可以举个“n级实对称(反对称,上三角形)矩阵”的例子吗？这里感谢！！ 线性代数中Fn*n中全体对称矩阵（反对称，上三角）构成的线性...
\u4ec0\u4e48\u662f\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u80fd\u4e3e\u4e2a\u4f8b\u5b50\u5417\uff1f\u6ee1\u8db3A^T=-A\u7684\u5b9e\u77e9\u9635A\u5c31\u53eb\u5b9e\u53cd\u5bf9\u79f0\u9635\u3002
\u6bd4\u5982
0 1 2
-1 0 -3
-2 3 0
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\u5b83\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a1.A\u7684\u7279\u5f81\u503c\u662f\u96f6\u6216\u7eaf\u865a\u6570\uff1b2.|A|\u662f\u4e00\u4e2a\u975e\u8d1f\u5b9e\u6570\u7684\u5e73\u65b9\uff1b3.A\u7684\u79e9\u662f\u5076\u6570\uff0c\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u96f6 \u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u82e5\u77e9\u9635A\u6ee1\u8db3\u6761\u4ef6A=-AT\uff0c\u5219\u79f0A\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\u3002\u7531\u5b9a\u4e49\u77e5\u53cd\u5bf9\u79f0\u77e9\u9635\u4e00\u5b9a\u662f\u65b9\u9635\uff0c\u800c\u4e14\u4f4d\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u4e24\u4fa7\u5bf9\u79f0\u4f4d\u7f6e\u4e0a\u7684\u5143\u7d20\u5fc5\u7b26\u53f7\u76f8\u53cd\uff0c\u5373 \uff0c\u5176\u4e2di\u3001j\u4e3a\u4efb\u610f\u4e0d\u5927\u4e8e\u77e9\u9635\u7ef4\u6570\u7684\u5b9e\u6570\u3002
\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\u6027\u8d281\uff1a\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u503c\u4e3a0\u3002
\u6027\u8d282\uff1a\u5f53A\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u65f6\uff0c\u5bf9\u4e8e \u6709XTAX =0\u3002
\u6027\u8d283\uff1a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f\u96f6\u6216\u7eaf\u865a\u6570\u3002
\u6027\u8d284\uff1a\u82e5A\u4e3a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cA\u7684\u7279\u5f81\u503c\u03bb= bi(b\u22600)\u6240\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u03b1+\u03b2i\u4e2d\u5b9e\u90e8\u4e0e\u865a\u90e8\u5bf9\u5e94\u7684\u5411\u91cf\u03b1\u3001\u03b2\u76f8\u4e92\u6b63\u4ea4 \u3002
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\u89e3\u51b3\u65b9\u68481\uff1a
\u7ef4\u6570\uff1an(n+1)/2. \u57fa\uff1a\u5bf9\u89d2\u7ebf\u5143\u662f1\uff0c\u5176\u4f59\u5168\u662f0\u7684\u5bf9\u79f0\u9635\uff0c\u5171n\u4e2a\uff1b\u7b2ci\u884c\u7b2cj\u5217\u548c\u7b2cj\u884c\u7b2ci\u5217\u4e3a1\uff0c\u5176\u4f59\u4e3a0\u7684\u5bf9\u79f0\u9635\uff08i\u548cj\u4e0d\u76f8\u7b49\uff09\uff0c\u5171n(n-1)/2\u4e2a\uff0c\u76f8\u52a0\u4e3an(n+1)/2\u4e2a\u3002
\u89e3\u51b3\u65b9\u68482\uff1a
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\u7b54\uff1a\u76f4\u89c2\u7406\u89e3\uff0cn\u9636\u5bf9\u79f0\u77e9\u9635\u7684\u4e0a\u4e09\u89d2\u90e8\u5206\u662f\u5b8c\u5168\u81ea\u7531\u7684\uff0c\u81ea\u7531\u5ea6\u662f1+2+...+n\uff0c\u4e5f\u5c31\u662f\u5f20\u6210\u7684\u7a7a\u95f4\u7684\u7ef4\u6570 \u4e25\u683c\u8bc1\u660e\u5c31\u662f\u9020\u4e00\u7ec4\u57fa\u51fa\u6765\u6309\u5b9a\u4e49\u8bc1
n\u9636\u5168\u4f53\u5bf9\u79f0\u77e9\u9635\u6240\u6210\u7684\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u6570\u600e\u4e48\u6c42
\u7b54\uff1a\u4f60\u597d\uff01\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u8fd9\u4e2a\u7ebf\u6027\u7a7a\u95f4\u7684\u4e00\u7ec4\u57fa\uff0c\u6240\u4ee5\u5b83\u7684\u7ef4\u6570\u4e2dn(n+1)/2\u3002\u7ecf\u6d4e\u6570\u5b66\u56e2\u961f\u5e2e\u4f60\u89e3\u7b54\uff0c\u8bf7\u53ca\u65f6\u91c7\u7eb3\u3002\u8c22\u8c22\uff01
\u95ee\u5218\u8001\u5e08\uff0c\u6240\u6709n\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u6784\u6210\u6570\u57dfP\u4e0a\u7684\u7ebf\u6027\u7a7a...
