一道证明题：A为实矩阵，A+A转置=E，证明A可逆 证明题： 设方阵A满足A3=0，试证明E-A可逆，且（E-A...
\u7ebf\u4ee3\u9898\uff1aA\u7684\u4f34\u968f\u77e9\u9635\u7b49\u4e8eA\u7684\u8f6c\u7f6e\u77e9\u9635\uff0c\u5982\u4f55\u8bc1\u660eA\u662f\u53ef\u9006\u77e9\u9635\uff1f\u89e3\uff1a\u672c\u9898\u5229\u7528\u4e86\u53ef\u9006\u77e9\u9635\u7684\u6027\u8d28\u6c42\u89e3\u3002
\u672c\u9898\u8fdb\u884c\u8bc1\u660e\u5e94\u5f53\u5177\u6709\u4e00\u4e2a\u524d\u7f6e\u6761\u4ef6A \u2260 0\u3002
\u6545\u5047\u8bben = 2\u65f6,\u8bbe\u77e9\u9635A =
a b
c d
\u5219\u4f34\u968f\u77e9\u9635A* =
d -b
-c a
\u7531\u8f6c\u7f6eA\u2018 = A*\u5f97a = d,b = -c.
\u5f53\u8ba8\u8bba\u9650\u5236\u4e3a\u5b9e\u77e9\u9635,\u884c\u5217\u5f0f|A| = a²+b² > 0,A\u53ef\u9006.
\u590d\u77e9\u9635\u65f6\u6709\u53cd\u4f8b:
1 i
-i 1
n > 2\u65f6,\u65e0\u8bba\u5728\u54ea\u4e2a\u57df\u4e0a,\u547d\u9898\u603b\u662f\u6210\u7acb\u7684,\u8bc1\u660e\u5982\u4e0b.
\u82e5A\u7684\u79e9r(A) < n-1,\u4f34\u968f\u77e9\u9635A*\u662f\u7531A\u7684n-1\u9636\u5b50\u5f0f\u6784\u9020,\u6709A* = 0,\u4e0eA \u2260 0\u4ece\u800c\u8f6c\u7f6e\u77e9\u9635A' \u2260 0\u77db\u76fe\u3002
\u82e5r(A) = n-1,\u7531AA* = |A|\u00b7E = 0,\u53ca\u4e0d\u7b49\u5f0fr(A)+r(A*)-n \u2264 r(AA*),\u6709r(A*) \u2264 1 < r(A) = r(A')\u3002
\u4e8e\u662fr(A) < n\u65f6\u603b\u6709A* \u2260 A'.\u5373\u7531A* = A'\u53ef\u63a8\u51faA\u53ef\u9006\u3002
\u6269\u5c55\u8d44\u6599\uff1a
1\u3001\u53ef\u9006\u77e9\u9635\u4e00\u5b9a\u662f\u65b9\u9635\u3002
2\u3001\u5982\u679c\u77e9\u9635A\u662f\u53ef\u9006\u7684\uff0c\u5176\u9006\u77e9\u9635\u662f\u552f\u4e00\u7684\u3002
3\u3001A\u7684\u9006\u77e9\u9635\u7684\u9006\u77e9\u9635\u8fd8\u662fA\u3002\u8bb0\u4f5c\uff08A-1\uff09-1=A\u3002
