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The determinant of a complex matrix and Gershgorin circles
Publikationstyp
Journal Article
Date Issued
2019
Sprache
English
Author(s)
Institut
TORE-URI
Volume
35
Issue
1
Start Page
181
End Page
186
Citation
Electronic Journal of Linear Algebra - ELA (35): 181-186 (2019)
Publisher DOI
Scopus ID
Publisher
Soc.
Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affrmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.
Subjects
Gershgorin circle
Determinant
Minkowski product
DDC Class
004: Informatik