Eigenvalues of the Sum of Matrices from Unitary Similarity Orbits

Abstract

Let A and B be n × n complex matrices. Characterization is given for the set E(A, B) of eigenvalues of matrices of the form U * AU + V * BV for some unitary matrices U and V . Consequences of the results are discussed and computer algorithms and programs are designed to generate the set E (A, B). The results refine those of Wielandt on normal matrices. Extensions of the results to the sum of matrices from three or more unitary similarity orbits are also considered.2000 Mathematics Subject Classification. 15A18.

A normal variation of the Horn problem: the rank 1 case

“…We end the introduction with noting that Wielandt [11] characterized when a complex number ν is the eigenvalue of the sum of two normal matrices with prescribed eigenvalues. This was further pursued in [9]. Using standard Householder techniques, one can easily put any matrix in upper Hessenberg form (see, e.g, [3], Section 7.4, Algorithm 7.4.2), which is the content of the following auxiliary result.…”

The sum of unitary similarity orbits containing only special operators

“…In the finite dimensional case, researchers determined the ranks, determinants, eigenvalues and singular values of matrices in k j=1 U(A j ); see [8,9,11,13] and their references. When A 1 , .…”