What are eigen values and expansions?

Question

What are eigen values, vectors and expansions and as an algorithm designer how can I use them?

EDIT: I want to know how YOU have used it in your program so that I get an idea. Thanks.

Answer 1

check out http://mathworld.wolfram.com/Eigenvalue.html

Using eigen values in algorithms will need you to be proficient with the math involved. I'm absolutely the wrong person to be talking about math: I puke on it.

cheers, jrh.

Answer 2

Eigen values and vectors are used in matrix computation as finding of reverse matrix. So if you need to write math code, precomputing them can speed some operations.

In short, you need them if you do matrix algebra, linear algebra etc.

Answer 3

Eigen vectors and corresponding eigen values are mainly used to switch between different coordinate systems. This might simplify problems and computations enormously by moving the problem sphere to from one coordinate system to another.

This new coordinates system has the eigen vectors as its base vectors, i.e. they "span" this coordinate system. Since they can be normalized, the transformation matrix from the first coordinate system is "orthonormal", that is the eigen vectors have magnitude 1 and are perpendicular to each other.

In the transformed coordinate system, a linear operation A (matrix) is pure diagonal. See Spectral Theorem, and Eigendecomposition for more information.

A quick implication is for example that you can from a general quadratic curve:

ax^2 + 2bxy + cy^2 + 2dx + 2fy + g = 0

rewrite it as

AX^2 + BY^2 + C = 0

where X and Y are counted along the direction of the eigen vectors.

Cheers !

Answer 4

they're used for a lot more than matrix algebra. examples include:

• the asymptotic state distribution of a hidden markov model is given by the left eigenvector associated with the eigenvalue of unity from the state transition matrix.
• one of the best & fastest methods of finding community structure in a network is to construct what's called the modularity matrix (which basically is how "surprising" is a connection between two nodes), and then the signs of the elements of the eigenvector associated with the largest eigenvalue tell you how to partition the network into two communities
• in principle component analysis you essentially select the eigenvectors associated with the k largest eigenvalues from the n>=k dimensional covariance matrix of your data and project your data down to the k dimensional subspace. the use of the largest eigenvalues ensures that you're retaining the dimensions that are most significant to the data, since they are the ones that have the greatest variance.
• many methods of image recognition (e.g. facial recognition) rely on building an eigenbasis from known data (a large set of faces) and seeing how difficult it is to reconstruct a target image using the eigenbasis -- if it's easy, then the target image is likely to be from the set the eigenbasis describes (i.e. eigenfaces easily reconstruct faces, but not cars).
• if you're in to scientific computing, the eigenvectors of a quantum hamiltonian are those states that are stable, in that if a system is in an eigenstate at time t1, then at time t2>t1, if it hasn't been disturbed, it will still be in that eigenstate. also, the eigenvector associated with the smallest eigenvalue of a hamiltonian is the ground state of a system.

Answer 5

Using the notation favored by physicists, if we have an operator H, then |x> is an eigenstate of H if and only if

H|x> = h|x>

where we call h the eigenvalue associated with the eigenvector |x> under H.

(Here the state of the system can be represented by a matrix, making this math isomorphic with all the other expressions already linked.)

Which brings us to the uses of these things once they have been discovered:

The full set of eigenvectors of a system under a given operator form an orthagonal spanning set for they system. This set may be a basis if there is no degeneracy. This is very useful because it allows extremely compact expressions of arbitrary (non eigen-) states of the system.