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?
Best Answer
I don't think $x_1$ must be positive.
A counter example might be a positive definite matrix $A = [1 \space -0.2 ; \space -0.2 \space 1]$ with its inverse matrix $A^{-1}$ having $A_{11}, A_{12} > 0$.
- Edit: Sorry. A counter example might be a normalized covariance matrix
$ A= \left( \begin{array}{ccc} 1 & 0.6292 & 0.6747 & 0.7208 \\ 0.6292 & 1 & 0.3914 & 0.0315 \\ 0.6747 & 0.3914 & 1 & 0.6387 \\ 0.7208 & 0.0315 & 0.6387 & 1 \end{array} \right) $.