Notation
We will use the following standard notation throughout this book.
Basics
| \(\mathbb{R}^{}\) | real numbers |
| \(\mathbb{R}^{}_{+}\) | nonnegative real |
| \(\mathbb{R}^{}_{++}\) | positive real |
| \(\mathbb{Z}\) | integers |
| \(\mathbb{N}\) | nonnegative integers |
| \(\mathbb{N}_{+}\) | positive integers |
| \(\mathbb{R}^{n}\) | \(n\)-D column vector |
| \(\mathbb{R}^{n}_{+}\) | nonnegative orthant |
| \(\mathbb{R}^{n}_{++}\) | positive orthant |
| \(e_i\) | standard basic vector |
| \(\Delta_n := \{x \in \mathbb{R}^n_{+} \mid \sum x_i = 1 \}\) | standard simplex |
Matrices
| \(\mathbb{R}^{m \times n}\) | \(m \times n\) real matrices |
| \(\mathbb{S}^{n}\) | \(n\times n\) symmetric matrices |
| \(\mathbb{S}^{n}_{+}\) | \(n\times n\) positive semidefinite matrices |
| \(\mathbb{S}^{n}_{++}\) | \(n\times n\) positive definite matrices |
| \(\langle A, B \rangle\) or \(\bullet\) | inner product in \(\mathbb{R}^{m \times n}\) |
| \(\mathrm{tr}(A)\) | trace of \(A \in \mathbb{R}^{n \times n}\) |
| \(A^\top\) | matrix transpose |
| \(\det(A)\) | matrix determinant |
| \(\mathrm{rank}(A)\) | rank of a matrix |
| \(\mathrm{diag}(A)\) | diagonal of a matrix \(A\) as a vector |
| \(\mathrm{Diag}(a)\) | turning a vector into a diagonal matrix |
| \(\mathrm{BlkDiag}(A,B,\dots)\) | block diagonal matrix with blocks \(A,B,\dots\) |
| \(\succeq 0\) and \(\preceq 0\) | positive / negative semidefinite |
| \(\succ 0\) and \(\prec 0\) | positive / negative definite |
| \(\lambda_{\max}\) and \(\lambda_{\min}\) | maximum / minimum eigenvalue |
| \(\sigma_{\max}\) and \(\sigma_{\min}\) | maximum / minimum singular value |
| \(\mathrm{vec}(A)\) | vectorization of \(A \in \mathbb{R}^{m \times n}\) |
| \(\mathrm{svec}(A)\) | symmetric vectorization of \(A \in \mathbb{S}^{n}\) |
| \(\Vert A \Vert_\mathrm{F}\) | Frobenius norm |
| \(\mathrm{Range}(A)\) | span of the column vectors |
| \(\mathrm{ker}(A)\) | right null space |
Geometry
| \(\Vert a \Vert_{p}\) | \(p\)-norm |
| \(\Vert a \Vert\) | \(2\)-norm |
| \(B(o,r)\) | ball with center \(o\) and radius \(r\) |
| \(\mathrm{aff}(S)\) | affine hull of set \(S\) |
| \(\mathrm{conv}(S)\) | convex hull of set \(S\) |
| \(\mathrm{cone}(S)\) | conical hull of set \(S\) |
| \(\mathrm{int}(S)\) | interior of set \(S\) |
| \(\mathrm{ri}(S)\) | relative interior of set \(S\) |
| \(\partial S\) | boundary of set \(S\) |
| \(P^\circ\) | polar of convex body |
| \(P^{*}\) | dual of set \(P\) |
| \(\mathrm{O}(d)\) | orthogonal group of dimension \(d\) |
| \(\mathrm{SO}(d)\) | special orthogonal group of dimension \(d\) |
| \(\mathcal{S}^{d-1}\) | unit sphere in \(\mathbb{R}^{d}\) |
Optimization
| KKT | Karush–Kuhn–Tucker |
| LP | linear program |
| QP | quadratic program |
| SOCP | second-order cone program |
| SDP | semidefinite program |
Algebra
| \(\mathbb{R}[x]\) | polynomial ring in \(x\) with real coefficients |
| \(\deg\) | degree of a monomial / polynomial |
| \(\mathbb{R}[x]_d\) | polynomials in \(x\) of degree up to \(d\) |
| \([x]_d\) | vector of monomials of degree up to \(d\) |
| \([\![x ]\!]_d\) | vector of monomials of degree \(d\) |