Procrustes problems and inverse eigenproblems for multilevel block α ‐circulants

Summary Let n = ( n 1 , n 2 ,…, n k ) and α = ( α 1 , α 2 ,…, α k ) be integer k ‐tuples with α i ∈{1,2,…, n i −1} andn i ≥ 2 for all i = 1,2,…, k . Multilevel block α ‐circulants are ( k + 1)‐level block matrices, where the first k levels have the block α i ‐circulant structure with ordersn 1 , n 2 , … , n k ≥ 2 and the entries in the ( k + 1)‐st level are unstructured rectangular matrices with the same sized 1 × d 2( d 1 , d 2 ≥ 1 ) . 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( α 1 , n 1 ) > 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. Whend 1 = d 2 ≥ 1 , 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.