# LMIs in Control/pages/Fundamentals of Matrix and LMIs/Notion of Matrix Positivity

## Notation of Positivity

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

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

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

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

negative definite, ${\displaystyle (-A>0)}$ .

indefinite if ${\displaystyle A}$  is neither positive semidefinite nor negative semidefinite.

## Properties of Positive Matricies

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