.. _chap_notation: Notation ======== The notation used throughout this book is summarized below. Numbers ------- - :math:x: A scalar - :math:\mathbf{x}: A vector - :math:\mathbf{X}: A matrix - :math:\mathsf{X}: A tensor - :math:\mathbf{I}: An identity matrix - :math:x_i, :math:[\mathbf{x}]_i: The :math:i^\mathrm{th} element of vector :math:\mathbf{x} - :math:x_{ij}, :math:[\mathbf{X}]_{ij}: The element of matrix :math:\mathbf{X} at row :math:i and column :math:j Set Theory ---------- - :math:\mathcal{X}: A set - :math:\mathbb{Z}: The set of integers - :math:\mathbb{R}: The set of real numbers - :math:\mathbb{R}^n: The set of :math:n-dimensional vectors of real numbers - :math:\mathbb{R}^{a\times b}: The set of matrices of real numbers with :math:a rows and :math:b columns - :math:\mathcal{A}\cup\mathcal{B}: Union of sets :math:\mathcal{A} and :math:\mathcal{B} - :math:\mathcal{A}\cap\mathcal{B}: Intersection of sets :math:\mathcal{A} and :math:\mathcal{B} - :math:\mathcal{A}\setminus\mathcal{B}: Subtraction of set :math:\mathcal{B} from set :math:\mathcal{A} Functions and Operators ----------------------- - :math:f(\cdot): A function - :math:\log(\cdot): The natural logarithm - :math:\exp(\cdot): The exponential function - :math:\mathbf{1}_\mathcal{X}: The indicator function - :math:\mathbf{(\cdot)}^\top: Transpose of a vector or a matrix - :math:\mathbf{X}^{-1}: Inverse of matrix :math:\mathbf{X} - :math:\odot: Hadamard (elementwise) product - :math:[\cdot, \cdot]: Concatenation - :math:\lvert \mathcal{X} \rvert: Cardinality of set :math:\mathcal{X} - :math:\|\cdot\|_p: :math:\ell_p norm - :math:\|\cdot\|: :math:\ell_2 norm - :math:\langle \mathbf{x}, \mathbf{y} \rangle: Dot product of vectors :math:\mathbf{x} and :math:\mathbf{y} - :math:\sum: Series addition - :math:\prod: Series multiplication Calculus -------- - :math:\frac{dy}{dx}: Derivative of :math:y with respect to :math:x - :math:\frac{\partial y}{\partial x}: Partial derivative of :math:y with respect to :math:x - :math:\nabla_{\mathbf{x}} y: Gradient of :math:y with respect to :math:\mathbf{x} - :math:\int_a^b f(x) \;dx: Definite integral of :math:f from :math:a to :math:b with respect to :math:x - :math:\int f(x) \;dx: Indefinite integral of :math:f with respect to :math:x Probability and Information Theory ---------------------------------- - :math:P(\cdot): Probability distribution - :math:z \sim P: Random variable :math:z has probability distribution :math:P - :math:P(X \mid Y): Conditional probability of :math:X \mid Y - :math:p(x): Probability density function - :math:{E}_{x} [f(x)]: Expectation of :math:f with respect to :math:x - :math:X \perp Y: Random variables :math:X and :math:Y are independent - :math:X \perp Y \mid Z: Random variables :math:X and :math:Y are conditionally independent given random variable :math:Z - :math:\mathrm{Var}(X): Variance of random variable :math:X - :math:\sigma_X: Standard deviation of random variable :math:X - :math:\mathrm{Cov}(X, Y): Covariance of random variables :math:X and :math:Y - :math:\rho(X, Y): Correlation of random variables :math:X and :math:Y - :math:H(X): Entropy of random variable :math:X - :math:D_{\mathrm{KL}}(P\|Q): KL-divergence of distributions :math:P and :math:Q Complexity ---------- - :math:\mathcal{O}: Big O notation Discussions __ ------------------------------------------------- |image0| .. |image0| image:: ../img/qr_notation.svg