I am looking for analytical expressions for the the singular values of a Toeplitz matrix. If possible for a general Toeplitz matrix but I would also take results for a tridiagonal Toeplitz matrix

\begin{equation} A = \begin{pmatrix} a & b \\ c & a & b \\ & \ddots & \ddots & \ddots \\ & & c & a\end{pmatrix} \end{equation}

While I have references for the eigenvalues of such a matrix, e.g. the book by Smith, I am struggling to find expressions for the singular values.

I am aware that if $c=b$ and $A$ is real and symmetric, the singular values are basically the eigenvalues (absolute values of the eigenvalues to be precise). But what if that is not the case?

Smith, G. D., Numerical solution of partial differential equations. Finite difference methods. 2nd ed, Oxford Applied Mathematics and Computing Science Series. Oxford: Clarendon Press. XII, 304 p. hbk: \textsterling 9.00; pbk: \textsterling 4.95 (1978). ZBL0389.65040.

  • $\begingroup$ Thanks for the reference! Seems interesting but at least at first glance, I did not see an expression for the singular values. I will take another look but may be you could point me to the right equation in the paper? $\endgroup$
    – Daniel
    Mar 27 '21 at 8:41
  • $\begingroup$ I misread. For singular value to equal the eigenvalues, isn't positive semidefiniteness required? $\endgroup$ Mar 27 '21 at 9:38
  • $\begingroup$ Yes, you may be right. But that is probably the case if a, b, c are non-zero. $\endgroup$
    – Daniel
    Mar 27 '21 at 10:23
  • $\begingroup$ Not so sure. You need symmetry and diagonal dominance. Think of Gershgorin. $\endgroup$ Mar 27 '21 at 11:00

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