On the continuity of Birkhoff–James ε-orthogonality sets

摘要

Consider two matrices A, B ∈ C~(n×m) with B ≠0, a matrix norm ‖?‖ and a real parameter ? ∈ [0, 1). The Birkhoff–James ε-orthogonality set of A with respect to B, F_(‖?‖)~?(A; B)={μ∈C: ‖A-λB‖≥√1 -?~2‖B‖ |μ-λ|,?λ∈C}, is a compact and convex subset of the complex plane that has been recently introduced by the authors, as a natural generalization of the classical numerical range of square matrices. In this note, we derive the continuity of F_(‖?‖)~?A;BT with respect to A or ?.