The Maximal Numerical Range of a Quadratic Matrix

Let n be a positive integer and let \(M_n (\mathbb {C})\) denote the algebra of all complex n-by-n matrices. A matrix \(A\in M_n (\mathbb {C})\) is called quadratic if it satisfies some non-trivial quadratic equation \((A-\alpha I)(A-\beta I) = 0\) , where I denotes the \(n\times n\) identity matrix. In this paper, we give an explicit formula for the maximal numerical range of quadratic matrices.