Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai, Geary, and Kadanoff

The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper confirms and makes more precise a conjecture by Dai, Geary, and Kadanoff of 2009 in a particular case.