Cauchy–Schwarz Inequality and Riccati Equation for Positive Semidefinite Matrices

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.