Positive-Definite Matrix

Definition

$M\succ 0\iff \forall \mathbf{z}\ne 0,{\mathbf{z}}^{⊺}M\mathbf{z}>0$

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

Facts

Let $M\succ 0$, then

• $A^{-1}\succ 0$
• $\mathrm{rank}(CA{C}^{⊺})=\mathrm{rank}(C)$ where $C$ is a $p\times n$ matrix.
• $\mathrm{rank}(C)=p\Rightarrow CM{C}^{⊺}\succ 0$
• The diagonal elements of $M$ are positive.
• For a Symmetric Matrix $B$, $\exists t,M-tB\succ 0$ for sufficiently small $|t|$
• $\forall \mathbf{b},\underset{\mathbf{h}:\mathbf{h}\ne 0}{\sup }\frac{({\mathbf{h}}^{⊺}\mathbf{b}{)}^2}{{\mathbf{h}}^{⊺}M\mathbf{h}}={\mathbf{b}}^{⊺}{A}^{-1}\mathbf{b}$
• $\exists M^{1/2}\succ 0\text{s.t.}(M^{1/2}{)}^2=M$

$M\succ 0\iff \exists R\text{s.t.}A=R{R}^{⊺}$ where $R$ is a non-singular matrix.

$M\succ 0\iff$ all the leading minor determinants of $M$ are positive.

$\mathrm{rank}(X)=p\Rightarrow X^{⊺}X\succ 0$