4\u3001\u53ef\u9006\u77e9\u9635A\u7684\u8f6c\u7f6e\u77e9\u9635AT\u4e5f\u53ef\u9006\uff0c\u5e76\u4e14\uff08AT\uff09-1=\uff08A-1\uff09T (\u8f6c\u7f6e\u7684\u9006\u7b49\u4e8e\u9006\u7684\u8f6c\u7f6e\uff09
5\u3001\u82e5\u77e9\u9635A\u53ef\u9006\uff0c\u5219\u77e9\u9635A\u6ee1\u8db3\u6d88\u53bb\u5f8b\u3002\u5373AB=O\uff08\u6216BA=O\uff09\uff0c\u5219B=O\uff0cAB=AC\uff08\u6216BA=CA\uff09\uff0c\u5219B=C\u3002
6\u3001\u4e24\u4e2a\u53ef\u9006\u77e9\u9635\u7684\u4e58\u79ef\u4f9d\u7136\u53ef\u9006\u3002
7\u3001\u77e9\u9635\u53ef\u9006\u5f53\u4e14\u4ec5\u5f53\u5b83\u662f\u6ee1\u79e9\u77e9\u9635\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1- \u53ef\u9006\u77e9\u9635
\u8bbe\u65b9\u9635A\u6ee1\u8db3A3=0\uff0c\u8bd5\u8bc1\u660eE-A\u53ef\u9006\uff0c\u4e14\uff08E-A\uff09-1=E+A+A2 \uff0c\u8bc1\u660e\u8fc7\u7a0b\u5982\u4e0b\uff1a
E-A^3=E
\u5de6\u7aef\u56e0\u5f0f\u5206\u89e3\u6709\uff08E-A\uff09\uff08E+A+A^2\uff09=E
\u4ece\u800cE-A\u53ef\u9006\u4e14\uff08E-A\uff09^-1=E+A+A^2
\u5c06\u4e00\u4e2a\u77e9\u9635\u5206\u89e3\u4e3a\u6bd4\u8f83\u7b80\u5355\u7684\u6216\u5177\u6709\u67d0\u79cd\u7279\u6027\u7684\u82e5\u5e72\u77e9\u9635\u7684\u548c\u6216\u4e58\u79ef\uff0c\u77e9\u9635\u7684\u5206\u89e3\u6cd5\u4e00\u822c\u6709\u4e09\u89d2\u5206\u89e3\u3001\u8c31\u5206\u89e3\u3001\u5947\u5f02\u503c\u5206\u89e3\u3001\u6ee1\u79e9\u5206\u89e3\u7b49\u3002
\u53ef\u9006\u77e9\u9635\u7684\u6027\u8d28\u5b9a\u7406
1\u3001\u53ef\u9006\u77e9\u9635\u4e00\u5b9a\u662f\u65b9\u9635\u3002
2\u3001\u5982\u679c\u77e9\u9635A\u662f\u53ef\u9006\u7684\uff0c\u5176\u9006\u77e9\u9635\u662f\u552f\u4e00\u56de\u7684\u3002
3\u3001A\u7684\u9006\u77e9\u9635\u7684\u9006\u77e9\u9635\u8fd8\u662fA\u3002\u8bb0\u4f5c\uff08A-1\uff09-1=A\u3002
