### Essentially Hermitian matrices revisited

Stephen W. Drury
2006 The Electronic Journal of Linear Algebra
The following case of the Determinantal Conjecture of Marcus and de Oliveira is established. Let A and C be hermitian n × n matrices with prescribed eigenvalues a 1 , . . . , an and c 1 , . . . , cn, respectively. Let κ be a non-real unimodular complex number, B = κC, b j = κc j for j = 1, . . . , n. Then where Sn denotes the group of all permutations of {1, . . . , n} and co the convex hull taken in the complex plane. * http://math.technion.ac.il/iic/ela ELA Essentially Hermitian Matrices
more » ... itian Matrices Revisited 291 4. The Second Order Term -Equal Roots Case. We suppose that we are in the second case of Proposition 2.1 to be designated case 3 in the sequel. Proposition 4.1. With the hypotheses and notations of Proposition 2.1, for P an arbitrary skew hermitian matrix we obtain for suitable scalars C 1 and C 2 Proof. We have that (2.1) has a double root ρ and hence ρ is a real. In particular, ρ = κ −1 . We have So the first part of the second order term involves since tr(Z) = 0. The second part of the second order term involves clearly has a real trace. On the other hand, we have analogous to (3.2) and it follows that κρ(1 − κρ) −2 and κρ(1 − κρ) −2 tr(Z 2 ) are tangentially aligned. The Second Order Term -Conclusion . Surprisingly, case 2 is the easiest of the three cases to settle. Proposition 5.1. Assume that we are in case 2 arising in the proof of Proposition 3.1. It is possible to choose a skew hermitian matrix P such that E 1 ZE 1 and E 2 ZE 2 are simultaneously rank one and tr(E 1 Z) = − tr(E 2 Z) = 0. Consequently, the entire second order term is tangential and the underlying extreme point is flat. http://math.technion.ac.il/iic/ela 2 is not real (possible since m ≥ 3 is odd) we see that tr(W k ) = 0 for k = 1, 2, . . . , m − 1. Therefore we deduce