New approaches for solving linear confluent Vandermonde systems and inverse of their corresponding matrices via Taylor's expansion

TL;DR

This paper introduces a new direct method using Taylor's expansion to derive explicit formulas for the inverse of confluent Vandermonde matrices, which are useful in various scientific fields.

Contribution

It provides the first explicit and compact formulas for inverting confluent Vandermonde matrices using a novel Taylor expansion approach.

Findings

Explicit inverse formulas for confluent Vandermonde matrices

Method demonstrated with multiple examples

Potential applications across mathematics and sciences

Abstract

In this paper, we present a novel method to compute an explicit formula for the inverse of the confluent Vandermonde matrices. Our proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit and compact formulas not exist in the large literature on this essential topic. Several examples are presented to highlight the results. Our main tools are some elementary basic linear algebra.