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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:

$X{X}^{*}\ge X{Y}^{*}{\left(Y{Y}^{*}\right)}^{-1}Y{X}^{*},$

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 $U{U}^{*}=I:$

$U{A}^{-1}{U}^{*}\le \left(\left(m+M\right)I-UA{U}^{*}\right)/\left(mM\right)\le {\left(m+M\right)}^{2}{\left(UA{U}^{*}\right)}^{-1}/\left(4mM\right)·$

Reviewer: K.H.Kim

MSC:
 15A45 Miscellaneous inequalities involving matrices 26D15 Inequalities for sums, series and integrals of real functions
References:
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