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What is an eigenvalue?

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Re: What is an eigenvalue?

#111

Post by ravi-delia »

Waterluvian wrote: post #42

Something that frustrates me, and maybe I’m just confessing my stupidity, is the extra layer of indirection in any discipline when things are named after people and not the thing’s characteristics. My doctor once told me “if you learn enough Latin, a lot of names in medicine will hint at what they are, so you have less to memorize.” I find that these names often lend a sense of complexity to concepts that turn out to…

I recently embarked on a journey to come up with a math vocabulary for Toki Pona, a lovely little artistic conlang which deserves better than what I'm doing to it. In Toki Pona, words are build up from simpler ones to describe a thing as it is. A friend is 'jan pona', a person who is good (to me, the speaker). So I've had to come up with names which describe math topics.

It's awful.

You know how many same-xs there are?! Eigenvalue, eigenvector, homomorphism, isomorphism, homeomorphism, homotopic. Which one gets to actually be "same shape"? Worse are when well meaning mathematicians use descriptive names anyway. Open and closed are not mutually exclusive, giving rise to the awful clopen (and don't pretend like ajar helps. an ajar door is an open door!). Groups, rings, and fields all sort of bring to mind the objects they describe, but only after you know the archetypal examples. Math is the study of giving the same name to different things, and that gives rise to more names than there are short descriptions.

So do you know what I did? Whenever I could, I used a real person's name. It freed up a limited vocabulary, and gave enough wiggle room to translate most undergrad math without too much loss. I suspect a similar thing is in play with math. Maybe the category theory people have abstractions to usefully describe "same-functions" without confusion. But in general, things are named poorly because it's genuinely a hard task.

Re: What is an eigenvalue?

#112

Post by constantcrying »

ravi-delia wrote: post #106
OJFord wrote: post #88

Earlier quoted context omitted.

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

If you want, but you can do better. I believe, for instance, that at least continuous functions on the reals have a countable basis. Might even be as strong as measurable, not sure about that. That's how, for instance, fourier transforms work.

The fourier Transform essentially means that every L^2 (the square of the function has a finite integral) is completely "described" by an l^2 series (a series of numbers whose sum of squares converges), which is about the greatest piece of magic in mathematics. One very important piece here is that the term "function" is somewhat of a lie (since the result couldn't be true if it weren't).

The result for measurable functions (not almost functions) shouldn't be true (I think). I am not even sure it is true for L^1 almost functions.

Re: What is an eigenvalue?

#113

Post by lisper »

oifjsidjf wrote: post #102
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?

Honestly I forgot the details, but basical the ENTIRE field of DSP stands on this fact. Basicaly there exist some functions into which you can feed in sound waves and the output is guaranteed to still be a sound wave. If you'd feed in a sound wave and if the function would corrupt it you would not be able to do any digital signal processing, since the output must be a wave. Sound(wave) in -> Sound(wave) out, guarante…

> there exist some functions into which you can feed in sound waves and the output is guaranteed to still be a sound wave.

That in and of itself does not seem like a particularly insightful observation. It's just obvious that such functions exist. I can think of three of them off the top of my head: time delay, wave addition, and multiplication by a scalar. There must be something more to it than that.

Re: What is an eigenvalue?

#115

Post by canadianfella »

oifjsidjf wrote: post #102
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?

Honestly I forgot the details, but basical the ENTIRE field of DSP stands on this fact. Basicaly there exist some functions into which you can feed in sound waves and the output is guaranteed to still be a sound wave. If you'd feed in a sound wave and if the function would corrupt it you would not be able to do any digital signal processing, since the output must be a wave. Sound(wave) in -> Sound(wave) out, guarante…

No post body was provided.

Re: What is an eigenvalue?

#116

Post by constantcrying »

Waterluvian wrote: post #42

Something that frustrates me, and maybe I’m just confessing my stupidity, is the extra layer of indirection in any discipline when things are named after people and not the thing’s characteristics. My doctor once told me “if you learn enough Latin, a lot of names in medicine will hint at what they are, so you have less to memorize.” I find that these names often lend a sense of complexity to concepts that turn out to…

It is sometimes very hard to name things well. The name either becomes so unspecific that it is just as useless, or it gets so long that nobody will use it.

This gets worse the "deeper" the math goes, but for me it never was a real problem, as you usually learn the definition together with the name.

Re: What is an eigenvalue?

#117

Post by cogman10 »

constantcrying wrote: post #116
Waterluvian wrote: post #42

Something that frustrates me, and maybe I’m just confessing my stupidity, is the extra layer of indirection in any discipline when things are named after people and not the thing’s characteristics. My doctor once told me “if you learn enough Latin, a lot of names in medicine will hint at what they are, so you have less to memorize.” I find that these names often lend a sense of complexity to concepts that turn out to…

It is sometimes very hard to name things well. The name either becomes so unspecific that it is just as useless, or it gets so long that nobody will use it. This gets worse the "deeper" the math goes, but for me it never was a real problem, as you usually learn the definition together with the name.

You see this sort of thing crop up in chemistry.

For really simple compounds, names are more or less settled and consistent (with some exceptions).

But as soon as your compound starts to get more complex (think organic chemistry) all the sudden, it becomes nigh impossible to consistently name things. There are tons of compounds with the same chemical formula that are regionally named differently. Even worse, there are tons of compounds with the same chemical formula that are actually different things due to how the compound is arranged. (Good ole carbon chains).

Re: What is an eigenvalue?

#119

Post by marviel »

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

After watching the 3blueonebrown video linked below in the comments, I'm inclined to agree with you -- nice way of putting it.

Re: What is an eigenvalue?

#120

Post by alanbernstein »

contravariant wrote: post #68
alanbernstein wrote: post #27

Earlier quoted context omitted.

Complex exponentials are the eigenfunctions of the Fourier transform. In other words, frequency component values are the eigenvalues. https://en.m.wikipedia.org/wiki/Eigenfunction#Vibrating_stri...

That makes no sense, the Fourier transform of a complex exponential is a delta function.

Hmm, you're right, that should have been obvious. Thanks for the correction.
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