## Abstract

We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas in 1988–1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a ‘value’ of an outcome determined by a given weight function φ. An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite d× d matrix and its square blocks (sub-matrices) of different sizes. When φ≡ 1 , the weighted inequality becomes a ‘standard’ SHI; in general, the weighted version requires some assumptions upon φ. The SHI and its weighted version generalize a widely known ‘usual’ Hadamard inequality detC≤∏j=1dCjj.

Original language | English (US) |
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Pages (from-to) | 85-114 |

Number of pages | 30 |

Journal | Aequationes Mathematicae |

Volume | 96 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2022 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics