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 $A^{*}A=I$ .