LMIs in Control/Tools/Basic Matrix Theory

Basic Matrix Notation

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

$A={\begin{bmatrix}a_{11}&\dots &a_{1m}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{n\times m}$

Transpose of a Matrix

The transpose of $A$, denoted as $A^{T}$ or $A'$ is:

$A^{T}={\begin{bmatrix}a_{11}&\dots &a_{n1}\\\vdots &\ddots &\vdots \\a_{1m}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

Adjoint of a Matrix

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

$A^{*}={\begin{bmatrix}a_{11}^{*}&\dots &a_{n1}^{*}\\\vdots &\ddots &\vdots \\a_{1m}^{*}&\dots &a_{nm}^{*}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$

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

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

Important Properties of Matricies

Hermitian, Self-Adjoint, and Symmetric Matricies

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

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

Unitary Matricies

A square matrix $A\in \mathbb {C} ^{n\times n}$ 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.