The eigenvectors and diagonalizability of HST-matrices (Q1847347)

An \textit{HST-matrix} is an \(n\times n\)-matrix of the form \[ A_h=\begin{pmatrix} h & k_1 & k_2 & k_3 & \cdots & k_{n-2} & k_{n-1}\\ k_1& h & k_2 & k_3 & \cdots & k_{n-2} & k_{n-1}\\ k_1& k_2& h & k_3 & \cdots & k_{n-2} & k_{n-1}\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots &\vdots \\ k_1& k_2 & k_3 & k_4 & \cdots & k_{n-1} & h \end{pmatrix}, \] where \(h\), \(k_1,k_2,\ldots,k_{n-1}\in\mathbb C\). The author fills up a gap in a paper by A. A. Shah where eigenvalues of these matrices were computed. The question of diagonalizability of \(A_h\) is considered also: it is shown that \(A_h\) is diagonalizable iff \[ \sigma+k_i\neq0, \quad i=1,2,\dots,n-1 \] where \(\sigma=k_1+k_2+\cdots+k_{n-1}\).