I have a linear system $Ax=b$, where
- $A$ is symmetric, positive semidefinite, and positive. $A$ is a variance-covariance matrix.
- vector $b$ has elements $b_1>0$ and the rest $b_i<0$, for all $i \in \{2, \dots, N\}$.
Prove that the first component of the solution is positive, i.e., $x_1>0$.
Does anybody have any idea?