LMIs in Control/pages/Notion of Matrix Positivity

Notation of Positivity

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

positive semidefinite, $(A\geq 0)$ , if $x^{T}Ax\geq 0$ for all $x\in \mathbb {R} ^{n},x\neq \mathbf {0}$ .

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

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

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

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

For any matrix $M$ , $M^{T}M>0$ .
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 $M$ , $M^{T}AM>0$ .
If $A>0$ and $M>0$ , then $A+M>0$ .
If $A>0$ then $\mu A>0$ for any scalar $\mu >0$ .

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.