### Abstract

Vector and matrix inequalities play important r61es in various mathematical

and statistical contexts. The number of results and applications grows day by

day.

Our purpose is to survey Cauchy-Schwarz and Kantorovich-type vector and

matrix inequalities, to present related matrix results, and to give examples of

applications of Cauchy-Schwarz and Kantorovich-type matrix inequalities in

statistics and econometrics. The structure is as follows. Section 2 collects

Cauchy-Schwarz and Kantorovich-type vector inequalities, which have been

developed and applied widely. For this (vector) case we omit almost all

proofs. Section 3 contains three useful lemmas. We apply them in Section 4

to derive Cauchy-Schwarz and Kantorovich-type matrix inequalities. These arethe main results of the paper. Section 5 contains additional matrix inequalities which can be viewed as applications of the main results to sums of matrices, (upper left) submatrices and other matrix expressions. The last section

contains applications in statistics and econometrics.

Magnus and Neudecker (1991) and Wang and Chow (1994) can be

recommended as useful references for basic algebraic and statistical results.

All matrices, vectors and scalars considered throughout the paper will be real.

and statistical contexts. The number of results and applications grows day by

day.

Our purpose is to survey Cauchy-Schwarz and Kantorovich-type vector and

matrix inequalities, to present related matrix results, and to give examples of

applications of Cauchy-Schwarz and Kantorovich-type matrix inequalities in

statistics and econometrics. The structure is as follows. Section 2 collects

Cauchy-Schwarz and Kantorovich-type vector inequalities, which have been

developed and applied widely. For this (vector) case we omit almost all

proofs. Section 3 contains three useful lemmas. We apply them in Section 4

to derive Cauchy-Schwarz and Kantorovich-type matrix inequalities. These arethe main results of the paper. Section 5 contains additional matrix inequalities which can be viewed as applications of the main results to sums of matrices, (upper left) submatrices and other matrix expressions. The last section

contains applications in statistics and econometrics.

Magnus and Neudecker (1991) and Wang and Chow (1994) can be

recommended as useful references for basic algebraic and statistical results.

All matrices, vectors and scalars considered throughout the paper will be real.

Original language | English |
---|---|

Pages (from-to) | 55-73 |

Number of pages | 19 |

Journal | Statistical Papers |

Volume | 40 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 1999 |

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### Cite this

Liu, S., & Neudecker, H. (1999). A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities.

*Statistical Papers*,*40*(1), 55-73. https://doi.org/10.1007/BF02927110