ELLIPTIC NUMERICAL RANGES OF 4 4 MATRICES

Let A be an n n (complex) matrix. Recall that the numerical range W (A) of A is the set \ Ax, x: x Cⁿ, \|x\| = 1 \ in the plane, where, denotes the usual inner product in Cⁿ. In this paper a series of tests is given, allowing one to determine when the numerical range of a 4 4 matrix A is an elliptic disc.