线性代数:设三阶实对称矩阵A的秩为2，r1=r2=6是A的二重特征值。 设A是3阶实对称矩阵，秩r=2.若A的平方=A，则A的特征值...
3\u9636\u5b9e\u5bf9\u79f0\u77e9\u9635A\u7684\u79e9\u662f2\uff0c=6\u662f\u4e8c\u91cd\u7279\u5f81\u503ca1,a2,a3\u662f(A-6E)x=0\u7684\u6839\uff0c
\u4ee4P=
1 2 -1
1 1 2
0 1 -3
\u4e5f\u5c31\u662f(A-6E)P=0
\u6240\u4ee5A-6E\u7684\u884c\u5411\u91cf\u662fyP=0\u7684\u6839\uff0c\u7136\u540e\u6309\u7167\u666e\u901a\u7ebf\u6027\u65b9\u7a0b\u6c42\u89e3\u7684\u65b9\u6cd5\u6c42\u89e3\u5373\u53ef
\u4f46\u662fdet(P)=-3 -1 -2 +6\u4e0d\u7b49\u4e8e0
\u6240\u4ee5A-6E\u5fc5\u7136\u662f0\uff0c\u8fd9\u4e0d\u53ef\u80fd
\u800c\u4e14\u4e8c\u91cd\u7279\u5f81\u503c\u53ea\u53ef\u80fd\u6709\u4e24\u4e2a\u7ebf\u6027\u65e0\u5173\u7279\u5f81\u5411\u91cf\u3002a1,a2,a3\u597d\u50cf\u7ebf\u6027\u65e0\u5173\u3002\u4e5f\u8bb8\u662f\u6211\u8ba1\u7b97\u9519\u8bef\uff0c\u4f60\u6700\u597d\u9a8c\u8bc1\u4e00\u4e0b\uff0c\u5982\u679c\u786e\u5b9e\u7ebf\u6027\u65e0\u5173\uff0c\u9898\u76ee\u80af\u5b9a\u9519\u8bef
A^2=A\u8bf4\u660eA\u7684\u7279\u5f81\u503c\u03bb\u5fc5\u987b\u6ee1\u8db3\u03bb^2=\u03bb\uff0c\u6240\u4ee5\u03bb\u53ea\u80fd\u662f0\u62161
\u6ce8\u610fA\u53ef\u5bf9\u89d2\u5316\uff0c\u6b64\u65f6rank(A)\u5c31\u662fA\u7684\u975e\u96f6\u7279\u5f81\u503c\u4e2a\u6570\uff0c\u6240\u4ee5A\u7684\u7279\u5f81\u503c\u662f1,1,0
第二问PAP^{-1} 死算，懒得算了……╮(╯▽╰)╭
希望对你能有所帮助。
绛旓細璁 x涓轰换涓鐗瑰緛鍚戦噺锛宺涓哄搴旂壒寰佹牴銆A^2=A ==> A2x=Ax ==> (r^2-r)x=0 ==> r(r-1)=0 鎵浠 r=1 鎴 0 鍥犱负 R(A)=2, 鎵浠ョ壒寰佹牴蹇呯劧鏄 1锛1锛0
绛旓細鐢变簬瀹炲绉扮煩闃鐨勫睘浜庝笉鍚岀壒寰佸肩殑鐗瑰緛鍚戦噺姝ｄ氦 鎵浠ユ湁 a+a(a+1)+1=0,鍗 (a+1)^2=0 鎵浠 a=-1.鎵浠ノ=1鍜屛=0鎵瀵瑰簲鐨勭壒寰佸悜閲忓垎鍒负(1,-1,1)T鍙(-1,0,1)T 璁続鐨灞炰簬鐗瑰緛鍊-1鐨勭壒寰佸悜閲忎负 (x1,x2,x3)^T 鍒 x1-x2+x3=0 -x1 +x3=0 寰 (1,2,1)^T 浠 P= 1 ...
绛旓細A2=A鏄粈涔堬紵鎵撻敊浜嗗惂锛岄夯鐑︿慨鏀逛竴涓嬨傚鏋滄槸A^2=A 鍗矨^2-A=0 鍐欐垚鐗瑰緛鍊兼柟绋嬑籢2-位=0 鎵浠鍙兘鐨勭壒寰佸兼槸锛0鍜1 鍥犱负A鐨绉╂槸2锛屾墍浠ユ槸1,1,0 鏂规硶鎬荤粨涓涓嬪氨鏄 --- 鐢ㄧ粰鐨鐭╅樀鍏崇郴寮忥紝鍐欏嚭鐗瑰緛鍊兼柟绋嬶紝鐒跺悗瑙ｅ嚭鍙兘鐨勭壒寰佸硷紝杩欎簺鐗瑰緛鍊煎彧鏄彲鑳藉硷紝鏈夊嚑涓 锛屾湁娌℃湁閮芥槸涓嶇‘瀹...
