0.1. Common Notations Rules

Matrix and Vector Notations §

Entity	Represented by	Example
Set	Uppercase Letter	$S$
Matrix	Uppercase bold letter	$\text{A}$
Vector	Lowercase bold letter	$\mathbf{x}$
Scalar	Lowercase symbol	$\alpha$
Element of Vector	Lowercase letter with a subscript	$x_1$
Element of Matrix	Lowercase letter with subscript	$a_{23}$
Column of a Matrix	Lowercase bold letter with subscript	${\mathbf{a}}_1$
Row of a Matrix	Lowercase bold letter (with prime) with subscript (and transposed)	${\mathbf{a}}_1^{{\prime }^{\top }}$

All vectors are represented as column vector

Mathematical Operation Symbols §

Symbol	Operation Name	Example
${}^{\top }$	Transposition	${\mathbf{A}}^{\top }$
${}^{\ast }$	Adjoint	${\mathbf{A}}^{\ast }$
Tr$()$	Trace	Tr$(\mathbf{A})$
$\det ()$	Determinant	$\det (\mathbf{A})$
$⟨\cdot ,\cdot ⟩$	Inner Product	$⟨\mathbf{x},\mathbf{y}⟩$

Definiteness Symbols §

Symbol	Represents
$\succ 0$	Positive Definite
$⪰0$	Positive Semi-Definite
$\prec 0$	Negative Definite
$⪯0$	Negative Semi-Definite

Vector Space Symbols §

Symbol	Represents
${\mathbb{R}}^n$	Vector Space of $n$ dimensional real vectors
${\mathbb{C}}^n$	Vector Space of $n$ dimensional complex vectors

Norm Symbols §

Symbol	Represents
$\Vert \cdot \Vert$	Vector Norm
$\Vert \cdot {\Vert }_p$	$p$-Norm
$\Vert \cdot {\Vert }_2$	L2-Norm
$\Vert \cdot {\Vert }_1$	L1-Norm
$\Vert \cdot {\Vert }_{\infty }$	Infinity Norm
$\Vert \cdot {\Vert }_F$	Frobenius Norm