Extremal case in Marcus–Oliveira conjecture and beyond
Alexander Emilevich Guterman, Rute Lemos, Graça Soares · Linear and Multilinear Algebra · 2012
For A, C ∈ M n the C-determinantal range of A is the following set on the complex plane ▵ C (A) = {det(A − UCU*): UU* = I n }. For normal matrices A and C with eigenvalues α1, … , α n and γ1, … , γ n , respectively, Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973), pp. 1137–1149] and Oliveira [G.N. de Oliveira, Normal matrices (research problem), Linear Multilinear Algebra 12 (1982), pp. 153–154] conjectured that ▵ C (A) is a subset of the convex hull of the points , σ ∈ S n , where S n is the symmetric group of degree n. We investigate the extremal set of matrices for which the equality holds in the Marcus–Oliveira conjecture. We illustrate the use of the obtained results by two different applications. The first one deals with the equality case between the radius of ▵ C (A) and the radius of the convex hull of the points z σ, σ ∈ S n . The second one is the characterization of additive Frobenius endomorphisms for the determinantal range or radius on the space M n and on its real subspace of Hermitian matrices.