# 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?

• Characterization of endomorphisms becomes ez pz?
– user463026
Nov 18 '17 at 11:12
• @MolecularPrime ez pz??? Nov 18 '17 at 11:51
• @user42912 "easy peasy". Nov 18 '17 at 13:48

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 $$A \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} = \begin{pmatrix} F_{n+1} \\ F_n + F_{n+1} \end{pmatrix} = \begin{pmatrix} F_{n+1} \\ F_{n+2} \end{pmatrix}, \; A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}.$$ 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!

• This is how I was introduced to the concept - as a homework problem, a semester before anyone told me what an "eigenvalue" was. It was a great way to handle it, I immediately came up with my own word for it ("homothetic subspace") and started trying to generalize the method on my own. Surely computing powers of a matrix is by far the most fundamental and important application of diagonalization. Nov 18 '17 at 23:07

When addressing people who do not have the necessary mathematical prerequisites to understand the technical details, it is imperative that you locate the point of contact between their daily lives and the topic at hand. When you are successful, then it becomes impossible for them to question the relevance of your talk.

I would open with the famous movie of the Tacoma bridge suffering from wind induced oscillations which end with violent structural collapsed. I would mention that the problem could have been predicted and avoided through computation of the eigenvalues for the stiffness matrix.

I would mention signal analysis and digitalization of sound and images, a topic where Fourier analysis is central.

Time permitting I would mention research into the fundamental properties of matter, i.e., quantum mechanics, the study of which hinges on our understanding of unbounded linear operators, i.e., an extension of your topic.

Instead of technical details, explain that these topics can not be accessed, understood and advanced further, if they do not posses the mathematical skills that you can teach them.

• Very interesting answer. Do you know where I can find more about the stiffness matrix of this bridge? Nov 18 '17 at 12:02
• Sadly, no. I spoke of it in abstract terms, i.e. had the matrix been available and had we had the computing power, then the problem could have been avoided. In general, structures are modeled with a system of ordinary differential equations: $M\ddot{x} + D\dot{x} + Kx = F(t)$, where the state vector $x$ contains the positions of key points of the building, $M$ is a mass matrix, $D$ is a dampning matrix, $K$ is the stiffness matrix and $F$ is the driving force, i.e. the wind. It is just an expression of Newton's 3rd laws of motion. Nov 18 '17 at 12:37

When motivating the diagonalization of matrices, I like to emphasize the following points in order to build an intrinsic narrative of why we are doing such a thing:

1. A matrix $A$ is not a linear transformation---it is the representation of a linear transformation $T$ with respect to a given basis $\mathcal{B}$. Symbolically, $A = \mathcal{Mat}(T, \mathcal{B})$.

2. Given a different basis $\mathcal{B}'$, we get a different matrix representing this linear transformation, which is related by a change of basis matrix: $A = P^{-1} B P$ where $B = \mathcal{Mat}(T, \mathcal{B}')$ and $P = \mathcal{Mat}(Id, \mathcal{B}', \mathcal{B})$. For a non-trivial vector space, there are many possible bases we could choose and many possible matrix representations of a linear transformation. For a freshman in engineering/physics/..., connect this with the idea of choosing different coordinate systems when studying a physical problem. There's no "correct" coordinate system to use, only ones that are more convenient.

3. A natural question arises: what is the best basis we can pick to study the linear transformation $T$, i.e., in which basis is the matrix representation of $T$ simplest? Ask the students which matrices are easiest to multiply/invert/apply to vectors/etc. Invariably, this will be diagonal matrices. In particular, it is easier to see how a vector will be transformed under the effect of a diagonal matrix.

4. Thus our goal is to find a basis $\mathcal{B}'$ such that the matrix representation of $T$ is diagonal: $D = \mathcal{Mat}(T, \mathcal{B}')$. Again, this corresponds to finding the "best" coordinate system with which to study a problem.

5. Combining points (4) and (2), we arrive at $$A = P^{-1} D P.$$

6. After introducing the eigenvector equation $A \vec{v} = \lambda \vec{v}$ and $\det(A- \lambda I) =0$ and working a few examples, be sure to note that not all matrices are diagonalizeable!

A youtube video by 3blue1brown gives a nice graphical representation of diagonalization (and his entire "Essence of Linear Algebra" series makes excellent recommended viewing for students).

If you prefer to work with specific motivating examples (as opposed to showing how diagonalization of matrices is a natural instrinsic question to ask), considering finding the steady-state solution(s) of a Markov process. In particular, Google's classical PageRank algorithm is a $700B application of finding such a steady-state solution. When diagonalizing a matrix, we simply pick a coordinate system in which the corresponding linear endomorphism looks easy; we pick the eigenbasis. This is useful in physics, a basic example is the moment of inertia of a massive object. For an observer, in his coordinates the objects movement may be hard to grasp and describe, but in the coordinate system "of the object" the motion is easy to describe. The main application to diagonalizing a matrix that I can think of, aside from finding eigenvalues etc., is that you're rewriting the equations that form the matrix such that each equation depends only on one variable, thus making it easier to solve. There are a lot of applications, and what to talk about probably depends on their level of knowledge. Some examples • Very abstract motivation is to try to classify different matrices using similarity relation. • Diagonalization is a useful tool when studying linear ODEs such as$\dot{\vec{x}}=A\vec{x}$. • The theory of quadratic forms$\vec{x}^{T}A\vec{x}$becomes very simple with the diagonalization of the corresponding (symmetric) matrices. and there are of course many more to talk about. 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.