What exactly are eigen-things?

Question

[Wikipedia](http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) defines an eigenvector like this: *An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at most by a multiplicative scalar.* So basically in layman language: An eigenvector is a vector that when you multiply it by a square matrix, you get the same vector or the same vector multiplied by a scalar. There are a lot of terms which are related to this like eigenspaces and eigenvalues and eigenbases and such, which I don't quite understand, in fact, I don't understand at all. Can someone give an explanation connecting these terms? So that it is clear what they are and why they are related.

Answer 1

The relation Av=cv means the vector Av lies in subspace spanned by v. This is the reason of the word "eigen" that means "self" in German. So, ''eigenvector" means in "self-vector" in the sense that when you multiply by A the matrix multiplication leaves the vector in the same space generated by v.

Same for "eigenvalue" it means "self-value". For this reason the eigenvectors depend on the eigenvalues.

Best,

Tomás