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From real to complex sign pattern matrices

Published online by Cambridge University Press:  17 April 2009

Carolyn A. Eschenbach ,
Frank J. Hall  and
Zhongshan Li
Carolyn A. Eschenbach
Affiliation:
Department of Mathematics and Computer ScienceGeorgia State UniversityAtlanta GA 30303United States of America
Frank J. Hall
Affiliation:
Department of Mathematics and Computer ScienceGeorgia State UniversityAtlanta GA 30303United States of America
Zhongshan Li
Affiliation:
Department of Mathematics and Computer ScienceGeorgia State UniversityAtlanta GA 30303United States of America
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Abstract

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This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 159 - 172
Copyright
Copyright © Australian Mathematical Society 1998

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