LMIs in Control/pages/Notion of Matrix Positivity

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Notation of Positivity[edit]

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 Matricies[edit]

  • 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 Links[edit]