Motivation behind matrix diagonalisation I'm going to give a 50 minutes lecture about matrix diagonalization for first year college students and I would like to give some applications of it. I've been thinking about saying the calculation of matrix exponential become simpler in a diagonal matrix, but this subject is a higher level for them and I have only 10-15 minutes to talk about it.

So what do you think I can say to motivate them to study matrix diagonalization?
One application which I think is quite interesting and basic is to calculate the closed form of a recurrence relation.

Let's say we want to find a closed form (Binet's formula) of the Fibonacci sequence $(F_i)_{i \in \mathbb{N}}$. For this we can look at the equation
$$
A \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} = \begin{pmatrix} F_{n+1} \\ F_n + F_{n+1} \end{pmatrix} = \begin{pmatrix} F_{n+1} \\ F_{n+2} \end{pmatrix}, \; A = \begin{pmatrix} 0 &amp; 1 \\ 1 &amp; 1 \end{pmatrix}.
$$
Applying this for $F_0 = 1$ and $F_1 = 1$ repeatedly we get
$$
\begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} = A^n \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \; n \in \mathbb{N}.
$$
We can diagonalize $A=T^{-1}DT$ with a diagonal matrix $D$ and get
$$
A^n = (T^{-1}DT)^n = T^{-1}DTT^{-1} DT \dots T^{-1}DT = T^{-1} D^n T.
$$
With simple calculations we get $D^n, T$ and $T^{-1}$, so we easily get $A^n$ and $A^n \begin{pmatrix}0 \\ 1 \end{pmatrix}$ gives Binet's formula.

I hope this is an interesting application.

Kind regards!
I'm a bit physics biased, but here's my two cents. Solving an eigenvalue problem is equivalent to diagonalizing an operator, think quantum mechanics, where the time independent Schroedinger equation is $$\hat{H}\psi=E\psi$$ Where $\hat{H}$ is an operator. To find the eigenstates of the system you diagonalize the operator $\hat{H}$, in the sense that you look for the eigenvalues.

More generally, I'd say the practical interest of diagonalization lies in finding the eigenvalues and eigenvectors of an operator. The eigenfunctions of the wave equation are practically the basis for Fourier series decomposition.

These are all generalizations of eigenstuff to operators on functional spaces, but even in finite dimensions, you can diagonalize a system of a given number of degrees of freedom in an even potential to get decoupled harmonic oscillators, you can diagonalize a matrix to find the principal inertial axis of a solid.

A physicist rarely spends a day without diagonalizing an operator, I imagine it's the same for a mathematician.