Construction of a solution to the rank 2 Horn problem

Abstract

Given three sets of $n$ real eigenvalues satisfying the trace equality and the Horn inequalities, we know that there are $n\times n$ real symmetric matrices $A$ and $B$ so that $A$ has the first set of eigenvalues, $B$ has the second set of eigenvalues, and $A+B$ has the last set of eigenvalues. Under the condition that $B$ is a rank 2 matrix, we give a construction for the matrices $A$ and $B$. This construction is based on performing two orthogonal rank 1 updates on $A$. We end with a discussion of the relationship between this rank 2 Horn problem and the following similar problem: given a set of $n$ real eigenvalues, a set of $2$ real eigenvalues, and a set of $n+2$ real eigenvalues satisfying certain conditions, find an $(n+2)\times(n+2)$ real symmetric matrix such that the top left principal submatrix has the first set of eigenvalues, the bottom right principal submatrix has the second set of eigenvalues, and the full matrix has the last set of eigenvalues