TITLE: Geometric interpretation of complex eigenvalues
QUESTION [7 upvotes]: What is the geometric interpretation of complex eigenvalues?
For me it is clear that real eigenvalues of a matrix $A$ are associates to eigenvectors along which the matrix $A$ contracts or expands. Complex eigenvalues are associated intuitively (but not clearly) to me to eigenvectors along which the matrix $A$ rotates the space.

REPLY [4 votes]: If $A$ is real matrix and exists its (not real) eigenvalue $\lambda=a+bi$ and eigenvector $x=v+iw$ ($a,b \in \mathbb{R}$, $v,w$-real vectors), then:
$Ax=\lambda x$, so:
$A(v+iw)=\lambda(v+iw)$
$Av+iAw=av-bw+i(aw+bv)$
So $Av=av-bw$ and $Aw=ax+bw$. You can interpret this result this way: there exist $a,b \in \mathbb{R}$ and vectors $v,w$ that $A$ transforms $v$ to $av-bw$ and $w$ to $aw+bv$.