positive semidefinite matrix for 3x3 case

Question

Given a symmetric matrix $$ A = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_4 & x_5 \\ x_3 & x_5 & x_6\end{bmatrix} $$ we want to find the conditions on these 6 variables that make the matrix positive semidefinite. Now, a part of the explanation given is as follows: For any given vector $[z_1 \ z_2 \ z_3]^T$ we want $z^TAz \geq 0$. Then it states: if $x_1=0$ then we must have $x_2=x_3=0$ and matrix A is positive semidefinite only if $\begin{bmatrix}x_4 & x_5 \\ x_5 & x_6\end{bmatrix}$ is positive semidefinite. I don't understand any of this reasoning.

Answer 1

Consider $z = (t, 1, 0)$. Then $z^{T}Az \geq 0$ is equivalent to $x_{1}t^{2} + 2x_{2}t + x_{3} \geq 0$, for all $t\in \mathbb{R}$. Now if $x_{1} =0$, we have $2x_{2}t + x_{4} \geq 0$ for all $t$, which gives $x_{2} = 0$. We can get $x_{3}=0$ similarly using $z = (t, 0, 1)$.