Eigenvalues and Eigenvectors

What are they and why are they interesting

Etymology

The word "eigen" is supposed to mean "characteristic" or "peculiar" in German and Dutch. In linear algebra, it means characteristic vector or value. I will explain.

Eigenvector

Geometrically, a vector can be transformed by multiplying it with another vector. Depending on the multiplying vector, the result can be rotation, scaling, shear in linear transformation. There are, however, some vectors that stay on the same plain other than the fact that they are only scaled one way or the other. These are called "eigenvectors" meaning that they are scale vectors retaining the same characteristics of the original vector.

Algebraically, an eigenvector v is expressed as:

$$Av = \lambda v$$

where A is an n-dimensional array and lambda is the eigenvalue.

(to be continued...)