1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ker Positive Definite Matrix

  1. Mar 13, 2007 #1
    Let [tex]K = A^T C A[/tex], where C>0. Prove that kerK=cokerK=kerA, and rngK = corngK = corng A.

    I sort of need a kickstart to get going. I know that each element will be [tex]K_{ij} = v_i^T * C * v_j[/tex], so this is sort of like a Gram matrix, which in turn also means that the matrix is semi-positive definite. I am not quite what sure to do with the A matrix though. Clearly, if the columns of A are linearly independent, then the range of A has dimension m if A is an m x n matrix, and so the kernel for A will have a dimension of n-m. From here, I think I will have to show that the dimension of kerK (will be 0 is C is positive definite), is the same as kerA, and then show that the bases are linearly dependent.
     
  2. jcsd
  3. Mar 13, 2007 #2
    How does this look?

    If kerA is all A such that [tex]Ax=0[/tex] then kerK will be [tex]Kx = A^T C A x = 0[/tex] and thus shows that ker A is contained in ker K. Then taking [tex]Kx = 0[/tex] then [tex] 0 = x^T K x = x^T A^T C A x = y^T C y[/tex] where [tex]y = A x[/tex]. Since C>0 then y = 0, and [tex] x \in ker A[/tex].

    Now do the same thing for K transpose. [tex]0 = K^T x = (A^T C A)^T x = A^T C^T A x = y^T C^T y [/tex] where [tex]y = Ax[/tex]. If C>0 then C transpose must also be greater than the zero vector, and the portion that matters is the 'y' vector, which is the same as the previous one. This shows that cokerK = kerA, and in turn cokerK = kerK = kerA.
     
  4. Mar 13, 2007 #3
    Hmm... now for range, should I use fund thm of lin alg to show rngA and rngK have the same dimension, and then I would have to show their bases are linearly dependent, unless there is something tricky to do with the rank.

    *Also, the matrix K would not be sort of like a Gram matrix, it is a Gram matrix.
     
    Last edited: Mar 14, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Ker Positive Definite Matrix
Loading...