LMIs in Control/pages/Notion of Matrix Positivity

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  .


External LinksEdit