Introduction

This post serves as a reference for the mathematical notations used across my blog posts. If you ever find a symbol unfamiliar, this page is here to help.

Math Notations

Scalars

• A scalar is represented in italic, non-bold, and lowercase, like \(s\).

  • Example:

    \[s = 42\]

Vectors

• A vector is represented in non-italic, bold, and lowercase, like \(\mathbf{v}\).

  • Example:

    \[\begin{equation*} \mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \end{equation*}\]

• A general \(i\)-th element of a vector \(\mathbf{v}\) is represented in italic, non-bold, and lowercase with a subscript, like \(v_i\).

  • Example:

    \[\begin{equation*} \mathbf{v} = \begin{bmatrix} v_1 \\ \vdots \\ v_i\\ \vdots \end{bmatrix} \end{equation*}\]

• The norm of a vector \(\mathbf{v}\) is represented as \(\left\lvert\left\lvert \mathbf{v} \right\rvert\right\rvert\).

• A general \(i\)-th vector is represented in non-italic, bold, and lowercase with a subscript, like \(\mathbf{v}_i\).

• A mathematical way to say “a real vector \(\mathbf{v}\) is \(d\)-dimensional” is \(\mathbf{v} \in \mathbb{R}^{d}\).

Matrices

• A matrix is represented in non-italic, bold, and uppercase, like \(\mathbf{M}\).

  • Example:

    \[\begin{equation*} \mathbf{M} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \end{equation*}\]

• Element in the \(i\)-th row and \(j\)-th column of a matrix \(\mathbf{M}\) (\(p\times q\)) is denoted as \(m_{ij}\).

  • Example:

    \[\begin{equation*} \mathbf{M} = \begin{bmatrix} m_{11} & m_{12} & \cdots & m_{1q}\\ m_{21} & m_{22} & \cdots & m_{2q}\\ \vdots & \vdots & m_{ij} & \vdots\\ m_{p1} & m_{p2} & \cdots & m_{pq} \end{bmatrix} \end{equation*}\]

Sets

• Any general set is represented in a calligraphic font, like \(\mathcal{S}\).
• Set of real numbers is represented as \(\mathbb{R}\).