3
$\begingroup$

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?

  • 0
    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.2012-07-03

2 Answers 2