I struggle with the following proof from linear algebra:
Prove: [$A \in $ $ \mathbb{M}_{n \times n}(\mathbb{R}) $ is positive definite] $\implies$ $[\forall i \in \{1, ..., n\}: a_{i,i} > 0$]
Tried direct algebraic manipulations from $x^TAx$, but I did not derive any conclusions.
Any help/advice/solution very, very, very appreciated!!!
I'll post an answer so the question can be marked as answered. As discussed in the comments: if $x$ is the $i$th standard basis vector, then $$ x^T A x = a_{i,i} > 0. $$
(Here $a_{i,i}$ is the $i$th diagonal entry of $A$.)