# LMIs in Control/pages/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:

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

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

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