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.