Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/26598
Title: Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector
Authors: Li, CK
Sze, NS 
Keywords: Accretive-dissipative matrix
Determinantal inequality
Eigenvalues
Numerical ranges
Issue Date: 2014
Publisher: Academic Press Inc Elsevier Science
Source: Journal of mathematical analysis and applications, 2014, v. 410, no. 1, p. 487-491 How to cite?
Journal: Journal of Mathematical Analysis and Applications 
Abstract: Let A=(A11A12A21A22)∈Mn, where A11∈Mm with 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 A11 and A22 are 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.
URI: http://hdl.handle.net/10397/26598
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2013.08.040
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