 Author
 Year
 2014
 Title
 A block Hankel generalized confluent Vandermonde matrix
 Journal
 Linear Algebra and its Applications
 Volume
 455
 Pages (fromto)
 3272
 Document type
 Article
 Faculty
 Faculty of Economics and Business (FEB)
Faculty of Science (FNWI)  Institute
 Amsterdam School of Economics Research Institute (ASERI)
Kortewegde Vries Institute for Mathematics (KdVI)  Abstract

Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, q x q say, in which case the ith column is given by u(z(i)), where we write u(z) = (1, z,..., z(q1))(T). If all the zi (i = 1,, q) are different, the Vandermonde matrix is nonsingular, otherwise not. The latter case obviously takes place when all z are the same, z say, in which case one could speak of a confluent Vandermonde matrix. Nonsingularity is obtained if one considers the matrix V(z) whose ith column (i = 1,..., q) is given by the (i  1)th derivative u((i1)) (z)(T). We will consider generalizations of the confluent Vandermonde matrix V(z) by considering matrices obtained by using as building blocks the matrices M(z) = u(z)w(z), with u(z) as above and w(z) = (1, z,...,z(T1)), together with its derivatives M(k) (z). Specifically, we will look at matrices whose ijth block is given by M(i+j) (z), where the indices i, j by convention have initial value zero. These in general nonsquare matrices exhibit a blockHankel structure. We will answer a number of elementary questions for this matrix. What is the rank? What is the nullspace? Can the latter be parametrized in a simple way? Does it depend on z? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix M(z) and the number of derivatives M(k) (z) that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of matrix polynomials is needed. (C) 2014 Elsevier Inc. All rights reserved.
Keywords
Author Keywords:Hankel matrix; Confluent Vandermonde matrix; Matrix polynomial
KeyWords Plus:INVERSION
 URL
 go to publisher's site
 Language
 English
 Permalink
 http://hdl.handle.net/11245/1.431612
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library, or send a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.