Year: 2025
Type: article
Abstract: The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of two Hermitian matrices, given the eigenvalues of the summands.Let the summands be 𝑛 × 𝑛 matrices 𝐴 and 𝐵 with eigenvaluesIn 1962, A. Horn conjectured a set of sufficient and necessary constraints for ⃗ 𝛾 to be the spectrum of such a 𝐶.These all took the form of linear inequalities in the variables 𝛼 𝑖 , 𝛽 𝑗 , 𝛾 𝑘 .Some constraints are readily apparent, such asdue to the linearity of the trace, anddue to recasting the largest eigenvalue of a matrix 𝐴 as max{⃗ 𝑥 † 𝐴⃗ 𝑥 ∶ ∥⃗ 𝑥∥ = 1}.Other constraints are less obvious, but Horn's conjecture turned out to be true, though his list contained redundancies.Horn's inequalities all have the formwhere 𝐼, 𝐽, and 𝐾 are subsets of {1, ⋯, 𝑛} of the same cardinality.If 𝑆 is a subset of {1, ⋯, 𝑛} of cardinality 𝑟, we define a partition 𝜏(𝑆) as follows.First enumerate the elements of 𝑆 in order:Conjecture 1.1.Horn conjectured that the following list of inequalities was a necessary and sufficient set of constraints for 𝛾 to be the spectrum of a sum 𝐶 = 𝐴 + 𝐵: precisely those inequalities corresponding to 𝐼, 𝐽, 𝐾 for which 𝜏(𝐼), 𝜏(𝐽), and 𝜏(𝐾) are the eigenvalues of 𝑟 × 𝑟 Hermitian matrices, the third being the sum of the first two.In other words, Horn's inequalities
DOI: 10.1090/noti3148
Authors: Prakash Belkale, Joshua Kiers
Institution: University of North Carolina at Chapel Hill
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
OpenAlex ID: W4408410367