反对称行列式和对称行列式是什么 对称行列式与反对称行列式是怎么样的
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绛旓細瀵圭О琛屽垪寮忔槸琛屽垪寮忓=鍏惰浆缃鍒楀紡鍊硷紝鍙嶅绉拌鍒楀紡鏄鍒楀紡鍊=鍏惰浆缃鍒楀紡鍊肩殑璐熸暟銆傚洜涓哄鏋滅敤鍖栦笁瑙掑舰鐨勬柟娉曟潵瑙ｅ喅鐨勮瘽锛屽氨娑夊強鍒扮粰鏌愯鍑忓幓涓涓嬩竴琛岀殑锛4-位锛夊垎涔嬪嚑鐨勫嶆暟锛屾鏃朵綘涓嶇煡閬撐绘槸鍚︼紳4锛屾墍浠ヨ繖绉嶅彉鎹㈡槸涓嶅鐨勶紝涓鑸兘鏄妸鏌愪竴鍒楁垨鑰呰鍒掓帀2椤癸紝鍓╀笅涓椤逛笉涓0鐨勪笖鍚荤殑椤癸紝灏嗚...
绛旓細鏍规嵁鍙嶅绉鐭╅樀鐨勬ц川鏈: AT=-A, |A|=|AT|=|-A|=(-1)n|A| 濡傛灉n涓哄鏁, 鍒檤A|=0.濡傛灉n涓哄伓鏁,寰椾笉鍒琛屽垪寮鐨勫叿浣撳笺
绛旓細鍙嶅绉拌鍒楀紡鏄疉涓簄缁存柟闃碉紝鑻ユ湁A'=-A锛屽垯绉扮煩闃礎涓哄弽瀵圭О銆1銆佽鍒楀紡鏄竴涓氦鏇垮閲嶇嚎鎬у舰寮忥紝鑰屾垜浠氬父鐞嗚В鐨勬寮忕┖闂翠腑鐨勪綋绉篃鏄繖鏍蜂竴涓嚱鏁般傚崟浣嶇珛鏂逛綋浣撶Н涓1锛屾部鏌愭潯杈规墿澶鍊嶄綋绉氨鎵╁ぇc鍊嶏紝浜ゆ崲涓ゆ潯杈逛互鍚庝綋绉弽鍙枫2銆乶闃惰鍒楀紡鐨勬湰璐ㄦ槸n缁村悜閲忕┖闂碏n涓婄殑瑙勮寖n閲嶅弽瀵圭О绾挎у嚱鏁般侺apla...
绛旓細鍙嶅绉拌鍒楀紡鐨勫畾涔夋槸绫讳技鐨勶紝涔熸槸瀵瑰簲浣嶇疆涓婄殑鍏冪礌浜掍负鐩稿弽鏁銆備富瀵硅绾夸笂鐨勫厓绱犱负0銆傚浜庡弽瀵圭О鐭╅樀锛屽畠鐨勪富瀵硅绾夸笂鐨勫厓绱犲叏涓洪浂锛岃屼綅浜庝富瀵硅绾夸袱渚у绉扮殑鍏冨弽鍙枫傚弽瀵圭О鐭╅樀鍏锋湁寰堝鑹ソ鐨勬ц川锛屽鑻涓哄弽瀵圭О鐭╅樀锛屽垯A'锛屛籄鍧囦负鍙嶅绉扮煩闃碉紱鑻,B鍧囦负鍙嶅绉扮煩闃碉紝鍒橝卤B涔熶负鍙嶅绉扮煩闃点...
