pos def matrix with diagonal=1 -> to lower dimensional pos def matrix

Hello!

I have a nxn symmetric pos def matrix and the diagonal elements are all equal 1.

Is there a possibility to construct a n-1xn-1 matrix pos def matrix which contains all the entries from the nxn matrix except the diagonal 1 element?

I thought about constructing the n-1xn-1 matrix by deleting the diagonals and the the first upper diagonal of the nxn matrix, but the resulting matrices are not necessarily pos def...

Has anybody an idea? Or is it feasible at all?

Thank you!

Alex