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Matrix rank/inertia formulas for least-squares solutions with statistical applications. (English) Zbl 1333.15006
Summary: Least-squares solution (LSS) of a linear matrix equation and ordinary least-squares estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function \(\phi(Z) = Q - ZPZ'\) is given, where \(Z\) is taken as the LSS of the linear matrix equation \(AZ = B\). In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of \(\phi(Z)\) subject to the LSS of \(AZ = B\), and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.

MSC:
15A24 Matrix equations and identities
62J05 Linear regression; mixed models
15A16 Matrix exponential and similar functions of matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A18 Eigenvalues, singular values, and eigenvectors
62F30 Parametric inference under constraints
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