Linear matrix inequalities and control theory/pages/Notion of Matrix Positivity

Notation of Positivity
A symmetric matrix $$A\in\R^{n\times n}$$ is defined to be:

positive semidefinite, $$(A\ge 0)$$, if $$x^TAx\ge 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

positive definite, $$(A>0)$$, if $$x^TAx> 0 $$ for all $$x\in\R^n, x\neq \mathbf{0} $$.

negative semidefinite, $$(-A\ge 0)$$.

negative definite, $$(-A>0)$$.

indefinite if $$A$$ is neither positive semidefinite nor negative semidefinite.

Properties of Positive Matricies

 * For any matrix $$M$$, $$M^TM>0$$.
 * Positive definite matricies are invertible and the inverse is also positive definite.
 * A positive definite matrix $$A>0$$ has a square root, $$A^{1/2}>0$$, such that $$A^{1/2}A^{1/2}=A$$.
 * For a positive definite matrix $$A>0$$ and invertible $$M$$, $$M^TAM>0$$.
 * If $$A>0$$ and $$M>0$$, then $$A+M>0$$.
 * If $$A>0$$ then $$\mu A>0$$ for any scalar $$\mu>0$$.