绛旓細鍥犱负 R(A-2E)=1 鎵浠 A 鐨灞炰簬鐗瑰緛鍊2鐨绾挎鏃犲叧鐨勭壒寰佸悜閲忔湁 3-1=2 涓.鑰孉鏄瀹炲绉扮煩闃, k閲嶇壒寰佸兼湁k涓嚎鎬ф棤鍏崇殑鐗瑰緛鍚戦噺 鎵浠2鏄A鐨浜岄噸鐗瑰緛鍊.
绛旓細鍒╃敤瀹炲绉扮煩闃电殑灞炰簬涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺姝ｄ氦鐭ュ睘浜庣壒寰佸1鐨勭壒寰佸悜閲忔弧瓒 x1+x2-2x3=0瑙ｅ緱灞炰簬鐗瑰緛鍊1鐨勭壒寰佸悜閲 (1,-1,0)^T,(2,0,1)^T3涓壒寰佸悜閲忔瀯鎴愮煩闃礟鏈 A=Pdiag(1,1,-2)P^-1銆傚绉扮煩闃碉紙Symmetric Matrices锛夋槸鎸囦互涓诲瑙掔嚎涓哄绉拌酱锛屽悇鍏冪礌瀵瑰簲鐩哥瓑鐨勭煩闃点傚湪绾挎т唬鏁涓紝...
绛旓細鎴戝彧鍐欑瓟妗堜簡 绗竴棰楤.2 4.璁続,B,C,D鏄竴缁刵缁村悜閲忥紝鍏朵腑A,B,C绾挎鐩稿叧锛岄偅涔堬紙D. A,B,C,D绾挎х浉鍏 锛5.璁3闃跺疄瀵圭О鐭╅樀A鐨鐗瑰緛鍊煎垎鍒负1锛0锛-1锛屽垯锛 D. A璐熷畾 锛6.璁続涓哄疄瀵圭О鐭╅樀锛屼笖A鐨勫钩鏂圭瓑浜0鐭╅樀銆傞偅涔堬紙A. AAA=0 锛7.璁続 涓 mxn鐭╅樀锛岀З(A)=r<n 锛屽垯...
绛旓細鐢盧(A)=2鐭 0 鏄A鐨鐗瑰緛鍊 A 0 0 1 1 -1 1 = 0 0 -1 1 1 1 璇存槑 (0,1,-1)^T 鍜 (0,1,1)^T 鏄垎鍒睘浜庣壒寰佸-1鍜1鐨勭壒寰佸悜閲 鐢变簬瀹炲绉扮煩闃灞炰簬涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺姝ｄ氦 鎵浠ュ睘浜0鐨勭壒寰佸悜閲忔弧瓒 x2-x3=0, x2+x3=0 寰 (1,0,0)^T 鐢辫繖3涓悜閲...
绛旓細鍒╃敤瀹炲绉扮煩闃电殑灞炰簬涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺姝ｄ氦鐭ュ睘浜庣壒寰佸1鐨勭壒寰佸悜閲忔弧瓒 x1+x2-2x3=0瑙ｅ緱灞炰簬鐗瑰緛鍊1鐨勭壒寰佸悜閲 (1,-1,0)^T,(2,0,1)^T3涓壒寰佸悜閲忔瀯鎴愮煩闃礟鏈 A=Pdiag(1,1,-2)P^-1銆傚绉扮煩闃碉紙Symmetric Matrices锛夋槸鎸囦互涓诲瑙掔嚎涓哄绉拌酱锛屽悇鍏冪礌瀵瑰簲鐩哥瓑鐨勭煩闃点傚湪绾挎т唬鏁涓紝...
绛旓細瀹炲绉扮煩闃礎涓瀹氳兘鐩镐技瀵硅鍖栵紝鍗冲瓨鍦ㄥ彲閫嗙煩闃祊,浣垮緱P閫咥P =diag(位1锛屛2锛屛3).鍥犱负鍏剁З涓2锛屾墍浠ュ瑙掗樀涓篸iag(6,6,0)锛屽嵆鍏跺彟涓鐗瑰緛鍊间负0锛岀敱浜庡疄瀵圭О鐭╅樀涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺鐩镐簰姝ｄ氦锛屽彲璁惧睘浜庣壒寰佸0鐨勭壒寰佸悜閲忎负伪=(x1,x2,,x3)^T,鏁呂1伪^T=0,伪2伪^T=0锛屽彲瑙ｅ緱伪=(-1...
绛旓細鐢变簬灞炰簬涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺绾挎鏃犲叧 鎵浠 尾1,尾2 鏄 B 鐨勫垪鍚戦噺缁勭殑鏋佸ぇ鏃犲叧缁 鎵浠 r(B) = 2 尾1^T尾2 = 0 --瀹炲绉扮煩闃灞炰簬涓嶅悓鐗瑰緛鍊肩殑鐗瑰緛鍚戦噺姝ｄ氦
