The invariance of the upper triangular factor of a Vandermonde matrix under shifts of nodes. (Q1878739)

This short note shows that for any \(x\), the shift \[ b_1\to x+b_1,\, b_2\to x+b_2,\,b_n\to x+b_n \] has no influence on the \(DU\) part of the \(LDU\) decomposition of the Vandermonde matrix \(V_n(b_1,\ldots, b_n)\) into the product \((P_n[x-1]S_n)D_nP^T_n\), where \(S_n\) is the Stirling matrix of order \(n\), \(D_n=\text{diag}\{1, 2!,\ldots, (n-1)!\}\), and \(P_n\) is a lower-triangular matrix defined by \[ (P_n)_{ij}=\left(\begin{matrix} i-1\\j-1\end{matrix}\right),\quad i\geq j. \]