eigenvalue and eigenvector of a matrix (and why we bother)

These 2 links give a good review on it:
http://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx
https://www.math.hmc.edu/calculus/tutorials/eigenstuff/

say we've a matrix A, if we can satisfy this:
A*v_e = lambda*v_e

v_e = eigen vector of matrix A
lambda = eigen value of matrix A

example
|  2  7 |   | -1 |         | -1 |
| -1 -6 | * |  1 | = -5 *  |  1 |

why do we even need this?
see http://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors
in a nutshell, it allows us to transform from standard basis, which is sometimes computationally intensive to a different basis to work in, one which simplifies the calculations necessary"