On the shape of the numerical range of some tridiagonal matrices

Abstract: In this paper, we study the elliptical shape of the numerical range of tridiagonal matrices of orders $3,4$ and $5$ with zero entries on the main diagonal, a positive constant value $a$ along the super-diagonal, and sub-diagonal entries are \begin{align*} \begin{cases} r, sr^2, r^3, sr^4,\ldots, r^{n-1} & \mbox{ if $n$ is even, and } r, sr^2, r^3, sr^4,\ldots, sr^{n-1} & \mbox{ if $n$ is odd}, \end{cases} \end{align*} where $n$ denotes the order of the matrix, $s>0$ and $r$ is a real number. We further investigate the flat portions on the boundary of the numerical range of these $n\times n$ matrices in the special case $r=-1$. Some results given in \cite{chien2011numerical} are obtained as a particular case of our results.