Revisiting the Marcus–de Oliveira Conjecture

Abstract

The Marcus–de Oliveira determinantal conjecture claims that the determinant of the sum of two normal matrices A and B with the prescribed spectra σ(A)={a1,⋯,an} and σ(B)={b1,⋯,bn}, respectively, is contained in the convex hull of the points zσ=∏i=1n(ai+bσ(i)) for σ∈Sn, the symmetric group of degree n. The conjecture was independently proposed by Marvin Marcus in 1973 and de Oliveira in 1982, inspired by a result obtained by Miroslav Fiedler in 1971. We survey the main achievements relating to this open problem in matrix analysis. Some related results and questions that it has raised are also briefly reviewed. This overview aims to bring the attention of researchers to this problem and to stimulate the development of original approaches and techniques in the area. Ideally, this work may inspire further progress towards the solution of this long-standing conjecture.

Keywords

• Marcus–de Oliveira conjecture
• numerical range
• normal matrix
• unitary matrix
• determinant