Notation of PositivityEdit
A symmetric matrix is defined to be:
positive semidefinite, , if for all .
positive definite, , if for all .
negative semidefinite, .
negative definite, .
indefinite if is neither positive semidefinite nor negative semidefinite.
Properties of Positive MatriciesEdit
- For any matrix , .
- Positive definite matricies are invertible and the inverse is also positive definite.
- A positive definite matrix has a square root, , such that .
- For a positive definite matrix and invertible , .
- If and , then .
- If then for any scalar .
- 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.