Application of linear transformation and matrices

Question

I have been studying linear algebra for some time now, and I have seen some really interesting applications of linear transformation and matrices such as finding the integral $\int x^2e^x dx$. Does anyone have some other interesting or unexpected uses of linear algebra in analysis or any other field of mathematics?

Answer 1

One impressive example is the application of linear algebra to the study of Markov chains, which is a combination of probability and graph theory. In a nutshell, a Markov chain is a graph on which we "hop" between vertices. At any given moment, the probability of hopping from vertex $i$ to vertex $j$ is given by a (constant) probability $p_{i,j}$.

As it turns out, the "probability-matrix" $P$ whose $i,j$ entries are $p_{ij}$ encapsulates the behavior of a Markov chain very well. In particular, it turns out that the $i,j$ entry of $P^k$ is the probability that, in $k$ steps, we transition from vertex $i$ to vertex $j$.

An important result is that for a Markov chain (satisfying certain assumptions) with probability matrix $P$, the stationary distribution of the Markov chain can be computed as the eigenvector of $P^T$ corresponding to $\lambda = 1$. As it so happens, it is precisely this trick which is the main idea behind the PageRank algorithm (Google's first search algorithm, which is a big part of how the company became successful).