Connection between rank and positive definiteness
I would like to know, is there a connection between the rank of a matrix and whether it is positive definite? Specifically, if I can prove that a matrix is not full rank, then can I say that it is not positive definite? If so, why?
Thanks a lot for your help.
Solution 1:
Another fact that wasn't pointed out is that since the determinant is a product of the eigenvalues and both PD and ND matrices have either all strictly positive or all strictly negative eigenvalues you can deduce the determinant is non zero so the matrix is invertible (i.e. has full rank).