# Eigenvalues and eigenvectors

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

## Answers and Replies

Galileo
Homework Helper
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