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Procrustes problems and inverse eigenproblems for multilevel block α‐circulants
Authors:Wei‐Ru Xu  Guoliang Chen  Yi Gong
Affiliation:1. Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, China;2. Fundamental Department, Shanghai Customs College, Shanghai, China
Abstract:Let n = (n1,n2,…,nk) and α = (α1,α2,…,αk) be integer k‐tuples with αi∈{1,2,…,ni?1} and urn:x-wiley:nla:media:nla2060:nla2060-math-0001 for all i = 1,2,…,k. Multilevel block α ‐circulants are (k + 1)‐level block matrices, where the first k levels have the block αi‐circulant structure with orders urn:x-wiley:nla:media:nla2060:nla2060-math-0002 and the entries in the (k + 1)‐st level are unstructured rectangular matrices with the same size urn:x-wiley:nla:media:nla2060:nla2060-math-0003. When k = 1, Trench discussed on his paper "Inverse problems for unilevel block α‐circulants" the Procrustes problems and inverse problems of unilevel block α‐circulants and their approximations. But the results are not perfect for the case gcd( α , n ) > 1 (i.e., gcd(α1,n1) > 1). In this paper, we also discuss Procrustes problems for multilevel block α ‐circulants. Our results can further make up for the deficiency when k = 1. When urn:x-wiley:nla:media:nla2060:nla2060-math-0004, inverse eigenproblems for this kind of matrices are also solved. By using the related results, we can design an artificial Hopfield neural network system that possesses the prescribed equilibria, where the Jacobian matrix of this system has the constrained multilevel α ‐circulative structure. Finally, some examples are employed to illustrate the effectiveness of the proposed results. Copyright © 2016 John Wiley & Sons, Ltd.
Keywords:procrustes problem  inverse eigenproblem  block circulant  discrete Fourier transform  Moore–  Penrose generalized inverse  AMS Classification: 15A18  65F18  15A24  15A57
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