\u7b54\uff1a\u7531\u4e8e \u53cd\u5bf9\u79f0\u77e9\u9635 \u6ee1\u8db3 aij = - aji, \u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u5143\u7d20\u5168\u662f0 \u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u4ee5\u4e0b\u5143\u7d20\u7531\u4e3b\u5bf9\u89d2\u7ebf\u4ee5\u4e0a\u5143\u7d20\u552f\u4e00\u786e\u5b9a \u6240\u4ee5\u7ef4\u6570\u4e3a n-1 + n-2 + ...+ 2 + 1 = n(n-1)/2.
\u6c42n\u9636\u5168\u4f53\u5bf9\u79f0\u77e9\u9635\u6240\u6210\u7684\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u6570\u4e0e\u4e00\u7ec4\u57fa
\u7b54\uff1a1\u3001n\u9636\u5168\u4f53\u5bf9\u79f0\u77e9\u9635\u6240\u6210\u7684\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u6570\u662f (n^2 - n )/2 + n \u5176\u5b9e\u5c31\u662f\uff1a \u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u4e2a\u6570 + \u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u65b9\u7684\u5143\u7d20\u4e2a\u6570 \u8fd9\u4e9b\u5143\u7d20\u6240\u5728\u7684\u4f4d\u7f6e\uff0c\u552f\u4e00\u786e\u5b9a\u4e00\u4e2a\u5bf9\u79f0\u77e9\u9635\u3002 2\u3001\u6240\u4ee5\u6709\uff1a \u8bbe Eij \u4e3a \u7b2ci\u884c\u7b2cj\u5217\u4f4d\u7f6e\u662f1\u5176\u4f59\u90fd\u662f0\u7684n\u9636\u65b9\u9635 \u5219 n\u9636\u5168\u4f53...
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b d e
c e f 这是对称的
0 b c
-b 0 e
-c -e 0 这是反对称（反对称，对角线上元素一定为0）
a b c
0 d e
0 0 f这是上三角。
a,b,c,d,e,f取实数就好了，上述就是3阶的一般表示形式。
绛旓細b d e c e f 杩欐槸瀵圭О鐨 0 b c -b 0 e -c -e 0 杩欐槸鍙嶅绉帮紙鍙嶅绉帮紝瀵硅绾夸笂鍏冪礌涓瀹氫负0锛塧 b c 0 d e 0 0 f杩欐槸涓婁笁瑙掋俛,b,c,d,e,f鍙栧疄鏁板氨濂戒簡锛屼笂杩板氨鏄3闃剁殑涓鑸〃绀哄舰寮忋
绛旓細婊¤冻A^T=-A鐨勫疄鐭╅樀A灏卞彨瀹炲弽瀵圭О闃銆傛瘮濡 0 1 2 -1 0 -3 -2 3 0 鍏冪礌aij閮芥槸瀹炴暟锛屽苟涓攁ij=-aji(i锛宩=1锛2锛屸)锛宯鐨刵闃剁煩闃礎=(aij)銆傚畠鏈変互涓嬫ц川锛1.A鐨勭壒寰佸兼槸闆舵垨绾櫄鏁帮紱2.|A|鏄竴涓潪璐熷疄鏁扮殑骞虫柟锛3.A鐨勭З鏄伓鏁帮紝濂囨暟闃跺弽瀵圭О鐭╅樀鐨勮鍒楀紡绛変簬闆 銆
绛旓細鎬荤殑鏉ヨ锛N闃跺疄鍙嶅绉扮煩闃锛屽敖绠＄湅浼肩畝鍗曪紝鍗磋暣鍚潃涓板瘜鐨勬暟瀛︿箣缇庡拰骞挎硾鐨勫疄闄呭簲鐢ㄣ傚畠浠殑闈炶礋琛屽垪寮忋佸伓鏁扮З鍜岀壒娈婄浉浼兼爣鍑嗗瀷锛屾瘡涓椤圭壒鎬ч兘涓烘繁鍏ョ悊瑙ｆ暟瀛︿笘鐣屾墦寮浜嗕竴鎵囨柊绐椼傚湪鏈潵鐨勬暟瀛︽帰绱腑锛岃繖浜涚壒鎬ф棤鐤戝皢缁х画婵鍙戞垜浠殑鍏磋叮鍜屾濊冦
绛旓細棣栧厛瑕佺煡閬撳绉扮煩闃靛拰鍙嶅绉鐭╅樀鐨勫畾涔锛屽绉颁妇璇侊紝灏辨槸A鐨勮浆缃瓑浜嶢锛涘弽瀵圭О鐭╅樀灏辨槸B鐨勮浆缃瓑浜庯紞B锛岀敱浜庤瘉鏄庤繃绋嬭鐢ㄥ埌楂樼瓑鏁板璇佹槑绗﹀彿锛屽涓嬪浘鎵绀猴細瀵圭О鐭╅樀鐨勫熀鏈ц川锛1銆佹瘡涓疄鏂瑰舰鐭╅樀閮藉彲鍐欎綔涓涓疄瀵圭О鐭╅樀鐨勭Н锛屾瘡涓鏂瑰舰鐭╅樀閮藉彲鍐欎綔涓や釜澶嶅绉扮煩闃电殑绉2銆佽嫢瀵圭О鐭╅樀A鐨勬瘡涓厓绱犲潎涓...
