More on extremal ranks of the matrix expressions A-BX±X*B* with statistical applications

Through a Hermitian-type (skew-Hermitian-type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240:199-205), where A is Hermitian (skew-Hermitian), we show how to find a Hermitian (skew-Hermitian) matrix X such that the matrix expressions A - BX +/- X*B* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX +/- X*B* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, X beta, sigma(2)V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least-squares estimator and the best linear unbiased estimator of X beta, respectively. Copyright (C) 2007 John Wiley & Sons, Ltd.