Elliptic Numerical Ranges of 4 x 4 Matrices

Let A be a ntimesn complex matrix associated with f(z)=z
ⁿ
+a
₁
z
^n-1
+...+a
_n-1
z+a
_n
. Then the numerical range of A is defined to be W(A)={x* Ax:xisinC
^m
/|x|=1}. In this paper, we study some properties of the Kippenhahn curve C
_R
(A) firstly, and then discuss the necessary condition of W(A) (A is 4times4 reducible companion matrices) being an elliptic disc by making use of matrices Kronecker product and the Kippenhahn curve C
_R
(A).