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Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that, for any anti-circulant matrix, the eigenvalues (taken as $\mu$) of the anti-circulant matrix can be written as, \begin{equation} \mu = \pm \mid{\lambda_j}\mid \label{mu_alpha} \end{equation} where $\lambda_j$ is an eigenvalue of 1-circulant matrix with the same first row. This seems valid since any anti-circulant matrix should be symmetric resulting in real eigenvalues.

Can anyone send me a link to any reference which has this proof..? or can you please comment if you think that this should not be correct ?

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    [Please don't cross-post questions within minutes of each other](http://mathoverflow.net/questions/74703). Pick one (usually this one first), and try in the other if you haven't gotten answers after a good amount of time has passed.2011-09-07
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    Thanks for info.. J.M. I thought that these are two different communities, and posted the same at the same time since I need some urgent help with this proof.2011-09-07
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    A further question: If we have an irreducible and apperiodic circulant matrix $A$ with $N$ diffent rows. We can then define $N$ different anti-circulant matrices $\{B_n\}$ such that $B_n$ has $n-$th row of matrix $A$ as its first row. Q: Is there will exists at least one $B_n$ positive definite?2012-06-29
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    @yiwei Welcome to math.SE! Please ask new questions through the [Ask Question](http://math.stackexchange.com/questions/ask) page, and not via answers to other questions.2012-06-30

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