Let [tex]K = A^T C A[/tex], where C>0. Prove that kerK=cokerK=kerA, and rngK = corngK = corng A.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Ker Positive Definite Matrix

**Physics Forums | Science Articles, Homework Help, Discussion**