An eigenvector of a
matrix A is a
vector x such that:
Ax = λx
For some scalar λ, which is called the eigenvalue. That is, x does not change direction when multiplied by A. As an example case, consider the matrix:
[ 2 1 ]
[ 0 3 ]
The eigenvalues of this matrix are 2 and 3. For the first eigenvector, corresponding to eigenvalue 2, rearrange the above equation to:
(A - λI)x = 0
Which is just finding the null space of A-λI. That matrix is:
[ 0 1 ]
[ 0 1 ]
A basis for the null space of this matrix, and thus the eigenvalue, is:
[ 1 ]
[ 0 ]
Do the same for eigenvalue 3 to get:
[ 1 ]
[ 1 ]
These form a basis for the column space of A. In general, the eigenvectors will always form this basis, so long as A is non-degenerate.
For further information, see MIT's OpenCourseWare (http://ocw.mit.edu/18/18.06/f02/index.html), which contains a fantastic set of video lectures on the subject.