# A question concerning Toeplitz matrices

I am studying Toeplitz matrices. I have to find out the eigenvalues of the following Toeplitz matrix:

$$\begin{bmatrix} 2 & -8 & -24 \\ 3 & 2 & -8 \\ 1 & 3 & 2 \end{bmatrix}$$

Are there any different procedures to find out eigenvalues of Toeplitz matrices? Can't I use the general method of finding eigenvalues for them too? I need help with this. I need matlab code also. Edit: I was studying this paper http://www.sciencedirect.com/science/article/pii/0024379574900044

Thanks for help.

• I wouldn't call that a circulant matrix. In a circulant matrix, you get each row by pushing each entry in the row above it one place to the right, with the last entry in the upper row cycling around to become the first entry in the lower. Your example doesn't have that. What you have is actually a Toeplitz matrix, see en.wikipedia.org/wiki/Toeplitz_matrix – Gerry Myerson Jul 5 '12 at 3:30
• @GerryMyerson Can I add pdf file of paper that I am studying? – srijan Jul 5 '12 at 3:32
• @GerryMyerson I haven't studied them. I am sorry for my mistake. I am editing. – srijan Jul 5 '12 at 3:39
• The Wikipedia article says something about calculating LU-decompositions of Toeplitz matrices, that might be worth following up on. You might also try typing Toeplitz eigenvalue into a search engine to see what comes up. – Gerry Myerson Jul 5 '12 at 3:47
• For tri-diagonal Toeplitz matrices there is an explicit formula. See p. 6 of this paper arxiv.org/abs/1110.6620 – PAD Jul 5 '12 at 6:18

## 2 Answers

The following MATLAB code will find the eigenvalues of the matrix in your question:

A = toeplitz([2 3 1], [2 -8 -24]);
eigvalues = eig(A);


Of course you can use the same methods you would use for a non-Toeplitz matrix, and for something as small as this one there's no reason not to do so. You would only need a special algorithm for a matrix too large to handle by the usual methods. On the other hand, eigenvalues of large Toeplitz matrices can be numerically unstable, see e.g. http://people.maths.ox.ac.uk/trefethen/publication/PDF/1992_53.pdf.