| [ QuizWit ] in KIDS 글 쓴 이(By): iLUSiON (ivenomouth) 날 짜 (Date): 1998년 5월 4일 월요일 오전 08시 19분 37초 제 목(Title): [cap3] linear algebra 3 [ QuizWit ] in KIDS 글 쓴 이(By): guest (wiking) 날 짜 (Date): 1998년 5월 4일 월요일 오전 07시 54분 47초 제 목(Title): Re: linear algebra 3. Known Fact: A=(Aij = xi^(j-1)) where xi's are different is non-singular. Claim: rank(H) = rank(H2) where H2 = (1/(i+j)) Now I want to show H2 is positive-definite. First we observe <e|H2|e> = int[a in [0,1]](a*(sum(ei*a^(i-1)))^2). Using Known Fact, we know the intergral should be positive for all non zero e. As a result, rank(H) = n. [iLUSiON] hmmm... i am not sure if your integrand is correct. What happen to the index j A much nicer proof would be by noticing H2 is a matrix representation of an inner product <v,v> where the inner product is defined over a vector space spanned by polynomials on [0,1] then each x^i , i=0...n forms a basis and with L2 norm, <v,v> > 0 Then rank(H) = rank (H2) follows from Holder's inequality of L2 norm. (which is basically Cauchy-Schwartz inequality) 1010101010101010101010101000000010101010010101010101000000010001111110101001010 0101010011111010101000100010110101010101010101001010101010101010101111111101010 1011111100010101010101010101101010101011111101010111101010001110101010001010110 http://www.math.mcgill.ca/~chung |