关于矩阵A,B.那么关于AB=BA有何性质? 两个n阶矩阵A，B，那么AB与BA是什么关系
\u77e9\u9635A\uff0cB\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0bAB=BA \u6025\u77e9\u9635A\uff0cB\u5728\u7b80\u5355\u8ba1\u7b97\u4e00\u4e0b\u5373\u53ef\uff0c\u8be6\u60c5\u5982\u56fe\u6240\u793a
\u7b54\uff1a
\u4e24\u4e2an\u9636\u77e9\u9635A\uff0cB\uff0c
AB\u2260BA
|AB|=|BA|
\u2014\u2014\u7ebf\u4ee3\u516d\u7248p38
简单计算一下即可，答案如图所示
假如AB=BA,则A,B必定是同阶方阵.
(AB)^p=A^p*B^p，p∈N(AB)^p=AB*AB*......=A*BA*B*......=A*AB*B*......=AA*BB*......=AAA...AA*BBB...BB=A^p*B^p。
A=B.
绛旓細搴旇閫塁銆傚彲浠ユ湁涓ょ瑙ｉ噴鏂瑰紡銆倈AB|=|A|*|B|,鍥犳锛A锛孊涓嚦灏戞湁涓涓殑琛屽垪寮忎负0.AB=O,鍒檙(A)+r(B)<=n锛屾墍浠ヤ袱鑰呬腑蹇呮湁涓涓殑绉╁皬浜巒锛屽垯杩欎釜鐭╅樀鐨勮鍒楀紡涓0銆
绛旓細杩欐槸鐢变簬鐭╅樀鐗规畩鐨勮绠楁柟寮忓鑷寸殑锛岀煩闃电殑璁＄畻鏄竴琛屼笌涓鍒楀垎鍒浉涔樺啀鐩稿姞锛屾墍浠ュ綋AB=AC鏃朵笉鑳芥寜鏁扮殑璁＄畻绾﹀幓A锛濮戜笉鑳芥帹鍑
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝璇︽儏濡傚浘鎵绀
绛旓細AB=A-B <=> (I+A)(I-B)=I 浜庢槸(I+A)鍜(I-B)閮藉彲閫嗭紝(I-B)(I+A)=I 灞曞紑寰桞A=A-B锛鍗虫湁缁撹銆傛ゼ涓婄殑鍋氭硶渚濊禆浜嶢鍙嗭紝纰板埌A=B=0杩欑灏变笉琛屻
绛旓細璁A,B鍒嗗埆鏄痬*n鍜宯*m鐭╅樀锛屽垯AB鏄痬绾ф柟闃碉紝BA鏄痭绾ф柟闃点傛墍浠=n銆
绛旓細琛屽垪寮忕浉绛夛紝鐭╅樀涓嶄竴瀹氱浉绛夛細姣斿A = [1,0 ; 0,1] B=[1,1 ; 0,1] 鐭╅樀 A,B 涓嶇瓑锛屼絾锛殀A|=|B|=1 鐢憋細|AB|=|A||B|=|B||A|=|BA| 涓嶅彲浠ユ帹鍑猴細AB=BA .
绛旓細涓嶅繀瑕佸姞A=B銆備簨瀹炰笂锛岃嫢A锛孊閮戒负瀵圭О鐭╅樀銆傚垯 (AB)T=BTAT=BA 鍥犱负AB鏄绉扮煩闃碉紝鎵浠(AB)T=AB 鎵浠AB=BA 鍙嶄箣锛岃嫢AB=BA 鍒(AB)T=(BA)T AB=ATBT 鏁匒=AT锛孊=BT 涓や釜瀵圭О鐭╅樀鐨勭Н鏄绉扮煩闃碉紝褰撲笖浠呭綋涓よ呯殑涔樻硶鍙氦鎹備袱涓疄瀵圭О鐭╅樀涔樻硶鍙氦鎹㈠綋涓斾粎褰撲袱鑰呯殑鐗瑰緛绌洪棿鐩稿悓銆
绛旓細璇侊細棣栧厛鐢AB=A+B寰楋細AB-A-B+E=E 鍒(A-E)(B-E)=E,浠庤孉-E鍙 鍐嶇敱(A-E)(B-E)=E=(B-E)(A-E),鐭B=BA 鍦ㄧ嚎鎬т唬鏁板拰鐭╅樀璁轰腑锛屾湁涓や釜m脳n闃鐭╅樀A鍜B锛濡傛灉杩欎袱涓煩闃垫弧瓒矪=QAP锛圥鏄痭脳n闃跺彲閫嗙煩闃碉紝Q鏄痬脳m闃跺彲閫嗙煩闃碉級锛岄偅涔杩欎袱涓煩闃典箣闂存槸绛変环鍏崇郴銆備篃灏辨槸璇达紝...
绛旓細B=inv(A-2*eye(size(A))*A B = 3.0000 -8.0000 -6.0000 2.0000 -9.0000 -6.0000 -2.0000 12.0000 9.0000 AB=A+2B AB-2B=A (A-2E)B=A B=inv(A-2E)*A 鐭╅樀 鏄珮绛変唬鏁板涓殑甯歌宸ュ叿锛屼篃甯歌浜庣粺璁″垎鏋愮瓑搴旂敤鏁板瀛︾涓傚湪鐗╃悊瀛︿腑锛岀煩闃浜庣數璺銆佸姏瀛︺佸厜瀛﹀拰閲忓瓙...
绛旓細浠讳綍涓涓煩闃典笌鍗曚綅鐭╅樀鐨勪箻绉弧瓒充氦鎹㈠緥锛屽嵆A路I = I路A锛鍏朵腑I琛ㄧず鍗曚綅鐭╅樀銆傚瑙掔煩闃电殑浜ゆ崲锛氬鏋滀袱涓瑙掔煩闃电殑鍏冪礌婊¤冻浜ゆ崲鍏崇郴锛屽垯瀹冧滑鐨勪箻绉篃婊¤冻浜ゆ崲寰嬨備緥濡傦紝鑻鍜孊閮芥槸瀵硅鐭╅樀锛屼笖A鐨勫瑙掑厓绱犳寜鐓у崌搴忔帓鍒锛孊鐨勫瑙掑厓绱犳寜鐓ч檷搴忔帓鍒楋紝鍒AB = BA銆備氦鎹㈠瓙锛氳嫢涓や釜鐭╅樀A鍜孊鐨勪氦鎹㈠瓙[A锛...
