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Zbl 0706.15019
Marshall, A.W.; Olkin, I.
Matrix versions of the Cauchy and Kantorovich inequalities. (English)
[J] Aequationes Math. 40, No.1, 89-93 (1990). ISSN 0001-9054; ISSN 1420-8903/e

The authors prove this analogue of Cauchy's inequality for a $k\times n$ matrix X and a rank m, $m\times n$ matrix Y: $$ XX\sp*\ge XY\sp*(YY\sp*)\sp{-1}YX\sp*, $$ 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 $UU\sp*=I:$ $$ UA\sp{-1}U\sp*\le ((m+M)I-UAU\sp*)/(mM)\le (m+M)\sp 2(UAU\sp*)\sp{-1}/(4mM).$$
[K.H.Kim]

MSC 2000:: *15A45 Miscellaneous inequalities involving matrices
26D15 Inequalities for sums, series and integrals of real functions

Keywords: matrix versions; Cauchy's inequality; Kantorovich's inequality

Cited in: Zbl 0723.15017

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