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Notation 표기 시

$\R_+$ : 0과 양의 실수 집합

$\R$ : 실수 집합

$\R_{++}$ : 양의 실수 집합

$\R^k$ : $k \times k$ matrix

We denote $S_k^+$ as the set of symmetric positive semidefinite k×k matrices, and $S_k^{++}$ as the set of symmetric positive definite k×k matrices. The curled inequality symbol $≽$ (and its strict form $≻$) is used to denote generalized inequality: between vectors, it represents componentwise inequality; between symmetric matrices, it represents matrix inequality. With a subscript, the symbol $≽_K$ (or $≺_K$) denotes generalized inequality with respect to the cone K (explained in §2.4.1).

Our notation for describing functions deviates a bit from standard notation, but we hope it will cause no confusion. We use the notation $f: ℝ^p → ℝ^q$ to mean that f is an $ℝ^q$-valued function on some subset of $ℝ^p$, specifically, its domain, which we denote $dom f$. We can think of our use of the notation $f: ℝ^p → ℝ^q$ as a declaration of the function type, as in a computer language: $f: ℝ^p → ℝ^q$ means that the function f takes as argument a real p-vector and returns a real q-vector.

The set $\text{dom} \; f$, the domain of the function f, specifies the subset of $ℝ^p$ of points x for which $f(x)$ is defined. As an example, we describe the logarithm function as $\log: ℝ → ℝ$, with $\text{dom} \;\log = ℝ^{++}$. The notation $\log: ℝ → ℝ$ means that the logarithm function accepts and returns a real number; $dom log = ℝ^{++}$ means that the logarithm is defined only for positive numbers.

We use $ℝ^n$ as a generic finite-dimensional vector space. We will encounter several other finite-dimensional vector spaces, e.g., the space of polynomials of a variable with a given maximum degree, or the space $S_k$ of symmetric k×k matrices. By identifying a basis for a vector space, we can always identify it with $ℝ^n$ (where n is its dimension), and therefore the generic results, stated for the vector space $ℝ^n$, can be applied. We usually leave it to the reader to translate general results or statements to other vector spaces.

For example, any linear function $f: ℝ^n → ℝ$ can be represented in the form $f(x) = c^T x$, where $c ∈ ℝ^n$. The corresponding statement for the vector space $S_k$ can be found by choosing a basis and translating. This results in the statement: any linear function $f: S_k → ℝ$ can be represented in the form $f(X) = tr(CX)$, where $C ∈ S_k$.

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