OMC for scalar multiples of unitaries

✍ Scribed by S.W. Drury

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 133 KB
• Volume: 422
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2006.10.022

✦ Synopsis

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a 1 , . . . , a n and b 1 , . . . , b n , respectively. Then for any scalars t and s det(tA -sB) ∈ co ⎧ ⎨ ⎩ n j =1 (ta jsb σ (j) ); σ ∈ S n ⎫ ⎬ ⎭ ,

where S n denotes the group of all permutations of {1, . . . , n} and co the convex hull taken in the complex plane.