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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.

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  • $\begingroup$ Do you know about the real Jordan form? That makes clear your intuition. en.wikipedia.org/wiki/Jordan_normal_form#Real_matrices $\endgroup$
    – Lee Mosher
    Jul 30, 2014 at 18:00
  • $\begingroup$ I know it, but I do not have it geometric interpretation clear. $\endgroup$
    – user39115
    Jul 30, 2014 at 18:02
  • $\begingroup$ I may write an answer later but here is a good geometric description: math.purdue.edu/~bkrummel/ma265_lecture5_5.pdf $\endgroup$
    – Al.G.
    Jul 31, 2021 at 7:27
  • $\begingroup$ @Al.G. the link is dead, do you remember the title of the note? $\endgroup$ Nov 27, 2023 at 16:55
  • $\begingroup$ Sadly not, but there are other resources that explain the same. A quick googling lead me here: services.math.duke.edu/~jdr/1617f-1553/materials/11-09-web.pdf, which kinda explains what I was thinking about - a pair of complex eigenvalues means rotation by the complex value argument within a particular subspace (in contrast to real eigenvalues, which encode scaling of the axes). $\endgroup$
    – Al.G.
    Nov 27, 2023 at 17:10

1 Answer 1

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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 $aw+bv$.

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  • $\begingroup$ I think you want "A transforms" instead of "a transforms". Other than that, I really like this. $\endgroup$ Jul 30, 2014 at 18:40
  • $\begingroup$ I m getting some more of intuition, but I still lacking the geometric clear picture... $\endgroup$
    – user39115
    Jul 30, 2014 at 18:50

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