Higher numerical ranges of matrix polynomials

Let P(λ) P(λ) be an n n-square complex matrix polynomial, and 1≤k≤n 1≤k≤n be a positive integer. In this paper, some algebraic and geometrical properties of the k k-numerical range of P(λ) P(λ) are investigated. In particular, the relationship between the k k-numerical range of P(λ) P(λ) and the k k-numerical range of its companion linearization is stated. Moreover, the k k-numerical range of the basic A A-factor block circulant matrix, which is the block companion matrix of the matrixpolynomial P(λ)=λ m I n −A P(λ)=λmIn−A, is studied.