Overview
This glossary collects every symbol used in the book: what it denotes, its typical shape, and the chapter where it first appears. Symbols are grouped by category; within each group they appear in roughly first-use order.
Typographic Conventions
| Lowercase italic \(a, b, \lambda\) |
Scalar |
| Lowercase bold \(\mathbf{x}, \mathbf{v}\) |
Column vector |
| Uppercase italic \(A, B, R\) |
Matrix |
| Uppercase calligraphic \(\mathcal{C}, \mathcal{N}\) |
Set or subspace |
| Uppercase sans-serif \(\mathsf{T}\) |
Transformation (rare) |
| Hat \(\hat{\mathbf{x}}, \hat{\omega}\) |
Unit vector or skew-symmetric lift |
| Bar \(\bar{x}\) |
Sample mean |
| Tilde \(\tilde{\mathbf{x}}\) |
Homogeneous coordinate |
| Superscript \(\top\) |
Transpose (same as \(T\) in the book) |
| Superscript \(*\) |
Complex conjugate or optimal value |
| Superscript \(\dagger\) |
Moore-Penrose pseudoinverse |
Symbol Tables
Scalars and Fields
| \(\mathbb{R}\) |
Set of real numbers |
Ch. 1 |
| \(\mathbb{R}^n\) |
Real \(n\)-dimensional vector space |
Ch. 4 |
| \(\mathbb{R}^{m \times n}\) |
Set of real \(m \times n\) matrices |
Ch. 5 |
| \(\mathbb{C}\) |
Set of complex numbers |
Ch. 13 |
| \(\mathbb{Z}\) |
Set of integers |
Ch. 3 |
| \(a, b, c\) |
Generic scalars |
Ch. 1 |
| \(\alpha, \beta, \gamma\) |
Scalar coefficients or angles |
Ch. 4 |
| \(\lambda, \mu\) |
Eigenvalue or Lagrange multiplier |
Ch. 13 |
| \(\sigma\) |
Singular value or standard deviation |
Ch. 18 |
| \(\epsilon\) |
Small positive number; tolerance |
Ch. 3 |
| \(i, j, k, \ell\) |
Index variables |
Ch. 1 |
| \(m, n, p, r\) |
Dimension parameters; rank |
Ch. 2 |
Vectors
| \(\mathbf{x}, \mathbf{v}, \mathbf{w}\) |
\((n,1)\) |
Generic column vectors |
Ch. 4 |
| \(\mathbf{a}, \mathbf{b}\) |
\((n,1)\) |
Fixed vectors (problem data) |
Ch. 2 |
| \(\mathbf{e}_i\) |
\((n,1)\) |
\(i\)-th standard basis vector |
Ch. 4 |
| \(\mathbf{0}\) |
\((n,1)\) |
Zero vector |
Ch. 4 |
| \(\hat{\mathbf{x}}\) |
\((3,1)\) |
Unit vector: \(\hat{\mathbf{x}} = \mathbf{x}/\|\mathbf{x}\|\) |
Ch. 8 |
| \(\tilde{\mathbf{x}}\) |
\((n+1,1)\) |
Homogeneous form of \(\mathbf{x}\) |
Ch. 7 |
| \(\boldsymbol{\mu}\) |
\((n,1)\) |
Mean vector |
Ch. 19 |
| \(\mathbf{r}\) |
\((m,1)\) |
Residual vector \(A\mathbf{x}-\mathbf{b}\) |
Ch. 17 |
| \(\boldsymbol{\omega}\) |
\((3,1)\) |
Angular velocity or rotation axis |
Ch. 8 |
| \(\mathbf{t}\) |
\((3,1)\) |
Translation vector |
Ch. 9 |
| \(\boldsymbol{\delta}\) |
varies |
Upstream gradient in backprop |
Ch. 21 |
