Matrix Eigenvalues

Abstract

Given an $n\times n$ (real or complex) matrix A and an n-vector $\mathbf{x}$, suppose we construct the n-vector $A\mathbf{x}=:\mathbf{y}$, then A can be interpreted as a linear transformation which carries $\mathbf{x}$ to $\mathbf{y}$ in an n-dimensional space. In particular, if a non-zero vector $\mathbf{x}$ can be found such that A carries $\mathbf{x}$ to a collinear vector $\lambda \mathbf{x}$, i.e.

$$A\mathbf{x}=\lambda \mathbf{x}$$

then $\mathbf{x}$ is called an eigenvector of A and $\lambda$ its corresponding eigenvalue.