### Abstract

A generalized word in two letters A and B is an expression of the form W = A^{α1}B^{β1}A^{α2}B^{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 |

State | Published - Feb 1 2002 |

### Fingerprint

### Keywords

- Generalized word
- Positive definite matrices
- Projections

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Electronic Journal of Linear Algebra*,

*9*, 21-26.

**Positive eigenvalues and two-letter generalized words.** / Hillar, C.; Johnson, C. R.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Electronic Journal of Linear Algebra*, vol. 9, pp. 21-26.

}

TY - JOUR

T1 - Positive eigenvalues and two-letter generalized words

AU - Hillar, C.

AU - Johnson, C. R.

AU - Spitkovsky, Ilya

PY - 2002/2/1

Y1 - 2002/2/1

N2 - 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.

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.

KW - Generalized word

KW - Positive definite matrices

KW - Projections

UR - http://www.scopus.com/inward/record.url?scp=3042574957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042574957&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042574957

VL - 9

SP - 21

EP - 26

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

SN - 1081-3810

ER -