| [ QuizWit ] in KIDS 글 쓴 이(By): guest (wiking) 날 짜 (Date): 1998년 5월 3일 일요일 오후 04시 38분 56초 제 목(Title): Re: linear algebra 2. Here is my version. Given condition. [AB, C] = 0 & [A, C] = [B, C] (= X). Using [AB, C] = A[B, C] + [A, C]B = 0. We have AX = -XB. Now if Kernel(X) does not span the space, Then X'X should have non-zero eigen values while all eigen vectors can be orthonomalized. Let e be one of non-zero eigen-valued eigen vectors (eigenvalue = a). Then <e|X'AX|e> = -a<e|B|e>, which is contradictory to positive definiteness of A & B. Therefore Kernel(X) spans the whole space. PS> The above solution by another guest wronly assumed that X'AX is symmetric. |