What is a vector Hankel determinant
type: Item
instance of: scholarly article
title: What is a vector Hankel determinant
zbMATH Open document ID: 0940.15006
DOI: 10.1016/S0024-3795(97)10091-X
published in: Linear Algebra and its Applications
publication date: 2000-07-17T00:00:00Z
review text: The aim of the paper under review is to give, under some assumptions, a necessary and sufficient condition that the following system \[ \sum_{j=1}^n x_j a_{i,j} = b_i, \] where \(i=1,\dots,n\), and the \(a_{i,j}\)'s and the \(b_i\)'s are in a euclidean vector space \(V\) of dimension \(n\), has one and only one solution. The author focuses his attention on Hankel system, namely \[ \begin{aligned} x_1 c_0 + x_2c_1+ \cdots +x_nc_{n-1} &= b_1,\\ x_1 c_1 + x_2c_2+ \cdots +x_nc_n &= b_2,\\ & \vdots\\ x_1 c_{n-1} + x_2c_n+ \cdots +x_nc_{2n-2} &= b_n, \end{aligned} \] where \(c_i\), \(i=0,\dots,2n-2\) and \(b_i\), \(i=1,\dots,n\) belong to \(V\). Such system is a vector analogue of the Padé approximation. The author constructs a vector Hankel determinant associated to the previous system \[ \Delta=\left|\begin{matrix} c_0&\cdots&c_{n-1}\\ \vdots& &\vdots\\ c_{n-1}&\cdots&c_{2n-2}\end{matrix}\right|. \] This construction is based on the designant; the right designant [\textit{A. Heyting}, Math. Ann. 98, 465--490 (1927; JFM 53.0120.01)] of a square matrix with elements in a division algebra is the last number to be obtained in a process of reduction to triangular form. The author shows that \(\Delta\) is an element of \(V\) (Theorem 1) and that if \(\Delta\) exists then the Hankel system has one and only one solution \(\bar{x}\in C(V)^n\), \(C(V)\) being the universal real Clifford algebra associated to \(V\), if and only if \(\Delta\) is different from zero (Theorem 2). Most of the proofs involve as a cornerstone some properties of designants that the author has already used in a previous paper [Numer. Algorithms 7, No. 2--4, 225--251 (1994; Zbl 0812.65003)]. The reader is invited to look at the corresponding review of \textit{P. Wynn} (Zbl 0812.65003) on this specific topic.
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 65B05
zbMATH DE Number: 1354776
zbMATH Keywords: designants
zbMATH Keywords: Clifford algebra
zbMATH Keywords: Clifford group
zbMATH Keywords: Hankel determinants
zbMATH Keywords: Hankel system
zbMATH Keywords: linear algebraic equations
zbMATH Keywords: vector epsilon algorithm
zbMATH Keywords: JFM 53.0120.01
zbMATH Keywords: division algebra
zbMATH Keywords: reduction to triangular form
author: Ahmed Salam
MaRDI profile type: Publication
cites work: Q5775371
cites work: Extrapolation methods
cites work: Lanczos-type algorithms for solving systems of linear equations
cites work: Q3919827
cites work: Les déterminants sur un corps non commutatif
cites work: Matrix representations of Clifford algebras
cites work: Q3901606
cites work: A note on the \(\epsilon\)-algorithm
cites work: Determinants of matrices over noncommutative rings
cites work: A theory of noncommutative determinants and characteristic functions of graphs
cites work: Non-commutative extrapolation algorithms
cites work: Q4311182
cites work: An algebraic approach to the vector \(\varepsilon\)-algorithm
cites work: Vector continued fractions
Recommended article: On vector Hankel determinants
Recommended article: Unitary automorphisms of the space of Hankel matrices
Recommended article: Approximation and realization of Hankel matrices
Recommended article: Q3491385
Recommended article: On the reduction of the normal Hankel problem to two particular cases
Recommended article: Unitary automorphisms of the space of Hankel matrices. II: The case of even order
Recommended article: Teoremi sul determinante di Hankel.
Recommended article: Q3728159
Recommended article: A contribution to the normal Hankel problem
Recommended article: Solvability of the Hankel Determinant Problem for Real Sequences
description: scientific article; zbMATH DE number 1354776