The eigenvalue decomposition conjecture for Toeplitz matrices generated by $g_{r,s}$

Let $g_{r,s}(\theta)$ be the symbol defined in the paper, with $1\leq r\leq s$, and let $T_n(g_{r,s})$ be the associated Toeplitz matrix. Write $\lambda_j(T_n(g_{r,s}))$ for its eigenvalues, and let $\lambda_+(T_n(g_{r,s}))$ denote the subset of these eigenvalues that are real and positive. Set $\gamma=\operatorname{gcd}(r,s)$, and let $\omega$, $\beta_\gamma$, $n_\gamma$, $r_\gamma$, and $s_\gamma$ be the parameters defined in the source.

The eigenvalue decomposition conjecture. The positive eigenvalues satisfy

$$\lambda_+\left(T_n(g_{r,s})\right)^\omega = \left(\bigcup_{k=0}^{\gamma-\beta_\gamma}\lambda_+\left(T_{n_\gamma}(g_{r_\gamma,s_\gamma})\right)\right)\bigcup\left(\bigcup_{k=0}^{\beta_\gamma}\lambda_+\left(T_{n_\gamma+1}(g_{r_\gamma,s_\gamma})\right)\right).$$

Moreover, the eigenvalues of the smaller Toeplitz matrices on the right are exactly those of the corresponding matrices $B_{(n_\gamma)_\sigma}^{n_\gamma,r_\gamma,s_\gamma}$ and $B_{(n_\gamma+1)_\sigma}^{n_\gamma+1,r_\gamma,s_\gamma}$, respectively, possibly with multiplicity. The full spectrum of $T_n(g_{r,s})$ is obtained from $\lambda_+(T_n(g_{r,s}))$ by $\omega$ rotations, together with $n_0$ zero eigenvalues.

The conjecture gives a recursive description of the positive real spectrum and a construction of the full spectrum from it. The article presents numerical examples supporting the claim and provides a proposed algorithm for computing the relevant smaller matrices; its resolution is not established in the supplied text.