A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities

Shuangzhe Liu, Heinz Neudecker

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Vector and matrix inequalities play important r61es in various mathematical
and statistical contexts. The number of results and applications grows day by
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 languageEnglish
Pages (from-to)55-73
Number of pages19
JournalStatistical Papers
Issue number1
Publication statusPublished - 1 Jan 1999


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