# What are some applications of elementary linear algebra outside of math?

I’m TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I’m looking for applications of elementary linear algebra outside of mathematics that I might talk about in discussion section.

In our class, we cover the basics (linear transformations; matrices; subspaces of $\Bbb R^n$; rank-nullity), orthogonal matrices and the dot product (incl. least squares!), diagonalization, quadratic forms, and singular-value decomposition.

Showing my ignorance, the only application of these I know is the one that was presented in the linear algebra class I took: representing dynamical systems as Markov processes, and diagonalizing the matrix involved to get a nice formula for the $n$th state of the system. But surely there are more than these.

What are some applications of the linear algebra covered in a first course that can motivate the subject for students?

I was a teaching assistant in Linear Algebra previous semester and I collected a few applications to present to my students. This is one of them:

This algorithm is the “heart” of the search engine and sorts documents of the world-wide-web by their “importance” in decreasing order. For the sake of simplicity, let us look at a system only containing of four different websites. We draw an arrow from $i$ to $j$ if there is a link from $i$ to $j$. The goal is to compute a vector $\underline{x} \in \mathbb{R}^4$, where each entry $x_i$ represents the website’s importance. A bigger value means the website is more important. There are three criteria contributing to the $x_i$:

1. The more websites contain a link to $i$, the bigger $x_i$ gets.
2. Links from more important websites have a more relevant weight than those of less important websites.
3. Links from a website which contains many links to other websites (outlinks) have less weight.

Each website has exactly one “vote”. This vote is distributed uniformly to each of the website’s outlinks. This is known as Web-Democracy. It leads to a system of linear equations for $\underline{x}$. In our case, for

$$P = \begin{pmatrix} 0&0&1&1/2\\ 1/3&0&0&0\\ 1/3& 1/2&0&1/2\\ 1/3&1/2&0&0 \end{pmatrix}$$

the system of linear equations reads $\underline{x} = P \underline{x}$. The matrix $P$ is a stochastical matrix, hence $1$ is an eigenvalue of $P$. One of the corresponding eigenvectors is

$$\underline{x} = \begin{pmatrix} 12\\4\\9\\6 \end{pmatrix},$$

hence $x_1 > x_3 > x_4 > x_2$. Let

$$G = \alpha P + (1-\alpha)S,$$

where $S$ is a matrix corresponding to purely randomised browsing without links, i.e. all entries are $\frac{1}{N}$ if there are $N$ websites. The matrix $G$ is called the Google-matrix. The inventors of the PageRank algorithm, Sergey Brin and Larry Page, chose $\alpha = 0.85$. Note that $G$ is still a stochastical matrix. An eigenvector for the eigenvalue $1$ of $\underline{x} = G \underline{x}$ in our example would be (rounded)

$$\underline{x} = \begin{pmatrix} 18\\7\\14\\10 \end{pmatrix},$$

leading to the same ranking.