On a special generalized Vandermonde matrix and its LU factorization (Q1012917)

The authors consider a special generalized Vandermonde matrix of the form \[ V_{\{2;1,n-1\}}=\left[\begin{matrix} 1 & v_1 & v_1^2 & \cdots & v_1^{n-1}\\ 1 & v_2 & v_2^2 & \cdots & v_2^{n-1}\\ 0 & v_2 & 2v_2^2 & \cdots & (n-1)v_2^{n-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & v_2 & 2^{n-2}v_2^2 & \cdots & (n-1)^{n-2}v_2^{n-1} \end{matrix}\right], \] in which \(n\geq 2\) is a positive integer and \(v_1,v_2\) are two distinct points. They give an explicit formula of the LU factorization of \(V_{\{2;1,n-1\}}\) and express the factors \(L\) and \(U\) as products of 1-banded matrices. The latter result enables one to obtain a 1-banded factorization of \(V_{\{2;1,n-1\}}\) and thus a closed-form formula of the inverse \(V_{\{2;1,n-1\}}^{-1}\).