Spectral characterizations and integer powers of tridiagonal 2-Toeplitz matrices

In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices \(\textbf{B}_n\) ${\mathbf{B}}_n$ . First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence \(\{\textbf{B}_n\}_n\) ${\{{\mathbf{B}}_n\}}_n$ , which is described via two eigenvalue functions of a \(2\times 2\) $2\times 2$ matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices \(\textbf{B}_n\) ${\mathbf{B}}_n$ of even order, that can be turned into a numerical effective scheme for the computation of all the entries of \(\textbf{B}_n^l\) ${\mathbf{B}}_n^l$ , n even and l positive and small compared to n. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained.