Cauchy–Schwarz Inequality and Riccati Equation for Positive Semidefinite Matrices

Masatoshi Fujii ()
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Masatoshi Fujii: Osaka Kyoiku University

A chapter in Differential and Integral Inequalities, 2019, pp 341-350 from Springer

Abstract: Abstract By the use of the matrix geometric mean #, the matrix Cauchy–Schwarz inequality is given as Y ∗X ≤ X ∗X # U ∗Y ∗Y U for k × n matrices X and Y , where Y ∗X = U|Y ∗X| is a polar decomposition of Y ∗X with unitary U. In this note, we generalize Riccati equation as follows: X ∗A †X = B for positive semidefinite matrices, where A † is the Moore–Penrose generalized inverse of A. We consider when the matrix geometric mean A # B is a positive semidefinite solution of XA †X = B. For this, we discuss the case where the equality holds in the matrix Cauchy–Schwarz inequality.

Keywords: 47A64; 47A63; 15A09 (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-27407-8_9

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DOI: 10.1007/978-3-030-27407-8_9

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