I have read in the post, but there is something missing I'd like to know. I do believe it's not a duplicate post. I have a specified question, probably a trivial one.
First, let $(s_n)_{n\geq 0}$ be a sequence of real numbers, and define the $n$th Hankel matrix $H_n=(s_{i+j})_{0\leq i,j\leq n}$.
If it's assumed that $\det H_n>0$ for all $n\geq 0$, then by Sylvester's Criterion [note: $H_n$ is Hermitian as it's symmetric], we get that $H_n$ is positive definite. Does it then imply that $H_n$ is positive semi-definite as well?
TL;DR: Is a real positive definite matrix always a positive semi-definite matrix?
I would say yes, but I want someone to confirm it for me.
Best Answer
Yes.
If $x^TAx > 0$ for all $x \in \mathbb{R}^n \setminus \{\mathbf{0}\}$, then $x^TAx \geq 0$ for all $x \in \mathbb{R}^n \setminus \{\mathbf{0}\}.$
As we always have $\mathbf{0}^TA\mathbf{0} = 0$, it also follows that $x^TAx \geq 0$ for all $x \in \mathbb{R}^n.$