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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. 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. – J. M. Sep 7 '11 at 3:00
    
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. – Udara Sep 7 '11 at 6:09
    
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? – user34757 Jun 29 '12 at 16:36
    
@yiwei Welcome to math.SE! Please ask new questions through the Ask Question page, and not via answers to other questions. – Asaf Karagila Jun 30 '12 at 22:38

Edit. Presumably the matrix is real, otherwise the claim is false and it is easy to generate a counterexample by computer.

Let $A$ be a circulant matrix and $B$ be the anticirculant matrix such that $A$ and $B$ have identical first rows. Then $B=PA$ for some permutation matrix $P$ (more specifically, $P=1\oplus J_{n-1}$, where $J_{n-1}$ is the reversal matrix obtained by flipping $I_{n-1}$ from left to right). Hence $A$ and $B$ have identical singular values. However, circulant matrices and real anticirculant matrices (which are also real symmetric) are normal matrices, so they can be unitarily diagonalized and their singular values are the moduli of their eigenvalues. Therefore, the eigenvalues $A$ and $B$ have identical moduli. Finally, as real anticirculant matrices are real symmetric, $B$ has real eigenvalues. Hence the assertion.

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Thanks for your explanation. Is there any reference (book, research paper) which I can look into this proof? – Udara Sep 7 '11 at 6:08
    
I don't know about such references, sorry. – user1551 Sep 7 '11 at 7:31

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