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.

Answer

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.

Attribution
Source : Link , Question Author : providence , Answer Author : Pedro

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