The symplectic and algebraic geometry of Hornʹs problem Original Research Article

One version of Hornʹs problem asks for which λ,μ,ν does Hλ+Hμ+Hν=0 have solutions, where Hλ,μ,ν are Hermitian matrices with spectra λ,μ,ν. This turns out to be a moment map condition in Hamiltonian geometry. Many of the results around Hornʹs problem proven with great effort “by hand” are in fact simple consequences of the modern machinery of symplectic geometry, and the subtler ones provable via the connection to geometric invariant theory. We give an overview of this theory (which was not available to Horn), including all definitions, and how it can be used in linear algebra.