| \(\nabla_\mathbf{x} f\) |
\((n,1)\) |
Gradient of scalar \(f\) w.r.t. \(\mathbf{x}\) |
Ch. 21 |
Matrices
| \(A, B, C\) |
\((m,n)\) |
Generic matrices |
Ch. 5 |
| \(I\) or \(I_n\) |
\((n,n)\) |
Identity matrix |
Ch. 5 |
| \(O\) |
\((m,n)\) |
Zero matrix |
Ch. 5 |
| \(A^T\) |
\((n,m)\) |
Transpose of \(A\) |
Ch. 5 |
| \(A^{-1}\) |
\((n,n)\) |
Inverse of square \(A\) |
Ch. 5 |
| \(A^\dagger\) |
\((n,m)\) |
Moore-Penrose pseudoinverse |
Ch. 17 |
| \(A^{-T}\) |
\((n,n)\) |
\((A^{-1})^T = (A^T)^{-1}\) |
Ch. 5 |
| \(\bar{A}\) |
\((m,n)\) |
Complex conjugate of \(A\) |
Ch. 13 |
| \(A^H\) or \(A^*\) |
\((n,m)\) |
Conjugate transpose (Hermitian) |
Ch. 13 |
| \(\hat\omega\) or \([\omega]_\times\) |
\((3,3)\) |
Skew-symmetric lift of \(\omega \in \mathbb{R}^3\) |
Ch. 8 |
| \(R\) |
\((3,3)\) |
Rotation matrix in \(SO(3)\) |
Ch. 8 |
| \(T\) or \([R|\mathbf{t}]\) |
\((4,4)\) or \((3,4)\) |
Rigid-body transform in \(SE(3)\) |
Ch. 9 |
| \(K\) |
\((3,3)\) |
Camera intrinsic matrix |
Ch. 10 |
| \(\Pi\) |
\((3,4)\) |
Camera projection matrix |
Ch. 10 |
| \(F\) |
\((3,3)\) |
Fundamental matrix |
Ch. 25 |
| \(E\) |
\((3,3)\) |
Essential matrix |
Ch. 25 |
| \(\Sigma\) |
\((n,n)\) |
Covariance matrix (PD, symmetric) |
Ch. 19 |
| \(W\) |
varies |
Weight matrix (neural network) |
Ch. 21 |
| \(H\) |
\((n,n)\) |
Hessian matrix |
Ch. 21 |
| \(J_f\) |
\((m,n)\) |
Jacobian of \(f:\mathbb{R}^n\to\mathbb{R}^m\) |
Ch. 22 |
| \(\Lambda\) |
\((n,n)\) |
Diagonal eigenvalue matrix |
Ch. 14 |
| \(\Sigma\) (SVD) |
\((m,n)\) |
Diagonal singular-value matrix |
Ch. 18 |
Decompositions
| \(A = LU\) |
LU factorization (\(L\) lower, \(U\) upper triangular) |
App. B |
| \(A = LDL^T\) |
Symmetric indefinite factorization |
App. B |
| \(A = LL^T\) |
Cholesky factorization (\(L\) lower triangular) |
App. B |
| \(A = QR\) |
QR factorization (\(Q\) orthogonal, \(R\) upper triangular) |
App. B |
| \(A = Q\Lambda Q^T\) |
Eigendecomposition (symmetric \(A\)) |
Ch. 14 |
| \(A = U\Sigma V^T\) |
Singular value decomposition |
Ch. 18 |
| \(A = V\Lambda V^{-1}\) |
Eigendecomposition (general \(A\)) |
Ch. 14 |
Operations and Functions
| \(\mathbf{u} \cdot \mathbf{v}\) or \(\langle \mathbf{u},\mathbf{v} \rangle\) |
Dot (inner) product |
Ch. 4 |
| \(A\mathbf{x}\) |
Matrix-vector product |
Ch. 5 |
| \(AB\) |
Matrix-matrix product |
Ch. 5 |
| \(A \odot B\) |
Hadamard (element-wise) product |
Ch. 5 |
| \(A \otimes B\) |
Kronecker product |
Ch. 5 |
| \(\det(A)\) |
Determinant |
Ch. 12 |
| \(\operatorname{tr}(A)\) |
