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Matrix versions of the Cauchy and Kantorovich inequalities. (English) Zbl 0706.15019

The authors prove this analogue of Cauchy’s inequality for a k×n matrix X and a rank m, m×n matrix Y:

XX * XY * (YY * ) -1 YX * ,

and this analogue of Kantorovich’s inequality, where A is Hermitian positive definite, m, M are upper and lower bounds on the eigenvalues, and U is a rectangular matrix such that UU * =I:

UA -1 U * ((m+M)I-UAU * )/(mM)(m+M) 2 (UAU * ) -1 /(4mM)·

Reviewer: K.H.Kim

MSC:
15A45Miscellaneous inequalities involving matrices
26D15Inequalities for sums, series and integrals of real functions
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