4\u3001\u53ef\u9006\u77e9\u9635A\u7684\u8f6c\u7f6e\u77e9\u9635AT\u4e5f\u53ef\u9006\uff0c\u5e76\u4e14\uff08AT\uff09-1=\uff08A-1\uff09T (\u8f6c\u7f6e\u7684\u9006\u7b49\u4e8e\u9006\u7684\u8f6c\u7f6e\uff09
5\u3001\u82e5\u77e9\u9635A\u53ef\u9006\uff0c\u5219\u77e9\u9635A\u6ee1\u8db3\u6d88\u53bb\u5f8b\u3002\u5373AB=O\uff08\u6216BA=O\uff09\uff0c\u5219B=O\uff0cAB=AC\uff08\u6216BA=CA\uff09\uff0c\u5219B=C\u3002
6\u3001\u4e24\u4e2a\u7b54\u53ef\u9006\u77e9\u9635\u7684\u4e58\u79ef\u4f9d\u7136\u53ef\u9006\u3002
符号说明：右上一撇表示转置，对于复数和复向量，头上一杠表示共轭 有什么问题希望及时反馈
利用这个公式，A的转置=A^(n-1).代入得A(E+A^(n-2))=E.因而A可逆
⒂⒘
绛旓細绗﹀彿璇存槑锛氬彸涓婁竴鎾囪〃绀鸿浆缃紝瀵逛簬澶嶆暟鍜屽鍚戦噺锛屽ご涓婁竴鏉犺〃绀哄叡杞湁浠涔堥棶棰樺笇鏈涘強鏃跺弽棣
绛旓細鏄剧劧褰揂鏄浜ら樀鐨勬椂鍊(A伪,A尾)=(伪,尾)鍙嶈繃鏉, 浠=A^TA, M鏄竴涓绉伴樀 鍙栁=尾=e_i寰楀埌M(i,i)=1, 杩欓噷e_i鏄崟浣嶉樀鐨勭i鍒 瀵逛簬i鈮爅, 鍙栁=e_i, 尾=e_j, 寰楀埌M(i,j)=0鎵浠=I
绛旓細x^T(A^TA)x = (Ax)^T(Ax)>=0 鑰屽疄瀵圭О鐭╅樀鐨勭壒寰佸奸兘鏄疄鏁 鎵浠ュ疄瀵圭О鐭╅樀 A^TA 鐨勭壒寰佸奸兘鏄潪璐熷疄鏁
绛旓細璇佹槑A^TAx=0锛屼笌Ax=0鍚岃В鍗冲彲銆侫x=0锛岄偅涔圓^TAx=0 A^TAx=0锛岄偅涔坸^TA^TAx=0锛岄偅涔(Ax,Ax)=0,閭ｄ箞Ax=0 鏁匒^TAx=0锛屼笌Ax=0鍚岃В
绛旓細鎶鐭╅樀A璁惧嚭鏉ワ紝鐒跺悗璁(A^T)A=A(A^T锛夐偅涓瀹氫細鍙戠幇aij=aji锛屼篃灏辨槸A涓哄绉伴樀
绛旓細鍥炵瓟锛氳A鍏朵腑鐨勪竴琛屼负 a1,a2,,,,an 鍒 AA^T=0 寰楀嚭 a1^2+a2^2+a3^2++++an^2=0 鎵浠 a1=a2=a3======an=0 鎵浠 A涓0鐭╅樀
绛旓細寮曠敤: " 宸︿箻A-1寰 A'=A-1A'A鈭碅'涓哄绉鐭╅樀 "杩欎笉瀵, 涓. A涓嶄竴瀹氬彲閫 浜. 鍗充娇A鍙嗕篃鎺ㄤ笉鍑篈'瀵圭О 鎴戝杩欓鏈夊叴瓒, 浣嗕笉浼氳В, 鎰熻棰樼洰缁欑殑鏉′欢涓嶈冻, 棰樼洰鏉ユ簮鏄摢閲?鑰冪爺棰樻垜閮芥湁, 杩欓鏄摢骞寸殑? 鏁板嚑?
绛旓細蹇呰鎬ф樉鐒躲備粎璇佸厖鍒嗘э紝璁AA'=A^2 鍒欒冭檻 tr(A-A')(A-A')'=2tr(AA')-2trA^2=0 浠庤孉-A'=0锛屽嵆A瀹炲绉般傛敞锛氭澶勪粎鐢ㄥ埌濡備笅浜嬪疄锛岀煩闃礎=B鐨勫厖瑕佹潯浠舵槸 tr(A-B)(A-B)'=0
绛旓細鎴戣寰楀彲浠ラ嗙敤鍑幈-姹夊瘑灏旈】瀹氱悊,浠涓虹壒寰佸,p涓虹壒寰佸悜閲,鍒A*p=q*p.灏咥^2-3A+2E=0涓よ竟鍚屼箻p,鍒(q^2-3q+2)*p=0,涓攑闈0.鍒欏彲浠ヨВ鍑簈=1,2.鐗瑰緛鍊煎潎澶т簬0,鍒橝姝ｅ畾.杩欑绫诲瀷棰樼洰寰堝,褰撶劧杩樻湁鍏朵粬瑙ｆ硶,涓鏃跺洖蹇嗕笉璧锋潵浜.
绛旓細杩欎釜璇佹槑涓嶉毦锛屾浜や笁瑙掑垎瑙ｅ彲浠ユ牴鎹濆瘑鐗规浜ゅ寲杩囩▼寰楀埌锛屽敮涓鎬у彲浠ュ啀鍋囪瀛樺湪涓涓繖鏍风殑鍒嗚В锛屽彲浠ュ緢绠鍗曠殑璇佹槑鍒嗚В鐨勫敮涓鎬
