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What is an eigenvalue?

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Re: What is an eigenvalue?

#121

Post by brendanclune »

lisper wrote: post #113
oifjsidjf wrote: post #102

Earlier quoted context omitted.

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.

In math, the obvious things aren't always true and the true things are often not obvious.

Trivially, the identity f(x) = x satisfies the guarantee as well. What amounts to insightful observation is the definition and classification of these functions. In exploring their existence in various forms, we can begin to understand what properties these functions share.

So the interesting part is not that this class of function _exists_, because of course it does! Your intuition has led you to three possible candidates. But if we limit ourselves to only the functions that satisfy the condition _wave-in implies wave-out_ what do they look like as a whole? What do these guarantees buy us if we _know_ the result will be a wave? For example, f(g(x)) is also guaranteed to be _wave-in-wave-out_. Again, maybe obvious, but it's a building block we can use once we've proved it true.

Re: What is an eigenvalue?

#122

Post by Tainnor »

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

They don't have to be fixed, a scalar multiple is allowed too. e^ax is an eigenfunction of the derivative too.

Re: What is an eigenvalue?

#124

Post by c-baby »

brendanclune wrote: post #121
lisper wrote: post #113

Earlier quoted context omitted.

> 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.

In math, the obvious things aren't always true and the true things are often not obvious. Trivially, the identity f(x) = x satisfies the guarantee as well. What amounts to insightful observation is the definition and classification of these functions. In exploring their existence in various forms, we can begin to understand what properties these functions share. So the interesting part is not that this class of funct…

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

Re: What is an eigenvalue?

#125

Post by Tainnor »

jffry wrote: post #92
gmfawcett wrote: post #83

Earlier quoted context omitted.

Huh? It literally translates to "thought experiment" in English, which is exactly what it means.

I think you may have gotten whooshed by the joke - both "eigenvector" and "gedankenexperiment" are mashups of a German word and an English word

"Gedankenexperiment" is a fully German word, though, not a mashup.

Re: What is an eigenvalue?

#126

Post by Tainnor »

billfruit wrote: post #16

I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.

It used to be called "proper value" in English (you can still find that in old textbooks), but the (semi-)German word has basically entirely replaced it.

Re: What is an eigenvalue?

#127

Post by lp251 »

c-baby wrote: post #105
geysersam wrote: post #104

Earlier quoted context omitted.

It's even worse than you describe it! f needs to be linear, but the function in your example is not linear. However, there are quite interesting linear functions. Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds

5z + 2 is linear?

affine, not linear. describes a line that doesn't go through the origin. that pesky shift breaks linearity

5(2z) + 2 != 2(5z + 2)

Re: What is an eigenvalue?

#128

Post by bodhiandphysics »

cscheid wrote: post #32
bodhiandphysics wrote: post #29

Earlier quoted context omitted.

Also in computer science.. a web sites page rank is the eigenvalue of the connectivity matrix.

That's not true. The page rank is read from the eigenvector, and is the value associated with the given vertex (ie web page). There are as many page rank values as there are web pages, but only one eigenvector from which to read: the dominant eigenvector of the transition matrix, which is the one with the largest eigenvalue. So, only a single eigenvalue for the entire pagerank computation.

You’re right!!! Acch...

Re: What is an eigenvalue?

#129

Post by lisper »

c-baby wrote: post #124
brendanclune wrote: post #121

Earlier quoted context omitted.

In math, the obvious things aren't always true and the true things are often not obvious. Trivially, the identity f(x) = x satisfies the guarantee as well. What amounts to insightful observation is the definition and classification of these functions. In exploring their existence in various forms, we can begin to understand what properties these functions share. So the interesting part is not that this class of funct…

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

Yes, exactly.

This comment was downvoted into oblivion (I vouched it back from the dead), but I have no idea why. You hit the nail on the head.

Re: What is an eigenvalue?

#130

Post by rmeno12 »

Tainnor wrote: post #122
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

They don't have to be fixed, a scalar multiple is allowed too. e^ax is an eigenfunction of the derivative too.

In this case, we could say that all functions e^ax are eigenfunctions of the derivative, with the eigenvalues being a.
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