TITLE: Intuition for connectedness of positive definite matrices
QUESTION [10 upvotes]: I have read, even on this website, about connectedness of positive definite matrices. The proof is given in the form of convex equation. I wonder how to understand it intutively. e.g., as an analogy, a parabola is convex and continuous, so, its connectedness is obvious. How to understand positive definite matrices on this line. At first thought, I felt, in the vecor space, the next placed matrix might not be positive definite, as we need all leading principal minors to be positive. I felt, maybe this condition would be satisfied only after certain distance in the vector space. So, I felt these matrices would be disconnected. But of course I was wrong. I wonder how to build intuition for connectedness along these lines. Thanks.

REPLY [7 votes]: The most straightforward intuition comes from the fact that the set $P$ of
positive definite matrices is convex and convex sets are path connected by
straight lines.
If $\langle x, A_k x \rangle > 0$ for all $x \neq 0$, then for $\lambda \in [0,1]$, we have
$\langle x, (\lambda A_1 +(1-\lambda) A_2 )x \rangle = \lambda \langle x, A_1 x \rangle + (1-\lambda) \langle x, A_2 x \rangle > 0$, hence
$\lambda A_1 +(1-\lambda) A_2$ is positive definite.