# What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I’m looking for examples of mathematically interesting computations involving matrices.

Preferably these examples would be easy to implement in a computer program.

For instance, suppose

$$\begin{eqnarray} F_0&=&0\\ F_1&=&1\\ F_{n+1}&=&F_n+F_{n-1}, \end{eqnarray}$$
so that $F_n$ is the $n^{th}$ term in the Fibonacci sequence. If we set

$$A=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$

we see that

$$A^1=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} F_2 & F_1 \\ F_1 & F_0 \end{pmatrix},$$

and it can be shown that

$$A^n = \begin{pmatrix} F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{pmatrix}.$$

This example is “interesting” in that it provides a novel way to compute the Fibonacci sequence. It is also relatively easy to implement a simple program to verify the above.

Other examples like this will be much appreciated.

If $(a,b,c)$ is a Pythagorean triple (i.e. positive integers such that $a^2+b^2=c^2$), then
$$\underset{:=A}{\underbrace{\begin{pmatrix} 1 & -2 & 2\\ 2 & -1 & 2\\ 2 & -2 & 3 \end{pmatrix}}}\begin{pmatrix} a\\ b\\ c \end{pmatrix}$$
is also a Pythagorean triple. In addition, if the initial triple is primitive (i.e. $a$, $b$ and $c$ share no common divisor), then so is the result of the multiplication.

The same is true if we replace $A$ by one of the following matrices:

$$B:=\begin{pmatrix} 1 & 2 & 2\\ 2 & 1 & 2\\ 2 & 2 & 3 \end{pmatrix} \quad \text{or}\quad C:=\begin{pmatrix} -1 & 2 & 2\\ -2 & 1 & 2\\ -2 & 2 & 3 \end{pmatrix}.$$

Taking $x=(3,4,5)$ as initial triple, we can use the matrices $A$, $B$ and $C$ to construct a tree with all primitive Pythagorean triples (without repetition) as follows:

$$x\left\{\begin{matrix} Ax\left\{\begin{matrix} AAx\cdots\\ BAx\cdots\\ CAx\cdots \end{matrix}\right.\\ \\ Bx\left\{\begin{matrix} ABx\cdots\\ BBx\cdots\\ CBx\cdots \end{matrix}\right.\\ \\ Cx\left\{\begin{matrix} ACx\cdots\\ BCx\cdots\\ CCx\cdots \end{matrix}\right. \end{matrix}\right.$$

Source: Wikipedia’s page Tree of primitive Pythagorean triples.