How to understsand eigenvalues and eigenvectors.

Question

I know basic linear algebra (what is a matrix, what is a determinant, what is a square matrix, what is an inverse of a matrix, how to add/sub/multiple matrices etc.) But I am finding the concept of eigenvalue and eigenvector extremely hard as per this: http://math.mit.edu/linearalgebra/ila0601.pdf

Are there any beginner books I should study before this paper?

Answer 1

If you prefer a visual explanation, this gif might help:

The transformation matrix $\begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix}$ preserves the direction of vectors parallel to $\begin{smallmatrix} 1 \\ 1 \end{smallmatrix} $ (in blue) and $\begin{smallmatrix} 1 \\ -1 \end{smallmatrix} $ (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. Notice that the blue vectors are scaled by a factor of 3. This is their associated eigenvalue. The violet vectors are not scaled, so their eigenvalue is 1.

http://en.wikipedia.org/wiki/File:Eigenvectors.gif

Answer 2

For a $n\times n$ matrix $A$, viewed as a linear transformation on a vector space of dimension $n$ , an eigenvector of $A$ is just a nonzero vector $v$ on which $A$ acts by scaling. Ie, $Av = \lambda v$ for some scalar $\lambda$. This scalar $\lambda$ is the eigenvalue associated to the eigenvector $v$.

Some matrices have lots of eigenvectors. For example, the $n\times n$ identity matrix has every nonzero vector as an eigenvector, all of them with eigenvalue 1. A diagonal matrix has every basis vector as an eigenvector (ie, if $A$ is diagonal w.r.t. a basis $\{v_1,\ldots,v_n\}$, then each $v_i$ is an eigenvector for $A$, and its eigenvalue is just $A_{i,i}$).

Some matrices have no eigenvectors. For example, any rotation matrix does not have an eigenvector, since it acts on every nonzero vector by rotation around the origin, so no vectors are scaled.

If you have some specific questions I could try to answer those.