Positive eigenvalues and two-letter generalized words

C. Hillar; C. R. Johnson; I. M. Spitkovsky

doi:10.13001/1081-3810.1069

Positive eigenvalues and two-letter generalized words

C. Hillar, C. R. Johnson, I. M. Spitkovsky

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A generalized word in two letters A and B is an expression of the form W = A^α1B^β1A^α2B^β2 ... A^αN B^βN in which the exponents are nonzero real numbers. When independent positive definite matrices are substituted for A and B, it is of interest whether W necessarily has positive eigenvalues. This is known to be the case when N = 1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite di.erent (even for 2-by-2 matrices), and this is the focus of the present work.

Original language	English (US)
Pages (from-to)	21-26
Number of pages	6
Journal	Electronic Journal of Linear Algebra
Volume	9
DOIs	https://doi.org/10.13001/1081-3810.1069
State	Published - Feb 2002

Keywords

Generalized word
Positive definite matrices
Projections

ASJC Scopus subject areas

Algebra and Number Theory

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10.13001/1081-3810.1069

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abstract = "A generalized word in two letters A and B is an expression of the form W = Aα1Bβ1Aα2Bβ2 ... AαN BβN in which the exponents are nonzero real numbers. When independent positive definite matrices are substituted for A and B, it is of interest whether W necessarily has positive eigenvalues. This is known to be the case when N = 1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite di.erent (even for 2-by-2 matrices), and this is the focus of the present work.",

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author = "C. Hillar and Johnson, {C. R.} and Spitkovsky, {I. M.}",

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AB - A generalized word in two letters A and B is an expression of the form W = Aα1Bβ1Aα2Bβ2 ... AαN BβN in which the exponents are nonzero real numbers. When independent positive definite matrices are substituted for A and B, it is of interest whether W necessarily has positive eigenvalues. This is known to be the case when N = 1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite di.erent (even for 2-by-2 matrices), and this is the focus of the present work.

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Positive eigenvalues and two-letter generalized words

Abstract

Keywords

ASJC Scopus subject areas

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