Contents
- General Syntax
- Column and Row Vectors
- Different Bracket Types
- Use of Ellipses
- Some example matrices
- Nested matrices
- Matrices in Mathematical Expressions
- Conclusion
This blog post will guide you through the steps of creating matrices in LaTeX. It will start with the general syntax and then explain how to create row and column vectors, determinants, arbitrary sized matrices and nested matrices. It will conclude with several examples of real world matrices and the use of matrices in mathematical expressions.
General Syntax
In LaTeX, matrices are created using the “bmatrix” environment. The general syntax for a matrix is as follows:
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
Replace the placeholders (a, b, c, etc.) with your desired matrix elements. The ampersand (&) is used to separate columns, and the double backslash (\\) indicates the end of a row.
Column and Row Vectors
Both column and row vectors are essentially specialized forms of matrices, and you can use the “bmatrix” environment to represent them.
Column Vector
To create a column vector, you can use the “bmatrix” environment with a single column:
\begin{bmatrix}
a \\
b \\
c \\
\end{bmatrix}
Row Vector
Similarly, a row vector is a matrix with a single row:
\begin{bmatrix}
a & b & c
\end{bmatrix}
Different Bracket Types
LaTeX supports various bracket types for matrices.
Parentheses
To represent matrices with parentheses $(\cdot)$ you can use the “pmatrix” environment:
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
Determinants
To represent determinants, you can use the “vmatrix” or “Vmatrix” environments:
vmatrix: Vertical bars $\vert \cdot \vert$ as brackets
\begin{vmatrix}
a & b \\
c & d \\
\end{vmatrix}
Vmatrix: Double vertical $\Vert \cdot \Vert$ bars as brackets
\begin{Vmatrix}
a & b \\
c & d \\
\end{Vmatrix}
No brackets
To structure data in a matrix format without any brackets, you can use the “matrix” environment:
\begin{matrix}
a & b \\
c & d \\
\end{matrix}
Use of Ellipses
Sometimes you want to represent a range of elements in a matrix without explicitly listing them all. For example when you have an arbitrary $m \times n$ matrix and you want to show only a few representative elements.
For this, you can use ellipses. There are different commands for horizontal, vertical and diagonal ellipses
Horizontal
The \dots (or \ldots) command can be used to represent a range of skipped elements in a row. For example, here is row vector showing only the first two and last of $n$ elements
\begin{bmatrix}
a_1 & a_2 & \dots & a_n
\end{bmatrix}
Vertical
The \vdots command can be used to represent a range of skipped elements in a column. For example, here is column vector showing only the first two and last of $n$ elements
\begin{bmatrix}
a_1 \\
a_2 \\
\ldots \\
a_n
\end{bmatrix}
Diagonal
The \ddots command, often used in combination with \dots and \vdots can be used to skip columns and rows simultaneously.
\begin{bmatrix}
a_{11} & \dots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \dots & a_{nn} \\
\end{bmatrix}
Some example matrices
Matrix elements can be arbitrary LaTeX expressions and the matrix will automatically adjust to accommodate the size of the expression.
Example: 2x2 rotation matrix
R = \begin{bmatrix}
\cos \theta &-\sin \theta \\
\sin \theta &\cos \theta
\end{bmatrix}
Example: curl of a vector field expressed as a determinant
\nabla \times \mathbf {F} =
\begin{vmatrix}
\boldsymbol {\hat {\imath }} & {\boldsymbol {\hat {\jmath }}} & {\boldsymbol {\hat {k}}}
\\ frac {\partial }{\partial x} & \frac {\partial }{\partial y} & \frac {\partial }{\partial z}\\F_{x} & F_{y} & F_{z}
\end{vmatrix}
Example: N-point discrete Fourier transform matrix
W = \frac{1}{\sqrt{N}} \begin{bmatrix}
1 & 1 & 1 & 1 & \cdots & 1 \\
1 & \omega & \omega^2 & \omega^3 & \cdots & \omega^{N-1} \\
1 & \omega^2 & \omega^4 & \omega^6 & \cdots & \omega^{2(N-1)} \\
1 & \omega^3 & \omega^6 & \omega^9 & \cdots & \omega^{3(N-1)} \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
1 & \omega^{N-1} & \omega^{2(N-1)} & \omega^{3(N-1)} & \cdots & \omega^{(N-1)(N-1)}
\end{bmatrix}
where $\omega = e^{-2\pi i/N}$
Nested matrices
You can also nest matrices within matrices. Here’s an example of a block diagonal matrix that represents an important quantum computing logic gate called the CNOT gate:
\begin{bmatrix}
I_2 & 0 \\
0 & X
\end{bmatrix}
= \begin{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} & 0 \\
0 & \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\end{bmatrix}
where $I_2$ is the $2 \times 2$ identity matrix and $X$ is the Pauli-X matrix
You can nest matrices to any depth as needed. For example, the matrix for another quantum logic gate, the Toffoli gate is a block diagonal matrix with a nested matrix in the bottom right corner:
\begin{bmatrix}
I_4 & 0 \\
0 & CNOT
\end{bmatrix}
= \begin{bmatrix}
I_4 & 0 \\
0 & \begin{bmatrix}
I_2 & 0 \\
0 & X
\end{bmatrix}
\end{bmatrix}
= \begin{bmatrix}
I_4 & 0 \\
0 & \begin{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} & 0 \\
0 & \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\end{bmatrix}
\end{bmatrix}
where $I_4$ is the $4 \times 4$ identity matrix.
Matrices in Mathematical Expressions
Matrices can be incorporated into mathematical expressions just like any other variable. For example, here of the rotation of an arbitrary vector $\mathbf{v}$ by an angle $\theta$:
R\mathbf {v} =
\begin{bmatrix}
\cos \theta &-\sin \theta \\
\sin \theta &\cos \theta
\end{bmatrix}
\begin{bmatrix}
x\\y
\end{bmatrix}=
\begin{bmatrix}
x\cos \theta -y\sin \theta
\\x\sin \theta +y\cos \theta
\end{bmatrix}
Here is another example of the exponential of a diagonal matrix that demonstrates how matrices in LaTeX can be seamlessly integrated into a variety of mathematical expressions and how they play nicely with other LaTeX capabilities like superscripts and brackets.
\begin{aligned}
e^{\operatorname{diag}\left(
\begin{bmatrix}
a_1 \\
a_2
\end{bmatrix}
\right)} &= \exp\left(\begin{bmatrix}
a_1 & 0 \\
0 & a_2
\end{bmatrix}\right)
\\&= \sum_{i=0}^\infty \frac{1}{i!} \begin{bmatrix}
a_1 & 0 \\
0 & a_2
\end{bmatrix}^i
\\&= \sum_{i=0}^\infty \frac{1}{i!} \begin{bmatrix}
a_1^i & 0 \\
0 & a_2^i
\end{bmatrix}
\\&= \begin{bmatrix}
e^{a_1} & 0 \\
0 & e^{a_2}
\end{bmatrix}
\end{aligned}
Conclusion
In this blog post we covered the key features of representing matrices in LaTeX. You should now be able to create a variety of matrices and incorporate them in your LaTeX code.