Nonsymmetric Toeplitz matrices that commute with tridiagonal matrices (Q1898528)

A description of nonsymmetric Toeplitz matrices which commute with an irreducible tridiagonal matrix is presented. If \({\mathcal T}_s\) denotes the set of symmetric Toeplitz \(n\times n\) matrices which commute with an irreducible tridiagonal matrix, then for the sets \({\mathcal T}\) and \({\mathcal T}_H\) of Toeplitz and Hermitian Toeplitz \(n\times n\) matrices, respectively, which commute with an irreducible tridiagonal matrix the formulas \(T\in {\mathcal T} \Rightarrow T= D_\varepsilon T_s D_\varepsilon^{-1}\) for some \(\varepsilon\neq 0\) and \(T_H\in {\mathcal T}\Rightarrow T= D_\varepsilon Q_s D_\varepsilon^{-1}\) for some \(\varepsilon\), \(|\varepsilon|=1\), where \(T_s, Q_s\in {\mathcal T}_s\) and \(D_\varepsilon= \text{diag} (1, \varepsilon, \varepsilon^2, \dots, \varepsilon^{n- 1})\), are derived. A complete description of the set \(T\) for \(n= 2, 3, 4\) closes this paper.