The eigenvalues of \(r\)-periodic tridiagonal matrices by factorization of some recursive sequences
The authors claim the introduction of the concept of an \(r\)-periodic tridiagonal matrix and, partially, the spectral analysis for the cases when \(r=2,3,4\). However, the notion of tridiagonal \(k\)-Toeplitz matrix as it is known in the literature, an \(n\times n\) complex tridiagonal matrix \(A=(a_{ij})\) such that \[ a_{i+k,j+k}=a_{ij}\, , \quad \text{for \(i,j=1,2,\ldots,n-k\)}\, , \] was first studied by \textit{M. J. C. Gover} and \textit{S. Barnett} [IMA J. Numer. Anal. 5, 101--110 (1985; Zbl 0561.65016); J. Math. Anal. Appl. 123, 297--305 (1987; Zbl 0623.15017)]. The spectra of these matrices for \(k=2\) were found by \textit{M. J. C. Gover} [Linear Algebra Appl. 197--198, 63--78 (1994; Zbl 0833.15009)]. The general case was first established by the reviewer and \textit{J. Petronilho} [Linear Algebra Appl. 325, No. 1--3, 7--21 (2001; Zbl 0983.15007); Numer. Math. 100, No. 3, 457--482 (2005; Zbl 1076.15011)]. Unfortunately, these references are ignored in the paper under review. Moreover, there are several errors of different nature as well.
scientific article; zbMATH DE number 6860282