The numerical range of linear pencils of 2-by-2 matrices

Let A, B is an element of M-n. The numerical range of a linear pencil A lambda - B is defined by W(A lambda - B) = { t is an element of C : tw*Aw - w* Bw = 0 for some non-zero w is an element of C-n). The numerical range of a linear pencil of n-by-n matrices coincides with the union of the numerical ranges of linear pencils of 2-by-2 matrices. We emphasize on computation of the numerical range of linear pencils of 2-by-2 matrices. Let A, B is an element of M-2 with constant diagonals. We show that the boundary of W(A lambda - B) lies on a rational curve of degree at most 4. By using Lagrange's multiplier and implicit function, we describe the boundary of W(A; - B) for general 2-by-2 matices. (C) 2002 Elsevier Science Inc. All rights reserved.