# LMIs in Control/Tools/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 ${\displaystyle \mu >0}$.