LMIs in Control/Tools/Notion of Matrix Positivity

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A symmetric matrix $A \in ℝ^{n \times n}$ is defined to be:

positive semidefinite, $(A \geq 0)$ , if $x^{T} A x \geq 0$ for all $x \in ℝ^{n}, x \neq 𝟎$ .

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

negative semidefinite, $(- A \geq 0)$ .

negative definite, $(- A > 0)$ .

indefinite if $A$ is neither positive semidefinite nor negative semidefinite.

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 $M$ , $M^{T} A M > 0$ .
If $A > 0$ and $M > 0$ , then $A + M > 0$ .
If $A > 0$ then $μ A > 0$ for any scalar $μ > 0$ .

WIP, additional references to be added

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.