I am currently battling with a problem involving positive definite matrices and I would greatly appreciate some assistance.
Let $A$ be a positive definite, though not necessarily symmetric, matrix in $\mathbb{R}^{n \times n}$. I am trying to prove the inequality $a_{ii} \cdot a_{jj} > a_{ij} \cdot a_{ji}$ $\forall i \neq j$.
From the definition of positive definiteness, I understand that for all non-zero vectors $x$ in $\mathbb{R}^n$, we have $x^T A x > 0$.
I have attempted to use this property by considering $a_{ii} \cdot a_{jj} > a_{ij} \cdot a_{ji}$ $\iff$ $e_{i}^T A e_{i} \cdot e_{j}^T A e_{j}$ > $e_{i}^T A e_{j} \cdot e_{j}^T A e_{i}$ and aiming to simplify, but my efforts have been unsuccessful so far.
I would be grateful for any hints, insights, or proofs to guide me through this challenge.
Thank you in advance for your time and assistance.