What are eigen values?

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.g. xyz) coordinate system, the Eigenvalues can be thought of as a kind of multiplier across the unit vectors (Eigenvectors) [[1,0,0][0,1,0][0,0,1]]. This is the "stretching" analog mentioned in the article.

FWIW, Eigenvalues are not just a salient property of linear systems (i.e. matrices), but also of higher-order tensors.

Finding (or more-often approximating) these "characteristic scaling states" is a critical step in numerical analysis in everything from quantum mechanics, financial hedging strategies, and even consumer product marketing plans.

If you've ever represented a system as a series of Markov probability chains, every row of the "convergent/dominant" (if any) state contains an Eigenvalue.

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 does the coordinate transformation to diagonalize your matrix.

3. I think the paragraph beginning "For the common 3D isometric ..." is rather confused. I certainly am when reading it. Perhaps the problem is in my brain; what exactly do you mean? (Here's the nearest true thing I can think of to what that paragraph says: For many, but not all, linear transformations from a space to itself, there is an orthogonal coordinate system with respect to which the transformation's matrix is diagonal; then the eigenvectors are the axes of that coordinate system, and the eigenvalues are the amounts by which vectors along those axes get stretched.)

4. Tensors are just as linear as matrices. (You can do nonlinear things with a tensor, but then you can with a matrix too.)

5. The probabilities in a Markov chain's stationary state are not eigenvalues. (Well, I'm sure they're eigenvalues of something; any set of numbers can be made the eigenvalues of something; but they aren't, e.g., eigenvalues of the transition matrix.) What you may be thinking of is: a stationary state of a Markov chain is an eigenvector of the transition matrix, with eigenvalue 1; all eigenvalues have absolute value at most 1; if the Markov chain is ergodic (i.e., can get from any state to any other), then there is exactly one stationary state, exactly one eigenvector of eigenvalue 1, and all the other eigenvalues are strictly smaller. This is enough to guarantee convergence to the stationary state.

Also: If eigenvalues are as important in dealing with higher-order tensors as they are for the second-order case (i.e., matrices) then that's news to me. Tell me more?

This is ridiculous. I was just thinking last week that even though I took linear and did fairly well I never really grasped eigenvalues and eigenvectors since I see them coming up fairly frequently as far as math concepts go and don't instantly have an intuitive grasp. Literally the next week I see this posted on hn.

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 the mean from this, so our co-ordinates come from the middle of this cloud.

I can take this set of (x,y,z) points and and find the eigen (which means "characteristic") vectors and values for this set of points. The first eigenvector will point along the long axis of the "football" (right through the pointy bits) because this is the strongest, in some sense, single direction that is represented. It won't be perfectly aligned with a finite number of points...

The second and third eigenvectors will between them form a plane perpendicular to this. So essentially what you've done is found a co-ordinate system that is aligned with the data (modulo some details, this is essentially what a procedure called Principal Component Analysis (PCA) does.

Now that's the vectors, what about the values? The eigenvalues tell you the relative importance of each direction. If you stretch the "football" out, the first eigenvalue, which was associated with the first eigenvector pointing along the long axis, will be larger relative to the others. If you use a basketball, they'll all be the same magnitude.

Does that help?

The idea generalizes a lot from what I've told you, but there are a ton of places you can look up the details if you'd like.

Do you mean, what are eigenvalues used for?

I found this to be an incredibly clear and simple description of what eigenvalues are. I completely missed the point in University. I agree that it was missing a discussion of what they're used for.

You might also want to look at the eigenvalue section of the paper "An Introduction to the Conjugate Gradient Method without the Agonizing Pain" http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradi...