriu A i el seu polinomi caracter\u00EDstic Q(x), aquest s'anul\u00B7la en avaluar-lo en A.\u00C9s a dir, que Q(A)=0. \u00C9s immediat que dins el polinomi, la matriu A \u00E9s commutativa respecte a l'operaci\u00F3 producte. Una formulaci\u00F3 equivalent n'\u00E9s l'afirmaci\u00F3 que el polinomi caracter\u00EDstic de A \u00E9s un m\u00FAltiple del polinomi m\u00EDnim de A o, cosa que \u00E9s el mateix, que el polinomi caracter\u00EDstic de A \u00E9s un element de l'ideal principal de polinomis anul\u00B7ladors de A."@ca . . . . . . . "\u51F1\u840A\u2013\u54C8\u5BC6\u9813\u5B9A\u7406"@zh . "Teorema de Cayley-Hamilton"@es . . . . . . . . . . . . "Twierdzenie Cayleya-Hamiltona"@pl . . "1969"^^ . . . . . . "#FAFFFB"@en . . . . . . "En alg\u00E8bre lin\u00E9aire, le th\u00E9or\u00E8me de Cayley-Hamilton affirme que tout endomorphisme d'un espace vectoriel de dimension finie sur un corps commutatif quelconque annule son propre polyn\u00F4me caract\u00E9ristique. En termes de matrice, cela signifie que si A est une matrice carr\u00E9e d'ordre n et si est son polyn\u00F4me caract\u00E9ristique (polyn\u00F4me d'ind\u00E9termin\u00E9e X), alors en rempla\u00E7ant formellement X par la matrice A dans le polyn\u00F4me, le r\u00E9sultat est la matrice nulle : Le th\u00E9or\u00E8me de Cayley-Hamilton s'applique aussi \u00E0 des matrices carr\u00E9es \u00E0 coefficients dans un anneau commutatif quelconque. Un corollaire important du th\u00E9or\u00E8me de Cayley-Hamilton affirme que le polyn\u00F4me minimal d'une matrice donn\u00E9e est un diviseur de son polyn\u00F4me caract\u00E9ristique. Bien qu'il porte les noms des math\u00E9maticiens Arthur Cayley et William Hamilton, la premi\u00E8re d\u00E9monstration du th\u00E9or\u00E8me est donn\u00E9e par Ferdinand Georg Frobenius en 1878, Cayley l'ayant principalement utilis\u00E9 dans ses travaux, et Hamilton l'ayant d\u00E9montr\u00E9 en dimension 2."@fr . . "Inom linj\u00E4r algebra inneb\u00E4r Cayley\u2013Hamiltons sats (efter matematikerna Arthur Cayley och William Rowan Hamilton) att varje kvadratisk matris best\u00E5ende av komplexa eller reella tal uppfyller sin egen karakteristiska ekvation. Det vill s\u00E4ga: om \u00E4r en given n\u00D7n matris och In \u00E4r identitetsmatrisen med dimensionerna n\u00D7n, s\u00E5 definieras A:s karakteristiska ekvation som d\u00E4r \"det\" betecknar determinanten. Cayley\u2013Hamiltons sats inneb\u00E4r att om ers\u00E4tts med i den karakteristiska ekvationen erh\u00E5lls nollmatrisen:"@sv . . . . . . . "p/c120080"@en . . . "\u0645\u0628\u0631\u0647\u0646\u0629 \u0643\u0627\u064A\u0644\u064A-\u0647\u0627\u0645\u064A\u0644\u062A\u0648\u0646"@ar . "\uCF00\uC77C\uB9AC-\uD574\uBC00\uD134 \uC815\uB9AC"@ko . . "Em \u00E1lgebra linear, o teorema de Cayley-Hamilton (cujo nome faz refer\u00EAncia aos matem\u00E1ticos Arthur Cayley e William Hamilton) diz que o polin\u00F4mio m\u00EDnimo de uma matriz divide o seu polin\u00F4mio caracter\u00EDstico. Em outras palavras, seja uma matriz e o seu polin\u00F4mio caracter\u00EDstico, definido por: em que \u00E9 a fun\u00E7\u00E3o determinante e \u00E9 a matriz identidade de ordem Ent\u00E3o O teorema Cayley\u2013Hamilton \u00E9 equivalente \u00E0 afirma\u00E7\u00E3o de que o polin\u00F4mio m\u00EDnimo de uma matriz quadrada divide seu polin\u00F4mio caracter\u00EDstico."@pt . . . . . . . "Inom linj\u00E4r algebra inneb\u00E4r Cayley\u2013Hamiltons sats (efter matematikerna Arthur Cayley och William Rowan Hamilton) att varje kvadratisk matris best\u00E5ende av komplexa eller reella tal uppfyller sin egen karakteristiska ekvation. Det vill s\u00E4ga: om \u00E4r en given n\u00D7n matris och In \u00E4r identitetsmatrisen med dimensionerna n\u00D7n, s\u00E5 definieras A:s karakteristiska ekvation som d\u00E4r \"det\" betecknar determinanten. Cayley\u2013Hamiltons sats inneb\u00E4r att om ers\u00E4tts med i den karakteristiska ekvationen erh\u00E5lls nollmatrisen:"@sv . "6"^^ . . . . . . . . "Cayley\u2013Hamiltons sats"@sv . . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u0413\u0430\u0301\u043C\u0438\u043B\u044C\u0442\u043E\u043D\u0430 \u2014 \u041A\u044D\u0301\u043B\u0438 \u2014 \u043A\u043B\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u044B,\u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0430\u0435\u0442, \u0447\u0442\u043E \u043B\u044E\u0431\u0430\u044F \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u043D\u0430\u044F \u043C\u0430\u0442\u0440\u0438\u0446\u0430 \u0443\u0434\u043E\u0432\u043B\u0435\u0442\u0432\u043E\u0440\u044F\u0435\u0442 \u0441\u0432\u043E\u0435\u043C\u0443 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u043C\u0443 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u044E.\u041D\u0430\u0437\u0432\u0430\u043D\u043D\u0430\u044F \u0432 \u0447\u0435\u0441\u0442\u044C \u0423\u0438\u043B\u044C\u044F\u043C\u0430 \u0413\u0430\u043C\u0438\u043B\u044C\u0442\u043E\u043D\u0430 \u0438 \u0410\u0440\u0442\u0443\u0440\u0430 \u041A\u044D\u043B\u0438."@ru .