Joint Numerical Range of Matrix Polynomials

Some algebraic properties of the sharp points of the joint numerical range of a matrix polynomials are the main subject of this paper. We also consider isolated points of the joint numerical range of matrix polynomials.


1-Introduction:
Let AMn be the algebra of n  n complex matrices. The classical numerical range of A is the set of a complex numbers W(A)={x*Ax: xC n ,x*x=1} where C n vector space (over C) of complex n-vectors [6]. There has been many generalizations and applications of the classical numerical range, see, for example [6]. In the following, we consider a generalization of the classical numerical range. Suppose  [9]. This generalized joint numerical range has been discussed by [9]. On the other hand The joint numerical range of matrix polynomials, being a continuous image of the unit sphere, is compact and connected but not necessarily convex; see Binding and Li [3]. Its convex hull is denoted by co and it plays an important role in the study of damped vibration systems, with a finite number of degree of freedom [7] and it is useful in various theoretical and applied subjects (see [1,2,3,4 and 5]) and their references. The aim of this paper is to give some algebraic properties of the sharp points of the joint numerical range of matrix polynomials, we also consider an isolated point of the joint numerical range of ) ( p  .The rest of this paper is organized as follows: In section 2, we present definitions and some basic results which will be used in this paper. In section 3, we prove that if 0  is a sharp point of the joint numerical range of the linear pencil

2-Preliminaries:
In this section, we present some definitions and basic results on joint numerical ranges of matrix polynomials.

Definition 2.5 [6]
An n-by-n Hermitiat matrix A is said to be positive definite if

Definition 2.6 [6]
A matrix BMn is said to be positive semi-definite if x*Bx  0 for all xC n Definition 2.7 [6] The matrix adjoint A* of AMn(C) is define by A*=A -T where Ais the component-wise conjugate, and A T is the transpose of A.

Definition 2.8 [6]
The matrix AMn(C) is said to be Hermition if A=A* ,it is skew-Hermition if A=-A* and for any AMn

Definition 2.9 [6]
Let 0 p be an element of anon-empty set A, we say that 0 p is an

3-Properties of Sharp points
In the following, we will restrict ourselves to the definition of sharp points. The next theorem gives a connection of these points with respect to the origin as a joint numerical range of matrix polynomials.

Theorem 3.1
Suppose that o x is a unit vector such that have non-negative real parts for all x of the neighborhood . We see that and u is a unitary matrix. For

Theorem 3.2
Suppose o x is a unit vector such that

Proof:
Suppose there exists  >0, 1  and 2  such that

Proof:
By the equality It is not possible to have Since zero is a sharp point of such that for any complex number .Moreover, by the continuity of the functions  belong to S1(0,r1),S2(0,r2),…,Sm(0,rm) respectively. thus by equation arg(x*A1x)+arg( we have that each of ) ) ( arg( ),..., ) ( arg( . Hence the equation Q(,t)=0 in  has m roots by the fundamental theorem of algebra. The roots of the algebraic equation depends continuously on the coefficient, hence P(0)= A0=0