2025/09/30 by Calkin, Neil J., Corless, Robert M., Gonzalez-Vega, Laureano +2
#05C50 #34L15 #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2510.15919
Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let Sm[a,b] denote the set of all m× m symmetric matrices with entries in the real interval [a,b] and let Sm\a,b\ be the subset of Sm[a,b] of Bohemian matrices with population from only the extremal elements \a,b\. S. M. Fallat and J. J. Xing in 2012 proposed the following conjecture: the maximum spread in Sm[a,b] is attained by a rank 2 matrix in Sm\a,b\. X. Zhan had proved previously that the conjecture was true for Sm[-a,a] with a>0. We will show how to interpret this problem geometrically, via polynomial resultants, in order to be able to treat this conjecture from a computational point of view. This will allow us to prove that this conjecture is true for several formerly open cases.