On the multiplicative inverse eigenvalue problem

Abstract

Let A = ( a ij ) be an n-square matrix over an arbitrary field K, and let w 1,…, w n be elements of K. The following problem is well known: Under what conditions does there exist an n-square diagonal matrix D, over K, such that DA has eigenvalues w 1,…, w n ? Some solutions to the problem are known when K is the complex field C . It was proved by Friedland that if the principal minors of A are not zero, then the matrix D exists. The result that we present is the extension to an arbitrary algebraically closed field of the sufficient conditions established by Friedland.