Examples of the physical significance and importance of matrix diagonalization and eigenvalues for first year undergraduates? To a student of physics, who is only exposed to the techniques of mathematical physics and read classical mechanics at the undergraduate level, but not quantum mechanics yet, how can we explain the importance and significance of matrix diagonalization/similarity transformation, eigenvalues, etc.? How can I introduce the idea of matrix diagonalization in a natural way, in some physical context, not as an abstract mathematical procedure?
 A: This is not a physics application per se, but I remember that as a first year undergrad I was convinced in the usefulness of matrix diagonalization by the application to obtain a closed expression for the Fibonacci sequence, as explained for example here: https://austinrochford.com/posts/2014-04-23-diagonalization-fibonacci.html
For me, it shows that I can use eigenvectors to easily obtain a non-trivial result. When this is shown one can be convinced that it might be useful in more physically-motivated cases as well.
Later note: Such transfer matrix methods have also widespread uses in physics. As an example, you can consider a particle moving on a 1D lattice with $N$ sites, where for each lattice point it could be in sublattice A or B (think of a ladder-like lattice). We assign probabilities for the particle to move from A/B in site $n$ to A/B in site $n+1$ by the matrix
$$P=\pmatrix{p_{AA} & p_{AB}\\ p_{BA} & p_{BB}}$$
What is then the probability for a particle starting at site A at $n=1$ to reach site B at $n=N$? It is simply
$(P^n)_{AB}$, which can easily be found by diagonalizing $P$.
