# optimization with eigenvector composition

as the figure shows, I would like to minimize v, and also to solv the corresponding x vector. The solution has shown in the figure. I cannot solve x vector on my own. Help, with thanks...

• doesn't this text tell you what to do to find $x$ ? (in the sentence just above equation 5). – Carlo Beenakker Aug 10 at 11:47
• @CarloBeenakker It tells the solution, But the way to find the solution. – Ubo Chow Aug 10 at 16:14
• if $\Gamma$ is positive definite, equation (4) is equivalent to ${\rm det}\,(\lambda I- M)=0$ with $M$ from equation (5), so the $\lambda$ your are looking for is the smallest eigenvalue of $M$. If $z$ is the corresponding eigenvector you minimise $\nu(x)=\lambda$ for $x=\Gamma^{-1/2}z$. – Carlo Beenakker Aug 10 at 16:39
• emmmmmm... actually I shall understand M is equivalent to what is derived from equation (5). But I cannot deal with the very step where the corresponding eigenvector, that is x, is equal to Gamma with the power of -1/2 multiplied M... (\Gamma^(-1/2) times M) – Ubo Chow Aug 11 at 1:30
• the corresponding eigenvector is $z$, not $x$. – Carlo Beenakker Aug 11 at 9:22