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 $x^{T}Ax\ge 0$ for all $x\in {ℝ}^n,x\ne \mathrm{𝟎}$.

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

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 $M$, $M^{T}M>0$.
• 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 $\mu >0$.

External Links

• 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.