Determinantal conjecture

In mathematics, the determinantal conjecture of Marcus (1972) and de Oliveira (1982) asks whether the determinant of a sum A + B of two n-by-n normal complex matrices A and B lies in the convex hull of the n! points Π_i (λ(A)_i + λ(B)_σ(i)), where the numbers λ(A)_i and λ(B)_i are the eigenvalues of A and B, and σ is an element of the symmetric group S_n.

References

• de Oliveira, G.N. (1982), "Research problem: Normal matrices", Linear and Multilinear Algebra, 12: 153–154, doi:10.1080/03081087.1982.11882087
• Marcus, Marvin (1972), "Derivations, Plücker relations, and the numerical range", Indiana University Mathematics Journal, 22 (12): 1137–1149, doi:10.1512/iumj.1973.22.22094, ISSN 0022-2518, MR 0314862