Eigenvalues and inverse of a Toeplitz matrix

Given the Toeplitz matrix

$$X = \begin{pmatrix} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0 ~~~~0 ~~~~0 ~~~~\textbf{d}~~~~ \\ ~~~~\textbf{c} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0 ~~~~0 ~~~~0~~~~ \\ ~~~~\textbf{d} ~~~~\textbf{c} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0 ~~~~0~~~~ \\ ~~~~0 ~~~~\textbf{d} ~~~~\textbf{c} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0~~~~ \\ ~~~~0 ~~~~0 ~~~~\textbf{d} ~~~~\textbf{c} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d}~~~~ \\ ~~~~0 ~~~~0 ~~~~0 ~~~~\textbf{d} ~~~~\textbf{c} ~~~~\textbf{1} ~~~\textbf{c}~~~~ \\ ~~~~\textbf{d} ~~~~0 ~~~~0 ~~~~0 ~~~~\textbf{d} ~~~~\textbf{c} ~~~~\textbf{1}~~~~ \\ \end{pmatrix}$$

where $$d$$ and $$c$$ are different values between $$-1$$ and $$0$$. Furthermore,

$$2c + 2d = -1$$

My questions are to find expressions for

1. Second largest eigenvalue modulus (SLEM) or eigenvalues

2. Inverse of $$X$$

Thanks

• $c$ and $d$ will have to be in the interval $[-\frac{1}{2}, 0]$ in order the have the relation $2c+2d=-1$. – PAD Oct 10 '12 at 10:06

$1-\sqrt{2} c$ is the second largest eigenvalue. For $c=-\frac{1}{2}$ it gives the second largest modulus. According to maple this is the determinant: $$(2d-1)(-1+2c^2)(2c^4-4c^2+2dc^2-4d^3+2d-2d^2+1) .$$