What are eigenvalues and eigenvectors?

[orochimaru]

hi,
i have trouble understanding these two terms.
can anyone explain to me eigenvalues and eigenvectors in laymen terms?

Thks in advance!

 

[Galileo]

If you have a matrix A (or linear transformation, operator etc.) from the vector space V to itself acting no a vector v, then it will give another vector in the same space.
Generally this vector Ax will be some different vector, one that is linearly independent from v (it points in another direction). However if it is some scalar multiple of v (so $Av=\lambda v$ for some scalar $\lambda$ then v is called an eigenvector (the nullvector is ruled out as an eigenvector by definition) and $\lambda$ is its corresponding eigenvalue.

For example, if you take a vector in the plane R^2 and your linear transformation A is a rotation about the origin over 180 degrees, then every vector v will point in the opposite direction after the transformation, so Av=-v for all v. So every vector (not 0) is an eigenvector of A with eigenvalue -1.

 

[HallsofIvy]

You know, I presume, that any linear transformation can be written as a matrix so that applying the transformation to a vector is the same as multiplying the matrix and the vector.

Finding eigenvalues and eigenvectors is essentially finding for what vectors that matrix multiplication acts just like multiplying the vector by a number. It makes it possible to write the linear transformation as a sum of products of numbers,simplifying any problem involving that transformation.