Trace: \(\sum_i A_{ii}\) |
Ch. 12 |
| \(\operatorname{rank}(A)\) |
Column (or row) rank of \(A\) |
Ch. 6 |
| \(\operatorname{null}(A)\) |
Null space: \(\{\mathbf{x}: A\mathbf{x}=\mathbf{0}\}\) |
Ch. 6 |
| \(\operatorname{col}(A)\) |
Column space of \(A\) |
Ch. 6 |
| \(\operatorname{row}(A)\) |
Row space of \(A\) |
Ch. 6 |
| \(\ker(A)\) |
Kernel (same as null space) |
Ch. 6 |
| \(\operatorname{im}(A)\) |
Image (same as column space) |
Ch. 6 |
| \(\operatorname{span}\{v_1,\ldots,v_k\}\) |
Linear span of a set of vectors |
Ch. 4 |
| \(\operatorname{diag}(\mathbf{x})\) |
Diagonal matrix with \(\mathbf{x}\) on diagonal |
Ch. 5 |
| \(\operatorname{vec}(A)\) |
Vectorization: stack columns of \(A\) |
Ch. 5 |
| \(\exp(A)\) |
Matrix exponential |
Ch. 24 |
| \(\log(A)\) |
Matrix logarithm |
Ch. 24 |
| \(A^\wedge\) |
Lie algebra element lifted from vector |
Ch. 24 |
| \(A^\vee\) |
Lie algebra element lowered to vector |
Ch. 24 |
Norms
| \(\|\mathbf{x}\|\) or \(\|\mathbf{x}\|_2\) |
Euclidean (L2) norm |
Ch. 15 |
| \(\|\mathbf{x}\|_1\) |
L1 norm: \(\sum_i |x_i|\) |
Ch. 15 |
| \(\|\mathbf{x}\|_p\) |
Lp norm: \((\sum_i |x_i|^p)^{1/p}\) |
Ch. 15 |
| \(\|\mathbf{x}\|_\infty\) |
L-infinity norm: \(\max_i |x_i|\) |
Ch. 15 |
| \(\|A\|_F\) |
Frobenius norm: \(\sqrt{\operatorname{tr}(A^T A)}\) |
Ch. 15 |
| \(\|A\|_2\) |
Spectral (operator) norm: \(\sigma_{\max}(A)\) |
Ch. 15 |
| \(\|A\|_*\) |
Nuclear norm: \(\sum_i \sigma_i(A)\) |
Ch. 18 |
| \(d(\mathbf{u},\mathbf{v})\) |
Euclidean distance \(\|\mathbf{u}-\mathbf{v}\|\) |
Ch. 15 |
Calculus and Optimization
| \(\nabla_\mathbf{x} f\) |
\((n,1)\) |
Gradient of scalar \(f:\mathbb{R}^n\to\mathbb{R}\) |
Ch. 21 |
| \(\partial f / \partial W\) |
same as \(W\) |
Matrix gradient of scalar \(f\) w.r.t. \(W\) |
Ch. 21 |
| \(J_f\) |
\((m,n)\) |
Jacobian: \([J_f]_{ij} = \partial f_i / \partial x_j\) |
Ch. 22 |
| \(H_f\) |
\((n,n)\) |
Hessian: \([H_f]_{ij} = \partial^2 f / \partial x_i \partial x_j\) |
Ch. 21 |
| \(\delta = \nabla_\mathbf{y} L\) |
\((m,1)\) |
Upstream gradient in backprop |
Ch. 21 |
| \(df\) |
scalar |
Total differential of \(f\) |
Ch. 21 |
| \(dA\) |
same as \(A\) |
Total differential of matrix \(A\) |
Ch. 21 |
| \(\operatorname{argmin}_\mathbf{x} f(\mathbf{x})\) |
\((n,1)\) |
Minimizer of \(f\) |
Ch. 21 |
| \(\mathbf{x}^*\) |
\((n,1)\) |
Optimal solution |
Ch. 21 |
Probability and Statistics
| \(\boldsymbol{\mu}\) |
Mean vector |
Ch. 19 |
| \(\bar{x}\) |
Sample mean (scalar) |
Ch. 19 |
| \(\sigma^2\) |
Variance (scalar) |
