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What is an eigenvalue?

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Re: What is an eigenvalue?

#161

Post by kridsdale2 »

mturmon wrote: post #141
c-baby wrote: post #79

Earlier quoted context omitted.

Maybe I'm missing what's interesting about this, but a function like f(z) = 5z + 2 would output a wave with changed amplitude and phase when z = sin(x). That doesn't seem that interesting to me, so f(z) must have some other interesting properties?

I think what the comment is missing is that the class of linear f(z)'s is much larger than you might expect. Partly because it just is, and partly because we like it that way. In the land of analog signal processing: any combination of capacitors, inductors, and resistors [ https://soundcertified.com/wp-content/uploads/2020/04/speake... ] is linear. In the land of math abstractions of signal processing: differentiati…

Heavy Metal ruined the clean perfection of our fancy math devices, you say? That's pretty metal.

Re: What is an eigenvalue?

#162

Post by kridsdale2 »

docfort wrote: post #143
c-baby wrote: post #124

Earlier quoted context omitted.

Ok, but what is being asked is what interesting properties does such a function have. You still haven’t answered that question.

Let me try. Let’s say you have some operator F = crazyweirdblackbox. Like, it exponentiates and does a bunch more complicated things. You can apply this operator to some input function and transform it into some output function. The programming analogy would be some piece of code that works on a lambda and spits out another lambda. The finite-dimensional linear algebra equivalent is a matrix. You would ideally want F…

It's been 15 years since I was in my EE degree, and I learned all this stuff to the point of intuition. Since then I've been a C-monkey at megacorp and not doing any real math beyond simple geometry.

Thanks for tickling the old neurons.

Re: What is an eigenvalue?

#163

Post by kridsdale2 »

pclmulqdq wrote: post #146
kiernanmcgowan wrote: post #114

Eigen-things can also be thought of "fixed values" of a "thing" transformation. For example - the eigenfunction of a derivative is e^x since when you run the derivative function on e^x you get.... e^x

This is the best explanation of the eigenthings in my opinion. Too many people, this article included, reach for more advanced concepts to explain something as simple as "eigenvectors are how an operator transforms stuff and eigenvalues are how much."

God knows how I passed 4 years of engineering math without ever being able to describe to anyone what eigen* actually are, but I think that's actually the norm.

Re: What is an eigenvalue?

#164

Post by kridsdale2 »

vmilner wrote: post #86
raydiatian wrote: post #82

Earlier quoted context omitted.

Literally all you need.

I think anyone starting a lin alg course could do a lot worse than watch all his "Essence of Linear Algebra" series before starting - then watch the relevant (c. 15 min) episodes as you take each lecture.

My grades in undergrad would have been so much better if Youtube had existed. Oh well.

Re: What is an eigenvalue?

#165

Post by pclmulqdq »

kridsdale2 wrote: post #163
pclmulqdq wrote: post #146

Earlier quoted context omitted.

This is the best explanation of the eigenthings in my opinion. Too many people, this article included, reach for more advanced concepts to explain something as simple as "eigenvectors are how an operator transforms stuff and eigenvalues are how much."

God knows how I passed 4 years of engineering math without ever being able to describe to anyone what eigen* actually are, but I think that's actually the norm.

That's completely normal and very unfortunate. I only learned about the geometry of the eigen*s from an EE professor who loved math.

Unfortunately, the history of the eigen*s is rooted in the opaque and cryptic (but rigorous) linear algebra definition, and the non-rigorous but meaningful geometry came later.

Re: What is an eigenvalue?

#166

Post by raydiatian »

kridsdale2 wrote: post #164
vmilner wrote: post #86

Earlier quoted context omitted.

I think anyone starting a lin alg course could do a lot worse than watch all his "Essence of Linear Algebra" series before starting - then watch the relevant (c. 15 min) episodes as you take each lecture.

My grades in undergrad would have been so much better if Youtube had existed. Oh well.

One day, when 13 year olds can use VR to meaningfully bootstrap their understanding of differential equations, I’ll probably wish the metaverse was around when I was a kid.

Re: What is an eigenvalue?

#167

Post by sverona »

OJFord wrote: post #88
sverona wrote: post #57

Earlier quoted context omitted.

In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.

Or rather size-of-their-domain-dimensional?

The GP kind of implied functions R->R, but yes.
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