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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?

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Thanks. What about if A is positive definite? Is in that case first component positive? I can also "normalize" diagonal elements to be equal to 1 since they are variances, and rest of elements to be in interval [0,1], which would then be correlation coefficients. – Branka Jul 3 '12 at 11:01

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) $.

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IT seems that A is not only restricted to be positive definite, but also must have positive elements. – leonbloy Jul 5 '12 at 11:53
Thanks. Fixed it. – tatterdemalion Jul 5 '12 at 12:36

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