矩阵A的n次方 计算方法里面矩阵A的n次方怎么算
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1\u3001\u8ba1\u7b97A^2\uff0cA^3 \u627e\u89c4\u5f8b\uff0c\u7136\u540e\u7528\u5f52\u7eb3\u6cd5\u8bc1\u660e\u3002
2\u3001\u82e5r(A)=1\uff0c\u5219A=\u03b1\u03b2^T\uff0cA^n=(\u03b2^T\u03b1)^(n-1)A
\u6ce8:\u03b2^T\u03b1 =\u03b1^T\u03b2 = tr(\u03b1\u03b2^T)
3\u3001\u5206\u62c6\u6cd5\uff1aA=B+C\uff0cBC=CB\uff0c\u7528\u4e8c\u9879\u5f0f\u516c\u5f0f\u5c55\u5f00\u3002
\u9002\u7528\u4e8e B^n \u6613\u8ba1\u7b97\uff0cC\u7684\u4f4e\u6b21\u5e42\u4e3a\u96f6\uff1aC^2 \u6216 C^3 = 0
4\u3001\u7528\u5bf9\u89d2\u5316 A=P^-1diagP
A^n = P^-1diag^nP
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\u5bf9\u89d2\u6cd5\uff1a A=P^-1diagP\uff0cA^n = P^-1diag^nP\u3002
\u62c6\u5206\u6cd5\uff1aA=B+C,BC=CB,\u7528\u4e8c\u9879\u5f0f\u516c\u5f0f\u5c55\u5f00\uff0c\u9002\u7528\u4e8e B^n \u6613\u8ba1\u7b97,C\u7684\u4f4e\u6b21\u5e42\u4e3a\u96f6:C^2 \u6216 C^3 = 0\u3002
\u7279\u5f81\u503c\u6cd5\uff1a\u82e5r(A)=1,\u5219A=\u03b1\u03b2^T,A^n=(\u03b2^T\u03b1)^(n-1)A\uff0c\u6ce8:\u03b2^T\u03b1 =\u03b1^T\u03b2 = tr(\u03b1\u03b2^T)\u3002
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绛旓細涓昏鏈変互涓嬪嚑绉嶅姙娉曪細鏁板褰掔撼娉曪細璁$畻A^2,A^3鎵惧嚭鐭╅樀A鐨瑙勫緥锛屽亣璁続^(n-1)锛岀敤A^(n-1)鐨勬暟瀛﹀紡鏉ヨ瘉鏄嶢^n銆傚瑙掓硶锛 A=P^-1diagP锛孉^n = P^-1diag^nP銆傛媶鍒嗘硶锛欰=B+C,BC=CB,鐢ㄤ簩椤瑰紡鍏紡灞曞紑锛岄傜敤浜 B^n 鏄撹绠,C鐨勪綆娆″箓涓洪浂:C^2 鎴 C^3 = 0銆傜壒寰佸兼硶锛氳嫢r(A)...
绛旓細鐭╅樀鐨刵娆℃柟鍙互閫氳繃鐭╅樀涔樻硶鏉ヨ绠椼傝缁嗚В閲婂涓嬶細1. 鐭╅樀涔樻硶鐨勫畾涔変笌鎬ц川 鐭╅樀涔樻硶鏄竴绉嶇壒娈婄殑杩愮畻锛屽彧鏈夊綋涓や釜鐭╅樀鐨勭淮搴︾浉瀹规椂鎵嶈兘杩涜銆傚浜庝袱涓鐭╅樀A鍜孊锛屽畠浠殑涔樼НC鏄竴涓柊鐨勭煩闃碉紝鍏朵腑C鐨勬瘡涓厓绱犻兘鏄氳繃A鍜孊涓浉搴斾綅缃殑鍏冪礌鐩镐箻寰楀埌鐨勩傜煩闃典箻娉曟弧瓒崇粨鍚堝緥鍜屽垎閰嶅緥锛岃繖涓鸿绠鐭╅樀鐨骞...
绛旓細璺1锛 鑻(A)=1鍒橝鑳藉垎瑙d负涓琛屼笌涓鍒楃殑涓や釜鐭╅樀鐨涔樼Н锛岀敤缁撳悎寰嬪氨鍙互寰堟柟渚跨殑姹傚嚭A^n 鎬濊矾2锛 鑻鑳藉垎瑙f垚2涓煩闃电殑鍜孉 = B + C鑰屼笖BC = CB鍒橝^n = (B+C)^n鍙敤浜岄」寮忓畾鐞嗗睍寮锛屽綋鐒禕锛孋涔嬩腑鏈変竴涓殑鏂瑰瘑瑕佸敖蹇负0 ...
绛旓細浠=[a1,a2,...,an]^T, b=[b1,b2,...,bn]^T 閭d箞A=ab^T A^2=(ab^T)(ab^T)=a(b^Ta)b^T=(b^Ta)A ...A^n=(b^Ta)^{n-1}A
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绛旓細鍑嗙‘鏉ヨ搴旇鏄繖鏍风殑锛 姹鐭╅樀A涓悇鍏冪礌鐨勪箻鏂癸紙N娆℃柟锛夌殑鍛戒护鏄疉.^N,娉ㄦ剰搴曚笅鐨勨.鈥濓紱 A^N鍙繍琛屽彧鏄洜涓篈鏄柟闃,濡傛灉涓嶆槸鏂归樀灏变細鍑虹幇閿欒锛 姣斿A=[1,2,3,4];A^1.5;2^A;杩愯缁撴灉鏄嚭閿欑殑,姝g‘鐨勫啓娉曞簲璇ユ槸 A=[1,2,3,4];A.^1.5;2.^A;
绛旓細*A 鐢ㄩ掑綊瀹炵幇绠楁硶2锛歁atrix pow(Matrix A, int n) //姹侫^n { Matrix B;if(n==1) return A;else if(n % 2 == 0) { B = pow(A, n/2);return mul(B, B);} else { B = pow(A, n/2);return mul(A, mul(B, B));} } 鍏朵腑 mul(A,B)涓烘櫘閫鐭╅樀涔樻硶A*B ...
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绛旓細a^3=a^2*a=a*a=a 鎵浠^n=a