Matrices

Explore the fundamentals of matrices, including their notation, types, and essential operations. This section covers basic matrix environments, matrices with dots, multiplication, transposition, and inversion, providing a solid foundation for further applications.

Basic matrix environments

LaTeX provides several environments for typesetting matrices, each with different delimiters. These environments are part of the amsmath package, which must be included in the preamble of your LaTeX document:

\usepackage{amsmath}

The following examples demonstrate the various matrix environments provided by the amsmath package:

1. Plain matrix (No delimiters)
A plain matrix without any enclosing delimiters can be created using the matrix environment.

\[
\begin{matrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{matrix}
\]

Output:

plain matrix example with no delimiter

2. pmatrix (Matrix with parentheses)
Matrices enclosed in parentheses are created using the pmatrix environment:

\[
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\]

Output:

pmatrix example (matrix with parentheses)

3. bmatrix (Matrix with square brackets)
Matrices with square brackets are created using the bmatrix environment:

\[
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\]

Output:

bmatrix example (matrix with square brackets)

4. Bmatrix (Matrix with curly braces)
To enclose a matrix with curly braces, use the Bmatrix environment:

\[
\begin{Bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{Bmatrix}
\]

Output:

Bmatrix example (matrix with curly braces)

5. vmatrix (Matrix with single vertical bars)
Use the vmatrix environment to create matrices enclosed by single vertical bars, commonly used for determinants:

\[
\begin{vmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{vmatrix}
\]

Output:

vmatrix example (matrix with single vertical bars)

6. Vmatrix (Matrix with double vertical bars)
Double vertical bars are used with the Vmatrix environment, often to represent norms.

\[
\begin{Vmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{Vmatrix}
\]

Output:

Vmatrix example (matrix with double vertical bars)

Matrices with dots

In mathematics, large matrices or those following a recognizable pattern often use dots to indicate continuation. These dots help represent matrices concisely without explicitly listing every element. You can use the following commands to include dots in your matrix:

  • \cdots (Horizontal dots) → Indicate continuation across columns.
  • \vdots (Vertical dots) → Indicate continuation down rows.
  • \ddots (Diagonal dots) → Indicate continuation along the diagonal.

Here’s some examples of matrices with dots:

1. General square matrix (n×n)
An example of a general n×n matrix uses dots to indicate missing elements:

\[
A =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{bmatrix}
\]

Output:

general square matrix example using dots

2. Triangular matrix representation
Dots are frequently used in upper and lower triangular matrices:

\[
U =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & a_{22} & a_{23} & \cdots & a_{2n} \\
0 & 0 & a_{33} & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_{nn}
\end{bmatrix}
\]

Output:

upper triangular matrix example

3. Identity matrix representation
Dots help indicate the diagonal structure of an identity matrix:

\[
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{bmatrix}
\]

Output:

identity matrix example

Matrix multiplication

In LaTeX, the \times command is used to indicate multiplication, including between matrices. While matrices are usually written side by side to denote multiplication, explicitly using \times can sometimes make it clearer.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

% Defining two matrices A and B
\[
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\times
\begin{bmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{bmatrix}
= 
\begin{bmatrix}
a_{11} \times b_{11} + a_{12} \times b_{21} & a_{11} \times b_{12} + a_{12} \times b_{22} \\
a_{21} \times b_{11} + a_{22} \times b_{21} & a_{21} \times b_{12} + a_{22} \times b_{22}
\end{bmatrix}
\]

\end{document}

Output:

matrix multiplication example

Matrix transpose

The transpose of a matrix is obtained by swapping its rows and columns. It is denoted as 𝐴𝑇 or A ⊤. In LaTeX, the transpose of a matrix can be represented as:

\[
A^T
\]

or

\[
A^\top
\]

Here’s an example of transpose of a matrix:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
A =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{bmatrix}
\]

\[
A^T =
\begin{bmatrix}
a_{11} & a_{21} \\
a_{12} & a_{22} \\
a_{13} & a_{23}
\end{bmatrix}
\]

\end{document}

Output:

matrix transpose example

Matrix inverse

The inverse of a matrix 𝐴, denoted as 𝐴−1, represented in LaTeX as:

\[
A^{-1}
\]

Here’s an example of inverse of a matrix:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
A =
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
\]

\[
A^{-1} = \frac{1}{\det(A)}
\begin{bmatrix}
a_{22} & -a_{12} \\
-a_{21} & a_{11}
\end{bmatrix}
\]

\[
\det(A) = a_{11}a_{22} - a_{12}a_{21}
\]

\end{document}

Output:

matrix inverse example