The numerical range and decomposable numerical range of matrices
Natália Bebiano, Chi-Kwong Li, João da Providência · Linear and Multilinear Algebra · 1991
Let m and n be positive integers such that 1≤m≤n. Denote by the set of all n×m complex matrices. For a matrix , the mth decomposable numerical range of A is the set When m=1, it reduces to the classical numerical range of A which is denoted by W(A). It is known that is contaned in W(Cm (A)), where Cm (A) is the mth compound of A. In this paper we study the geometrical properties of and W(Cm (A)). Although the two sets are different in general, it is shown that they enjoy a lot of similar geometrical properties. For example, if the boundary of or W(Cm (A)) contains a point z, which is an eigenvalue of Cm (A) or a corner, then A is unitarily similar to a matrix of the form A 1⊕A 2 with satisfying z=detA 1. Moreover, we derive some sufficient conditions for a matrix A to be normal in terms of the geometrical properties of and W(Cm (A)). The matrices A for which or W(Cm (A)) is a line segment are also characterized.