Intuition for connectedness of positive definite matrices

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.

Solution 1:

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.

Solution 2:

A way is to take a positive definite matrix $M=PDP^{T}$ where $D$ is diagonal with all diagonal entries positive and $P$ is an orthogonal matrix of eigenvectors (use the Spectral Theorem). Now, consider the path $\mathcal{M}(t)$ given by

$$\mathcal{M}(t)=P\Big(tI+(1-t)D\Big)P^{T}$$

We have that $\mathcal{M}(0)=M$, $\mathcal{M}(1)=I$ and $\mathcal{M}(t)$ is positive definite for all $t\in[0,1]$. It follows that there is a path connecting each positive definite matrix to the identity, so they form a path-connected (and hence connected) set.