# LMIs in Control/Tools/Basic Matrix Theory

## Basic Matrix Notation

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

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

Transpose of a Matrix

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

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

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

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

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

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

## Important Properties of Matricies

A square matrix ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ is called Hermitian or self-adjoint if ${\displaystyle A=A^{*}}$.
If ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ is Hermitian then it is called symmetric.
A square matrix ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ is called unitary if ${\displaystyle A^{*}=A^{-1}}$ or ${\displaystyle A^{*}A=I}$.