绛旓細缁欏嚭鍩猴細aij=1锛宎ji=锛1锛屽叾浣欏厓绱犳槸0鐨勭煩闃垫槸涓涓弽瀵圭О闃点傚叾涓1<=i<=n锛宯>=j>i銆傝繖鏍风殑鐭╅樀鍏n(n锛1)/2涓紝杩欎簺鐭╅樀鏄嚎鎬ф棤鍏崇殑锛堟槗璇侊級锛屼笖姣忎竴涓弽瀵圭О闃甸兘鍙互鐢卞畠浠殑绾挎х粍鍚堢粰鍑猴紝鍥犳杩欐槸涓涓熀銆俷+(n-1)+(n-2)+鈥︹+2+1=n(n+1)/2缁 绗琲琛岀j鍒楋紙i<=j锛...
绛旓細杩欎釜棰樼洰闅惧害寰堥珮锛岃瘉鏄庤繃绋嬪鍥撅紝瑕佸熷姪杩炵画鍑芥暟鐨勫畾鍙锋ц川銆傜粡娴庢暟瀛﹀洟闃熷府浣犺В绛旓紝璇峰強鏃堕噰绾炽傝阿璋紒
绛旓細2銆瀹炲绉鐭╅樀A鐨勭壒寰佸奸兘鏄疄鏁帮紝鐗瑰緛鍚戦噺閮芥槸瀹炲悜閲忋3銆n闃跺疄瀵圭О鐭╅樀A蹇呭彲瀵硅鍖栵紝涓旂浉浼煎瑙掗樀涓婄殑鍏冪礌鍗充负鐭╅樀鏈韩鐗瑰緛鍊笺4銆佽嫢位0鍏锋湁k閲嶇壒寰佸笺蹇呮湁k涓嚎鎬ф棤鍏崇殑鐗瑰緛鍚戦噺锛屾垨鑰呰蹇呮湁绉﹔(位0E-A)=n-k锛屽叾涓璄涓哄崟浣嶇煩闃点傚绉扮煩闃垫ц川锛1銆佸浜庝换浣曟柟褰㈢煩闃礨锛孹+XT鏄绉扮煩闃点2...
绛旓細鏍规嵁瀹氫箟鍙婅浆缃繍绠楃殑鎬ц川鍙互濡傚浘璇佹槑杩欎袱涓煩闃靛垎鍒槸瀵圭О闃典笌鍙嶅绉闃点俕璇佹槑 锛氭寜瀹氫箟鏉ワ細锛1锛塧+a鐨則娆℃柟鏄闃电煩闃 瀵圭О闃电殑瀹氫箟涓猴細a^t=a 鏁卆+a^t=锛堬紙a+a^t锛塣shut锛塣t=(a+a^t)^t 鏁卆+a^t涓哄绉伴樀锛涳紙2锛塧鈥攁^t涓哄弽瀵圭О闃 鐢ㄥ畾涔夎瘉鏄 鍙嶅绉伴樀鐨勫畾涔変负:a^t=-a 锛...
绛旓細鏄剧劧涓嶅寘鎷紝鍥犱负鐭╅樀鍙嶅绉闃佃浆缃互鍚庯紝鍏冪礌鍙樺師鏉ョ殑璐熷硷紝褰撶劧涓嶆槸瀵圭О鐨勩
绛旓細銆愮瓟妗堛戯細鐢4-15棰樼煡瀹鍙嶅绉鐭╅樀A鐨勭壒寰佸间负0鎴栫函铏氭暟锛屾晠-1涓嶆槸A鐨勭壒寰佸硷紝鍗硘-E-A|=(-1)n|E+A|鈮0锛屼粠鑰屾湁|E+A|鈮0锛屾晠E+A涓哄彲閫嗙煩闃碉紝浜庢槸鐭(E+A)T=ET+AT=E-A浜︿负鍙嗙煩闃.$鍥犱负AT=-A锛屾墍浠BT=(E-A)(E+A)-1{(E-A)(E+A)-1}T=(E-A)(E+A)-1[(E+A)...
