Eigenvector and Eigenvalue

• Motivation 1. Say a $n\times n$ matrix describes a 3-D rotation. Then what is the axis of rotation?
• Motivation 2. An $n\times n$ matrix describes a linear transformation, but looking at each column only shows you the action of the basis vectors. What is a better way to characterize this linear transformation?

def. Eigenvector and eigenvalue. For matrix $A$, if applying $A\vec{v}$ simply scales $\vec{v}$ into say, $\stackrel{\text{ scalar }}{\overbrace{\lambda }}\vec{v}$, then $\vec{v}$ is an eigenvector of matrix $A$. In other words:

$$A\stackrel{\text{ also eigenvector }}{\overbrace{\vec{v}}}=\underset{\text{ eigenvalue }}{\underbrace{\lambda }}\underset{\text{ eigenvector }}{\underbrace{\vec{v}}}$$

thm. For diagonal matrix $D$, all basis vectors are eigenvectors.