A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities
Vector and matrix inequalities play important rules 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 are the 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. All matrices, vectors and scalars considered throughout the paper will be real.
scientific article; zbMATH DE number 1251499
scientific article