LMIs in Control/Tools/Basic Matrix Theory

Basic Matrix Notation

Consider the complex matrix $A\in {ℂ}^{n\times m}$.

$A=\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1m}\\ ⋮& \ddots & ⋮\\ a_{n1}& \dots & a_{nm}\end{array}\right]\in {ℂ}^{n\times m}$

Transpose of a Matrix

The transpose of $A$, denoted as $A^T$ or $A^{\prime }$ is:

$A^T=\left[\begin{array}{ccc}{a}_{11}& \dots & a_{n1}\\ ⋮& \ddots & ⋮\\ a_{1m}& \dots & a_{nm}\end{array}\right]\in {ℂ}^{m\times n}.$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $A$, denoted as $A^{*}$ is:

$A^{*}=\left[\begin{array}{ccc}{a}_{11}^{*}& \dots & a_{n1}^{*}\\ ⋮& \ddots & ⋮\\ a_{1m}^{*}& \dots & a_{nm}^{*}\end{array}\right]\in {ℂ}^{m\times n}.$

Where $a_{nm}^{*}$ is the complex conjugate of matrix element $a_{nm}$.

Notice that for a real matrix $A\in {ℝ}^{n\times m}$, $A^{*}=A^T$.

Important Properties of Matricies

Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $A\in {ℂ}^{n\times n}$ is called Hermitian or self-adjoint if $A=A^{*}$.

If $A\in {ℝ}^{n\times n}$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $A\in {ℂ}^{n\times n}$ is called unitary if $A^{*}=A^{-1}$ or $A^{*}A=I$.

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.