Symmetric functions and the Vandermonde matrix
type: Item
instance of: scholarly article
title: Symmetric functions and the Vandermonde matrix
zbMATH Open document ID: 1063.15023
author: Halil Oruç
author: Hakan K. Akmaz
published in: Journal of Computational and Applied Mathematics
publication date: 2004-10-12T00:00:00Z
review text: The authors discuss symmetric functions and related combinatorial numbers and their recurrences and relate this to the factorization of structured matrices, such as Vandermonde matrices. First definitions and properties are recalled for symmetric functions, defined as \[ \sigma_r(x_1,\ldots,x_n) = \sum_{1\leq i_1<\cdots<i_r\leq n} x_{i_1}\cdots x_{i_r}~~\text{and}~~ \tau_r(x_1,\ldots,x_n) = \sum_{\lambda_1+\cdots+\lambda_n=r} x_1^{\lambda_1}\cdots x_n^{\lambda_n}; \] the related Stirling numbers and \(q\)-Stirling numbers of the first and second kind are essentially obtained from these when \(x_i=i\), respectively \(x_i=q^{i-1}\) in \(\sigma_r\) and \(\tau_r\). Then it is shown how the symmetric functions appear in the expressions for the elements in the LDU factorization of a Vandermonde matrix and its inverse. Similarly, Stirling and \(q\)-Stirling numbers appear in factorizations of the corresponding Pascal and Stirling matrices. The properties of the symmetric functions are finally used to give a constructive proof for the fact that the \(L\) and the \(U\)-factors for the inverse of a Vandermonde matrix of size \(n+1\) can be written as the product of \(n\) bidiagonal matrices. This gives an \(O(n^2)\) algorithm for the solution of a Vandermonde system.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 2107316
zbMATH Keywords: Vandermonde matrix
zbMATH Keywords: symmetric functions
zbMATH Keywords: combinatorial numbers
zbMATH Keywords: triangular factorization
zbMATH Keywords: bidiagonal factorization
zbMATH Keywords: q-Stirling numbers
zbMATH Keywords: Pascal matrix
zbMATH Keywords: Stirling matrix
zbMATH Keywords: inverse matrix
zbMATH Keywords: LDU factorization
MaRDI profile type: Publication
full work available at URL: j.cam.2004.01.032
OpenAlex ID: W2156010507
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DOI: 10.1016/J.CAM.2004.01.032
description: scientific article; zbMATH DE number 2107316