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What are eigen values?

physlink.com

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Re: What are eigen values?

#21

Post by JabavuAdams »

baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

Eigenvectors are "important" directions with respect to some linear transform (operator). The eigenvalues are constant scaling factors that relate input vectors (inputs to the operator) to output vectors.

Knowing that an operator merely scales (but doesn't rotate) certain input vectors is an important property to characterize it.

Imagine some matrix describing the stresses (strains?) in a continuous medium of material, about a certain point. If you can find special directions from that point where the forces are just scaling, then you know you're just dealing with tension/compression in those directions, not bending.

Same thing applies for fluid flows, it can be useful to find out where the flow is just speeding up / slowing down, but not twisting.

... helpful?

Re: What are eigen values?

#23

Post by eru »

maggie wrote: post #18
Oompa wrote: post #16

Earlier quoted context omitted.

This is the problem I had with Linear Algebra. The first half of the class felt like I was just being drilled definitions. But once all the definitions click, it's rather simple.

Honestly Linear Algebra is the next time you have to do a 'jump' in the regular sequence of math. You hit algebra, new concepts, you have to jump a little. Same with calculus. Linear algebra is that next time.

And it gets easier. Linear Algebra is no more complicated than calculus --- rather less so. But it tends to be more abstract.

Re: What are eigen values?

#24

Post by eru »

gjm11 wrote: post #13
bonsaitree wrote: post #11

Earlier quoted context omitted.

Yes. For a linear system (of any kind of equivalence) within an N-dimensional vector space, the Eigenvalues represent the scaling factors across those dimensions when the system's state is represented by an NxN sparse-diagonalized matrix (i.e. all values are 0 except for the main diagonal). Those non-zero values along the main diagonal are its Eigenvalues and its rows are Eigenvectors. For the common 3D isometric (e.…

I'm afraid there are several errors in that. 1. A linear mapping is not a "kind of equivalence" by any reasonable definition. For instance, the function that maps every vector to 0 is a linear mapping, and it has plenty of eigenvectors. (All with eigenvalue 0.) 2. The eigenvectors are not the rows of the diagonalized matrix. They are the rows (or columns, depending on just how you define things) of the matrix that do…

> 1. A linear mapping is not a "kind of equivalence" by any reasonable definition. For instance, the function that maps every vector to 0 is a linear mapping, and it has plenty of eigenvectors. (All with eigenvalue 0.)

Linear mappings are the homomorphisms between vector spaces. So they are a "kind of equivalence".

See for example http://en.wikibooks.org/wiki/Linear_Algebra/Definition_of_Ho...

Re: What are eigen values?

#26

Post by jimbokun »

Novash wrote: post #14
antiform wrote: post #5

Earlier quoted context omitted.

In linear algebra, for a given linear transformation (a certain kind of matrix which generally represents some operation) M, an eigenvalue of M is a scalar c such that given a non-zero vector v (called the eigenvector), Mv = cv, where multiplication on the left is matrix multiplication and multiplication on the right is multiplication of v by the scalar c. The definition is significant because it says that for a cert…

Look, I am good at Math. I even love Number Theory. But what you wrote scares me. Can I run away now? (I hope to one day be able to look at it and say 'my, that is so simple...' like I do with high school math)

That was the point of the "What are eigen values?" article. It was to give the intuition behind the scary mathematical definition.

I find this helpful, as it answers the all important questions 1) Why should I care? and 2) What is the basic problem that the scary math is trying to solve?

With answers to those questions in hand, you can return to the scary math and work out how it maps to those answers.

Re: What are eigen values?

#27

Post by srn »

Wow, that's the best explanation I've read. Very intuitive. Not to say I didn't know the original definition at some point but having such a concept tied to a concrete example makes it much easier to remember.

It makes me wish that most mathematical concepts were explained with some kind of real world analogy.

Re: What are eigen values?

#28

Post by gjm11 »

eru wrote: post #24
gjm11 wrote: post #13

Earlier quoted context omitted.

I'm afraid there are several errors in that. 1. A linear mapping is not a "kind of equivalence" by any reasonable definition. For instance, the function that maps every vector to 0 is a linear mapping, and it has plenty of eigenvectors. (All with eigenvalue 0.) 2. The eigenvectors are not the rows of the diagonalized matrix. They are the rows (or columns, depending on just how you define things) of the matrix that do…

> 1. A linear mapping is not a "kind of equivalence" by any reasonable definition. For instance, the function that maps every vector to 0 is a linear mapping, and it has plenty of eigenvectors. (All with eigenvalue 0.) Linear mappings are the homomorphisms between vector spaces. So they are a "kind of equivalence". See for example http://en.wikibooks.org/wiki/Linear_Algebra/Definition_of_Ho...

The zero mapping is a homomorphism. I would not call it a "kind of equivalence"; would you?

Re: What are eigen values?

#29

Post by CamperBob »

ska wrote: post #17
baddox wrote: post #2

After reading the article, I have but one question: What are eigen values? A quick lookup on Wikipedia reveals that they are used in linear algebra and, for example, matrix transformations. I think this article failed by trying to relate eigenvalues to everyday scenarios, similar to when beginning calculus books give examples involving total pressure on an underwater dam door or mass of a rod as examples for the usef…

Ok, others have tried a little bit of the math, but this is a difficult form for that. So here is a very intuitive (i.e. hand-wavy) way to think about the type of relationships we are talking about. Suppose you have a (finite) cloud of points in three dimensions. For example, a bunch of GPS measurements. Lets just imagine that they are roughly elliptical, stretched out like a football. For simplicity we'll subtract t…

So there's one part of the explanation on the page that I don't quite get: she uses the example of a coin being turned 360 degrees along some axis as leaving all possible vectors as eigenvectors:

"If you rotate a coin by 360 degrees you preserve all directions and so each direction is an eigenvector. Because no stretching has occurred, all of these eigenvectors have eigenvalue 1. Rotating the coin by 60 degrees destroys all directions and this transformation has no eigenvectors or eigenvalues at all."

But in the 60-degree case, why isn't the axis of rotation an eigenvector? Its direction isn't "destroyed," is it?...

Re: What are eigen values?

#30

Post by diiq »

I first encountered eigenmagic in machine learning --- we were interested in the eigenvectors of the adjacency matrix of a graph. When the space that the matrix lives in is so abstract, 'real world' examples don't make it any easier to visualize what's happening.

If you really want to understand them intuitively, I reccomend plotting a few hundred matrices and their respective eigen- values and vectors. When you feel like you can predict what the results will look like at each frequency, then the stories about rubber bands and shiny coins will make sense.

Or maybe you're faster, and the stories helped you make the leap --- in which case, ignore me!

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