Positive Semi-Definite Matrix

Definition

$M⪰0\iff \forall \mathbf{z}\ne 0,{\mathbf{z}}^{⊺}M\mathbf{z}\ge 0$

Matrix $M$, in which ${\mathbf{z}}^{⊺}M\mathbf{z}$ is positive or zero for every non-zero column vector $\mathbf{z}$ is a positive semi-definite matrix

Facts

Let $M⪰0$, then

• $\mathrm{tr}(A)\ge 0$
• $\mathrm{rank}(A)=r\iff \exists R_{n\times n},\mathrm{rank}(R)=r,A=R{R}^{⊺}$
• $\mathrm{rank}(A)=r\Rightarrow \exists S_{n\times r},\mathrm{rank}(s)=r,S^{⊺}AS=I_r$
• $X^{⊺}AX=O\Rightarrow AX=O$ where $X$ is $n\times p$ matrix.