Ch. 19 |
| \(\Sigma\) |
Covariance matrix (positive semi-definite) |
Ch. 19 |
| \(\mathcal{N}(\boldsymbol{\mu}, \Sigma)\) |
Multivariate Gaussian distribution |
Ch. 19 |
| \(p(\mathbf{x})\) |
Probability density function |
Ch. 19 |
| \(\mathbb{E}[\mathbf{x}]\) |
Expectation of \(\mathbf{x}\) |
Ch. 19 |
3D Geometry
| \(SO(3)\) |
– |
Special orthogonal group: \(\{R \in \mathbb{R}^{3\times3}: RR^T = I, \det R = 1\}\) |
Ch. 8 |
| \(SE(3)\) |
– |
Special Euclidean group: rigid-body transforms |
Ch. 9 |
| \(\mathfrak{so}(3)\) |
– |
Lie algebra of \(SO(3)\): skew-symmetric \(3\times3\) matrices |
Ch. 24 |
| \(\mathfrak{se}(3)\) |
– |
Lie algebra of \(SE(3)\) |
Ch. 24 |
| \(\operatorname{Exp}(\boldsymbol{\omega})\) |
\((3,3)\) |
Rodrigues exponential: rotation by \(\|\omega\|\) about \(\hat\omega\) |
Ch. 8 |
| \(\operatorname{Log}(R)\) |
\((3,1)\) |
Rotation vector corresponding to \(R\) |
Ch. 8 |
| \(\mathbf{p}^C\) |
\((3,1)\) |
Point \(\mathbf{p}\) in camera frame \(C\) |
Ch. 10 |
| \({}^C T_W\) |
\((4,4)\) |
Transform from world \(W\) to camera \(C\) |
Ch. 9 |
| \(K\) |
\((3,3)\) |
Camera intrinsic matrix (focal lengths, principal point) |
Ch. 10 |
| \(\pi(\cdot)\) |
\(\mathbb{R}^3 \to \mathbb{R}^2\) |
Perspective projection function |
Ch. 10 |
| \([\mathbf{x}]_\times\) |
\((3,3)\) |
Skew-symmetric matrix for cross product: \([\mathbf{x}]_\times \mathbf{y} = \mathbf{x}\times\mathbf{y}\) |
Ch. 8 |
Sets and Logic
| \(\in\) |
Element of |
Ch. 1 |
| \(\subset\) or \(\subseteq\) |
Subset |
Ch. 6 |
| \(\cup, \cap\) |
Union, intersection |
Ch. 6 |
| \(\emptyset\) |
Empty set |
Ch. 6 |
| \(\{x : P(x)\}\) |
Set of elements satisfying predicate \(P\) |
Ch. 6 |
| \(\forall\) |
For all |
Ch. 4 |
| \(\exists\) |
There exists |
Ch. 4 |
| \(\iff\) |
If and only if |
Ch. 4 |
| \(\implies\) |
Implies |
Ch. 4 |
| \(:=\) |
Defined as |
Ch. 1 |
| \(\approx\) |
Approximately equal |
Ch. 3 |
| \(\propto\) |
Proportional to |
Ch. 19 |
Summation and Index Notation
| \(\sum_{i=1}^n a_i\) |
Sum of \(a_1,\ldots,a_n\) |
Ch. 4 |
| \(\prod_{i=1}^n a_i\) |
Product of \(a_1,\ldots,a_n\) |
Ch. 12 |
| \(A_{ij}\) or \([A]_{ij}\) |
\((i,j)\) entry of matrix \(A\) |
Ch. 5 |
| \(A_{i:}\) |
\(i\)-th row of \(A\) |
Ch. 5 |
| \(A_{:j}\) |
\(j\)-th column of \(A\) |
Ch. 5 |
| \(\sum_k A_{ik} B_{kj}\) |
Matrix product entry \((i,j)\); also written \(A_{ik}B_{kj}\) (Einstein) |
App. F |
| \(\delta_{ij}\) |
Kronecker delta: 1 if \(i=j\), else 0 |
Ch. 5 |
| \(\epsilon_{ijk}\) |
Levi-Civita symbol |
Ch. 8 |