## So... what actually *is* an “eigenfunction”, anyway?

Imagine we had a magical machine that takes in mathematical equations, does a $\frac{d}{dt}$ to them, and then spits it out:

$$x(t) \to \color{Blue}{\frac{d}{dt}} \to y(t)$$

You might be tempted to write $y(t) = x'(t)$ or maybe even $y = x'$ or $y(t) = \frac{d}{dt}(x(t))$, but the first thing you want to correct is that attitude 😉 We're *not* actually talking about x(t) or y(t) at all. We're talking about *the machine*, $\frac{d}{dt} [ {\color{Red}\star} ] $. That's our actual object of focus.

What can we *do* with this machine? Well as any good scientist would do, first we probably want to test a couple different inputs to see what just *passes through* the machine unharmed, like how water passes through a turbine unharmed.

$$x(t) = t \to \color{Blue}{\frac{d}{dt}} \to y(t) = \frac{d}{dt} [ t ] = 1$$

That's not it.

$$x(t) = t^n \to \color{Blue}{\frac{d}{dt}} \to y(t) = \frac{d}{dt} [ t^n ] = n t^{n-1}$$

That's not it either. Hm. This might be tougher than I thought.

$$x(t) = \alpha t^n + \beta t^m \to \color{Blue}{\frac{d}{dt}} \to y(t) = \frac{d}{dt} [ \alpha t^n + \beta t^m ] = n \alpha t^{n-1} + m \beta t^{m-1} $$

Okay, that's also not it — but that's kind of interesting. Somehow, this system's output *scales up and down exactly* with an input that scales up and down. It also *preserves addition*. It's what we would call **linear**.

Let's try something that ain't a polynomial.

$$x(t) = e^t \to \color{Blue}{\frac{d}{dt}} \to y(t) = e^t \color{Red}{= x(t)}$$

Success! $e^t$ passes through our machine unchanged.

***

Eigenfunctions are the *tiniest* bit more abstract than that. They are things you throw into the machine, and they pass out, unchanged except for the *scaling*. We do have to be a bit careful with that word, however.

Not really in the sense of letting $x(t) = e^t$ and then doing

$$2000 \cdot x(t) \to \color{Blue}{\frac{d}{dt}} \to y(t) = 2000e^t \color{Red}{= 2000 \cdot x(t)}$$

because we actually get *that* through linearity anyway, no need for a second special German name for it.

But suppose instead we did $x(t) = e^{5t}$. In that case,

$$x(t) = e^{5t} \to \color{Blue}{\frac{d}{dt}} \to y(t) = 5 \cdot e^{5t} \color{Red}{= 5 \cdot x(t)}$$

which is something we *don't* get “for free” from linearity. You can't do that with $t^{5n}$; your output is gonna be something times $t^{5n-1}$, which is a very different thing than $t^{5n}$!

***

Okay, big whoop. $e^{st}$ passes through the machine and is just $s \cdot e^{st}$ after, itself scaled up or down. Why do we care?

See if you can pick up what I'm putting down here:

- We have this pretty simple little thing that passes through the machine unharmed.
- The machine is kind of hard to predict for completely arbitrary things – not things like $t^5$ or even $sin(cos(tan(t)))$, although I couldn't figure out the second one anyway, but maybe even
*randomly changing*functions. Think like, an earthquake sweep, or a noisy reading from a digital thermometer. - But wait – if we can find a way to
**take things apart**, and**reconstruct them**as some combination of the simple little thing, we might be able to figure out what the machine will do to a lot more stuff than we would otherwise.

It turns out that we *can* in fact do that. You can actually take apart a whole bunch of different signals, reconstruct them as $e^{\alpha t} + e^{\beta t} + \dots$ , run *those* through the machine, and actually be able to predict the output, because all you're doing is **running a bunch of eigenfunctions, (things which pass through unchanged except for the scaling), through a machine, that preserves additivity**.

I'm sweeping a bunch of stuff under the rug here – things like convolutions, and Fourier transforms, and z-transforms and stuff – but that's the basic jist. One really interesting little catch is that in order to *really* represent everything we want to, we need to let $\alpha, \beta, etc \in \mathbb{C}$. That is to say, we need to let them be **complex numbers**.

But they'll teach you that in class. Or, rather, they'll talk about it in class, and then you'll figure it out when you do some homework problems. Good luck! ♥