Matrix completion problems over integral domains: The case with a diagonal of prescribed blocks

Abstract

Let R be an arbitrary integral domain, let ∧ = { λ 1 , … , λ n } be a multiset of elements of R , let σ be a permutation of { 1 , … , k } let n 1 , … , n k be positive integers such that n 1 + ⋯ + n k = n , and for r = 1 , … , k let A r ∈ R n r × n σ ( r ) . We are interested in the problem of finding a block matrix Q = Q rs r , s = 1 k ∈ R n × n with spectrum Λ and such that Q r σ ( r ) = A r for r = 1 , … , k . Cravo and Silva completely characterized the existence of such a matrix when R is a field. In this work we construct a solution matrix Q that solves the problem when R is an integral domain with two exceptions: (i) k = 2 ; (ii) k ≥ 3 , σ ( r ) = r and n r > n / 2 for some r . What makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains, which includes the important case of integers. Furthermore, we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix (we point out that Cravo and Silva’s proof is not constructive).