The Numerical Range of 6 6 Irreducible Matrices

Ahmed M. Sabir College of Sciences University of Salahaddin Received on: 22/11/2006 Accepted on: 05/03/2007 ABASTRACT In this paper, we consider the problem of characterizing the numerical range of 6 by 6 irreducible matrices which have line segments on their boundary.


Introduction
Let Mn (C) be the algebra of n  n complex matrices. The numerical range of AMn is defined by W(A)={x*Ax: xC n ,x*x=1} [4], where x* the adjoint of x C is defined by x*= T x where x is the component-wise conjugate, and x T is the transpose of x [4]. As pointed out by many authors, for2 2 matrices A a complete description of the numerical range W(A) is well-Known. Namely, W(A) is an ellipse with foci at the eigenvalues 1, . In [4], of course, s=0 for normal A, and the ellipse in this case degenerates into a line segments connecting 1with 2 . On the other hand ,for 2 2 matrices A with coinciding eigenvalues the ellipse W(A) degenerates into a disk. For3 3 matrix A, this was first done by Kippenhahn. In [6] , his characterization is based on the factorability of the associated polynomial PA(x,y,z)=det(xReA+yImA+zI3) .This was improved in [5] by expressing the condition in terms of entries of A, also for 4 4 and 5 5 matrices A, this was improved in [2] by expressing the conditions in terms of entries of A. The aim of this paper is to give a sufficient and necessary condition for numerical range of 6 by 6 matrix with a line segment on its boundary .

Preliminaries
In the following, we give some definitions and results on W(A) that are useful in this study.

Definition 2.1 [4]
A matrix AMn(C) is said to be irreducible if either n=1 or n  2 and there does not exist a permutation matrix PMn(C) such that where B, D are nonempty square matrices.

Definition 2.2 [4]
A matrix BMn(c) such that x*Bx  0 for all xC n , is said to be positive semidefinite .

Proposition 2.3 [4]
The numerical range of AMn is always a compact convex set in C. It contains the spectrum (A) of A and is equal to the convex hull of (A) if A is normal.

Proposition 2.6 [4]
Suppose that B is a principal submatrix of A Mn(C). Then W(B)  W(A).
We now relate the numerical range of an n  n matrix to an algebraic curve of class n. The next proposition indicates how the characteristic polynomial of some pencil associated with the matrix arises in this connection.

Proposition 2.9 [6]
If A is a n  n matrix, then its numerical range W(A) is the convex hull of the real points of the curve C(A). The real part of the curve C(A) in the complex plane namely the set{a+ibC : a,bR and ax+by+z=0 is tangent to PA(x,y,z)=0}, will be denoted by CR(A) and is called the Kippenhahn curve of A.

Line segments of the Boundary of Numerical Range
In the following we will restrict ourselves to the irreducible matrix The next theorem gives conditions for the numerical range to have a line segments on its boundary . Proof: W(A) has a line segments on the boundary of CR(A) has a double or triple or quadripartite or tangent ux+vy+w=0 (u and v not both zero) . This corresponds to a root w of the equation det(uReA+vImA+zI6)=0 with multiplicity 2 or 3 or 4 or 5, which is the same as saying that uReA+vImA+zI6 has rank 4 or 3 or 2 or 1. This proves the equivalence of (a) and (b).
(b)  (c) : Suppose (b) holds then -w is an eigenvalue of uReA+vImA with multiplicity 5, and we get double or triple or quadripartite or tangent .Thus (b) implies (c). To prove (c)  (b), assume that -w is a multiple eigenvalue of uReA+vImA.If -w is of multiplicity 6 , then the Hermition uReA+vImA would be a scalar matrix. In this case, ReA and ImA would commute and hence A would be normal, contradicting the irreduciblity of A. Thus (c) impies (b). We begin by deriving a canonical form for an irreducible form for an irreducible 6  6 matrix with a line segment on the boundary of it is numerical range, if W(A) has a line segment on its boundary . After rotation,Shifting, and multiplication by a positive number, we may assume that a line segment stretches from 0 to i. Since W(A) is convex, it must be contained entirely in the right or left half-plane. Applying yet another rotation and translation, if necessary we may assume that W(A) is in the right half-plane. By theorem(3.1) we have rankA=1 or 2 or 3. Therefore we will discus these two cases, respectively.