On the eigenvalues of Toeplitz matrices with two off-diagonals
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Consider the Toeplitz matrix T_n(f) generated by the symbol f(θ)=f̂_r e^𝐢rθ+f̂_0+f̂_-s e^-𝐢sθ, where f̂_r, f̂_0, f̂_-s∈ℂ and 0<r<n, 0<s<n. For r=s=1 we have the classical tridiagonal Toeplitz matrices, for which the eigenvalues and eigenvectors are known. Similarly, the eigendecompositions are known for 1<r=s, when the generated matrices are “symmetrically sparse tridiagonal”. In the current paper we study the eigenvalues of T_n(f) for 1≤ r<s, which are “non-symmetrically sparse tridiagonal”. We propose an algorithm which constructs one or two ad hoc matrices smaller than T_n(f), whose eigenvalues are sufficient for determining the full spectrum of T_n(f). The algorithm is explained through use of a conjecture for which examples and numerical experiments are reported for supporting it and for clarifying the presentation. Open problems are briefly discussed.
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