Okay, so this hopefully is an easy question, but I'm not that much into linear algebra. Could someone help me realize the following:
A is symmetric, positive definite $n\times n$, x is $n\times 1$ and non-negative (if we can relax that assumption it would be great), $\iota\text{ is }n\times 1$ and only consists of 1's.
I wan't $B=[x\;\iota]^{T}A^{-1}[x\;\iota]=\left( \begin{array}{cc} x^TA^{-1}x & x^TA^{-1}\iota\\ \iota^T A^{-1}x & \iota^T A^{-1}\iota \\ \end{array} \right)$ to be positive definite. How can I show that?
Best regards, Henrik