A brief survey on the decomposable numerical range of matrices

Abstract

Let m and n be positive integers. Denote by the set of all n × m complex matrices. Given a matrix , its mth decomposable numerical range (1 ≤ m ≤ n) is the set Associated with the mth decomposable numerical range is the mth decomposable numerical radius of A which is denoted and defined by When m = 1, the concepts reduce to the classical numerical range and the classical numerical radius of A, respectively, which are well studied. The study of the mth decomposable numerical range and radius is related to many subjects such as the theories of determinant, unitary similarity, principal submatrices, exterior space and matrix inequalities, etc. In fact, this topic has attracted the attention of many authors in recent years, and many interesting results have been obtained. The purpose of this paper is to do a brief survey on the subject. Some new results are also presented.