# How to understsand eigenvalues and eigenvectors.

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?

• That is a chapter from a beginning book on linear algebra, and is expected to be a first look at eigenvectors and eigenvalues. So it is hard to recommend a different book. I recommend that you play around a bit more with them and their definitions, and try to formulate a precise question and ask it here if you remain stuck. May 28, 2014 at 19:36
• How much do you know about the geometric interpretations of linear algebra? Do you think of matrices as representing transformations on a vector space? Do you think of the determinant as the scale factor of volumes under that transformation? If those concepts are familiar, then perhaps I can recommend some references. May 28, 2014 at 19:45
• Oops...actually i think of matrices as convenient ways to represent coefficients in linear equations. I guess i need to start over from scratch. May 28, 2014 at 20:11
• It's best to actually think about matrices as ways of representing linear transformations. Using them to solve linear equations is sort of a neat trick that doesn't mesh as well with the rest of linear algebra. (Well, I think this because I think that thinking about linear algebra geometrically is the best way to learn it. If you think about linear algebra as the theory of solving systems of linear equations, then maybe thinking of matrices as ways of representing coefficients is better for you.) May 28, 2014 at 20:15
• Possible duplicate of What is the importance of eigenvalues/eigenvectors?
– user296602
Jun 2, 2016 at 23:50

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

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.