Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Chi Kwong Li, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

28 Citations (Scopus)

Abstract

Let A=(A11A12A21A22)∈Mn, where A11∈Mmwith m≤n/2, be such that the numerical range of A lies in the set {eiφz∈C:|z|≤(≤z)tanα}, for some φ∈[0, 2π) and α∈[0, π/2). We obtain the optimal containment region for the generalized eigenvalue λ satisfyingλ(A1100A22)x=(0A12A210)xfor some nonzero x∈Cn, and the optimal eigenvalue containment region of the matrix Im-A11-1A12A22-1A21 in case A11and A22are invertible. From this result, one can show |det(A)|≤sec2m(α)×|det(A11)det(A22)|. In particular, if A is an accretive-dissipative matrix, then |det(A)|≤2m|det(A11)det(A22)|. These affirm some conjectures of Drury and Lin.
Original languageEnglish
Pages (from-to)487-491
Number of pages5
JournalJournal of Mathematical Analysis and Applications
Volume410
Issue number1
DOIs
Publication statusPublished - 1 Feb 2014

Keywords

  • Accretive-dissipative matrix
  • Determinantal inequality
  • Eigenvalues
  • Numerical ranges

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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