Consider a matrix $P\in \mathbb{R}^{n\times n}$ such that at most $m<n$ elements of each column are non zero and $P$ is symmetric. I would like to find the sufficient condition(s) such that $P^{1}$ is positive definite. Given the positions of non zero elements in $P$ what are their values such that $P$ is positive definite.

1$\begingroup$ do you need $P^{1}$ to be sparse? Otherwise it is trivial... $\endgroup$ – Dima Pasechnik Mar 31 '16 at 20:06

$\begingroup$ I need specific elements of $P$ to be non zero, Jacobi Rotations on a positive definite diagonal matrix might work as user251257 said. $\endgroup$ – Cauchy Apr 1 '16 at 1:54
If $P$ is an invertible real symmetric matrix, $P^{1}$ is positive definite iff $P$ is positive definite. There are many equivalent conditions to positive definiteness.

$\begingroup$ That is true but not sufficient for the sparsity . So How do I generate a positive definite sparse symmetric matrix? $\endgroup$ – Cauchy Mar 31 '16 at 6:43

$\begingroup$ @Cauchy: Google is your friend. Apply random Jacobi Rotations on a positive definite diagonal matrix... $\endgroup$ – user251257 Mar 31 '16 at 14:55