### Abstract

We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(e^{pA}#e^{pB})^{ 2 p}≤Tr e^{A+B} holds for all p>0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p^{-1}TrX log Y^{ p> 2}X^{p}Y^{ p 2}≤Tr X(log X+log Y)≤p^{-1}TrX log X^{ p 2}Y^{p}X^{ p 2} for all p>0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.

Original language | English |
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Pages (from-to) | 153-185 |

Number of pages | 33 |

Journal | Linear Algebra and Its Applications |

Volume | 181 |

Issue number | C |

DOIs | |

Publication status | Published - Mar 1 1993 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

*Linear Algebra and Its Applications*,

*181*(C), 153-185. https://doi.org/10.1016/0024-3795(93)90029-N

**The Golden-Thompson trace inequality is complemented.** / Hiai, Fumio; Petz, D.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 181, no. C, pp. 153-185. https://doi.org/10.1016/0024-3795(93)90029-N

}

TY - JOUR

T1 - The Golden-Thompson trace inequality is complemented

AU - Hiai, Fumio

AU - Petz, D.

PY - 1993/3/1

Y1 - 1993/3/1

N2 - We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB) 2 p≤Tr eA+B holds for all p>0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p-1TrX log Y p> 2XpY p 2≤Tr X(log X+log Y)≤p-1TrX log X p 2YpX p 2 for all p>0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.

AB - We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB) 2 p≤Tr eA+B holds for all p>0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p-1TrX log Y p> 2XpY p 2≤Tr X(log X+log Y)≤p-1TrX log X p 2YpX p 2 for all p>0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.

UR - http://www.scopus.com/inward/record.url?scp=2442597136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442597136&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(93)90029-N

DO - 10.1016/0024-3795(93)90029-N

M3 - Article

VL - 181

SP - 153

EP - 185

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -