More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications
Yonghui Liu, Yongge Tian · Numerical Linear Algebra with Applications · 2007
Abstract 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, Xβ, σ2V}, 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 Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.