Prerequisites & Notation

Before You Begin

This chapter assumes basic familiarity with the following. If any item feels unfamiliar, revisit the linked material first.

High-school algebra and basic set notation
Self-check: Can you define what a set, a function, and a field are?
Complex numbers: arithmetic, polar form, Euler's formula
Self-check: Can you compute $|1 + j|$ and $e^{j\pi/4}$ without hesitation?
Systems of linear equations and basic matrix operations (multiply, transpose)
Self-check: Can you multiply a $3 \times 2$ matrix by a $2 \times 1$ vector?
Summation notation and basic proof techniques (induction, contradiction)
Self-check: Can you prove that $\sum_{k=1}^{n} k = n(n+1)/2$ by induction?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{x}, \mathbf{y}, \mathbf{z}$	Column vectors in $\mathbb{C}^n$ (boldface lowercase)	s01
$V, W$	Vector spaces	s01
$\text{span}(\mathbf{v}_1, \ldots, \mathbf{v}_k)$	Linear span of a set of vectors	s01
$\dim(V)$	Dimension of vector space $V$	s01
$\langle \mathbf{x}, \mathbf{y} \rangle$	Inner product $\mathbf{y}^H \mathbf{x}$ (conjugate-linear in second argument)	s02
$\\|\mathbf{x}\\|$	Euclidean norm $\sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}$	s02
$\\|\mathbf{x}\\|_p$	$\ell_p$ norm: $(\sum_i \|x_i\|^p)^{1/p}$	s02
$\mathbf{A}, \mathbf{B}, \mathbf{H}$	Matrices (boldface uppercase)	s03
$\mathcal{R}(\mathbf{A})$	Range (column space) of $\mathbf{A}$	s03
$\mathcal{N}(\mathbf{A})$	Null space of $\mathbf{A}$	s03
$\text{rank}(\mathbf{A})$	Rank of matrix $\mathbf{A}$	s03
$(\cdot)^H$	Conjugate transpose (Hermitian transpose)	s03
$\mathbf{A} \succ 0$ , $\mathbf{A} \succeq 0$	Positive definite, positive semidefinite	s03
$\lambda_i(\mathbf{A})$	$i$ -th eigenvalue of $\mathbf{A}$ (ordered by magnitude unless stated otherwise)	s04
$\mathbf{Q} \mathbf{\Lambda} \mathbf{Q}^{-1}$	Eigendecomposition of a square matrix	s04
$R(\mathbf{A}, \mathbf{x})$	Rayleigh quotient $\mathbf{x}^H \mathbf{A} \mathbf{x} / \mathbf{x}^H \mathbf{x}$	s04
$\sigma_i(\mathbf{A})$	$i$ -th singular value of $\mathbf{A}$ (ordered $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$ )	s05
$\mathbf{U} \mathbf{\Sigma} \mathbf{V}^H$	Singular value decomposition	s05
$\text{tr}(\mathbf{A})$	Trace of $\mathbf{A}$ : $\sum_i a_{ii}$	s06
$\det(\mathbf{A})$	Determinant of $\mathbf{A}$	s06
$\otimes$	Kronecker product	s07
$\text{vec}(\mathbf{A})$	Column-wise vectorization of matrix $\mathbf{A}$	s07
$\frac{\partial f}{\partial \mathbf{x}}$	Gradient of scalar $f$ with respect to vector $\mathbf{x}$	s08
$\frac{\partial f}{\partial \mathbf{A}}$	Matrix derivative of scalar $f$ with respect to matrix $\mathbf{A}$	s08

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