设A是n阶方阵,Eij表示第(i,j)元素是1，其余元素是0的基础矩阵，则EijAEkj= eij表示第i个元素是1,第j个元素是-1,其余元素都是零的...
\u8bf7\u6559\u6570\u5b66\u9ad8\u624b\uff1aEij\u4e3a\uff08i\uff0cj\uff09-\u5143\u7d20\u4e3a1\uff0c\u800c\u5176\u4f59\u5143\u7d20\u90fd\u662f\u96f6\u7684n\u9636\u65b9\u9635\uff0c\u662f\u4ec0\u4e48\u610f\u601d\uff1f\u4f8b\u5982\uff1a1 0 0
0 0 0
0 0 0
\u5c31\u662fE11
\u4f8b\u5982\uff1a0 0 0
1 0 0
0 0 0
\u5c31\u662fE21
\u90a3\u4e48i\uff0cj\uff0cn\u662f\u5b9a\u503c\u5427\uff1f\uff1f\uff08\u5f53\u7136i\uff0cj\u4e5f\u53ef\u4ee5\u662f\u4e00\u7ec4\u6570\uff09\u3002(n\u7ef4\u662fn*n\u8fd8\u662fn*1\uff1f\uff09
\u5047\u5982i\uff0cj\u662f\u4e00\u4e2a\u6570\u3002n\u7ef4\u662fn*1\u5427\uff01
\u7a0b\u5e8f\uff1a
eij=zeros(n*1\uff09\uff1b
i= \uff1bj= \uff1b
eij\uff08i\uff09=1\uff1beij\uff08j\uff09=-1\uff1b\u5c31\u884c\u4e86
EijA表示第i行和A的第j行相同，但其他行的元素均为0的矩阵。
AEkj表示第j列和A的第k列相同，但其他列的元素均为0的矩阵。
那么EijAEkj，就是表示第i行第j列的元素为ajk其他元素为0的矩阵。其中ajk为A的第j行第k列的元素。
故EijAEkj=ajkEij
绛旓細閭ｄ箞EijAEkj锛灏辨槸琛ㄧず绗琲琛岀j鍒楃殑鍏冪礌涓篴jk鍏朵粬鍏冪礌涓0鐨勭煩闃銆傚叾涓璦jk涓篈鐨勭j琛岀k鍒楃殑鍏冪礌銆傛晠EijAEkj=ajkEij
绛旓細瑙ｇ瓟锛氳瘉鏄庯細锛1锛変护锛Eij琛ㄧず鍗曚綅闃典腑鐨勭i琛屽拰绗琷琛屽鎹紝鍒欑敱棰樻剰B=EijA锛岃孍ij鏄垵绛夌煩闃碉紝鏄彲閫嗙殑锛屽張A鏄鍙嗙殑锛屾牴鎹嗙煩闃电殑涔樼Н渚濈劧鏄彲閫嗙殑锛屽緱锛欱=AEij鍙嗭紟锛2锛夆埖B=EijA锛屸埓B-1=锛圗ijA锛-1=A-1?Eij-1=A-1Eij锛岋紙Eij鐨勯嗙煩闃典緷鐒朵负鏈韩锛変粠鑰岋細AB-1=A?A-1Eij=Eij锛
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绛旓細璁 Eij 涓 绗琲琛岀j鍒椾綅缃槸1鍏朵綑閮芥槸0鐨n闃舵柟闃 鍒 n闃跺叏浣撳绉扮煩闃垫墍鎴愮殑绾挎х┖闂寸殑涓缁勫熀涓:{ Eij,i,j = 1,2,...,n,i
绛旓細璁綞ij鏄i琛岀j鍒椾綅缃负1,鍏朵綑閮芥槸0鐨刵闃舵柟闃.鍒橢ijA = AEij EijA 鏄 绗琲琛屼负 aj1,aj2,...,ajn, 鍏朵綑琛岄兘鏄0鐨勬柟闃 AEij 鏄 绗琷鍒椾负 a1i,a2i,...,ani, 鍏朵綑鍒楅兘鏄0鐨勬柟闃 鎵浠ュ綋i鈮爅鏃, aij=0.鎵浠鏄竴涓瑙掔煩闃.璁綞(i,j)鏄鎹,j涓よ鐨勫垵绛夌煩闃.鐢盓(i,j)A=AE...
绛旓細璁綞ij鏄绗i琛岀j鍒椾綅缃负1,鍏朵綑閮芥槸0鐨n闃舵柟闃.鍒橢ijA = AEijEijA 鏄i琛屼负 aj1,aj2,...,ajn, 鍏朵綑琛岄兘鏄0鐨勬柟闃礎Eij 鏄j鍒椾负 a1i,a2i,...,ani, 鍏朵綑鍒楅兘鏄0鐨勬柟闃垫墍浠ュ綋i鈮爅鏃, aij=0.鎵浠A鏄涓涓瑙掔煩闃.璁綞(i,j)鏄鎹,j涓よ鐨勫垵绛夌煩闃.鐢盓(i,j)A=AE(i,j)鍙緱aii=...
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