绛旓細鍙嶅绉拌鍒楀紡灏辨槸涓诲瑙掑厓閮芥槸0锛屽叾瀹冨叧浜庝富瀵硅绾垮绉颁綅缃殑鍏冪礌绗﹀彿鐩稿弽銆備笅鍥炬柟娉曞彲浠ヨ瘉鏄庡鏁伴樁鍙嶅绉拌鍒楀紡閮界瓑浜0銆
绛旓細2. 鍏ㄩ浂琛屽垪寮忥細琛屽垪寮涓墍鏈夊厓绱犲潎涓洪浂锛岃鍒楀紡鐨勫间负0銆3. 鍗曚綅琛屽垪寮忥細琛屾暟绛変簬鍒楁暟锛屽瑙掔嚎涓婄殑鍏冪礌閮戒负1锛屽叾浠栧厓绱犲潎涓洪浂锛岃鍒楀紡鐨勫间负1銆4. 鐭╅樀琛屽垪寮忥細灏嗙煩闃佃浆鍖栦负琛屽垪寮忕殑褰㈠紡锛岃琛屽垪寮忎负鐭╅樀琛屽垪寮忥紝鍏跺间负鐭╅樀鐨勮鍒楀紡鍊笺5. 鐗瑰緛鍊艰鍒楀紡锛歯闃剁煩闃礎鐨勮鍒楀紡涓哄叾鐗瑰緛鍊肩殑涔樼Н锛屽嵆|A...
绛旓細棣栧厛锛鍙嶅绉拌鍒楀紡鏄鎸囦竴涓煩闃礎婊¤冻A=-A'锛岃繖閲岀殑A'鏄疉鐨勮浆缃傚湪浜旈樁鏂归樀A鐨勬渚嬩腑锛屾垜浠洿鎺ュ簲鐢ㄥ鏁伴樁鍙嶅绉拌鍒楀紡鐨勬ц川锛歞et[-A']=(-1)^5*det[A']=-det[A]锛屽洜姝わ紝琛屽垪寮忕殑鍊煎繀鐒朵负闆躲傜劧鑰岋紝杩欎釜瀹氱悊骞堕潪鍑┖鑰屾潵锛屽畠鐨勫熀纭鏄袱涓紩鐞嗐傚紩鐞嗕竴鍛婅瘔鎴戜滑锛岃鍒楀紡鐨勮浆缃瓑浜庡叾鏈韩锛屽嵆...
绛旓細鐢变簬濂囨暟闃鍙嶅绉鐭╅樀鐨琛屽垪寮涓0, 鑰寍A|=1 鏁卬涓哄伓鏁.鎵浠ュ湪琛屽垪寮弢A|涓湁 Aij = (-1)^(n-1)Aji = -Aji.灏嗚鍒楀紡瀹屽叏鍒嗘媶涓2^n涓鍒楀紡涔嬪拰 娉ㄦ剰鍒拌嫢鏈変袱鍒楀叏涓1, 鍒欒鍒楀紡涓0 瀵规煇鍒楀叏涓1鐨勮鍒楀紡鎸夋鍒楀睍寮, 琛屽垪寮忕瓑浜庢鍒楀厓绱犵殑浠ｆ暟浣欏瓙寮忎箣鍜 鎵浠 D = |A| + 鈭慉ij =...
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绛旓細琛屽垪寮忎腑鑻ュ叧浜庝富瀵硅绾垮绉扮殑鍏冪礌浠呯鍙风浉鍙嶏紝鍗砤ij=-aji锛屽垯琛屽垪寮忓彨鍋氭枩瀵圭О琛屽垪寮銆傚浜巒闃舵枩瀵圭О琛屽垪寮廳鏈夊叧绯诲紡d=(涓1)nd锛屼粠鑰屽彲鐭ュ鏁伴樁鐨勬枩瀵圭О琛屽垪寮忔绘槸绛変簬0銆傛枩瀵圭О琛屽垪寮(skew-symmetric determinant)鏄被浼间簬鏂滃绉扮煩闃电殑涓绉嶇壒娈婅鍒楀紡銆備竴涓鍒楀紡锛屽鏋滃叾涓诲瑙掔嚎涓婄殑鍏冪礌鍏ㄤ负闆讹紝鑰...
