Matrix rank/inertia formulas for least-squares solutions with statistical applications
نویسندگان
چکیده
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 φ(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 φ(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.
منابع مشابه
Ranks of the common solution to some quaternion matrix equations with applications
We derive the formulas of the maximal andminimal ranks of four real matrices $X_{1},X_{2},X_{3}$ and $X_{4}$in common solution $X=X_{1}+X_{2}i+X_{3}j+X_{4}k$ to quaternionmatrix equations $A_{1}X=C_{1},XB_{2}=C_{2},A_{3}XB_{3}=C_{3}$. Asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. We give the exp...
متن کاملOptimization problems on the rank and inertia of the Hermitian matrix expression A−BX − (BX)∗ with applications
We give in this paper some closed-form formulas for the maximal and minimal values of the rank and inertia of the Hermitian matrix expression A − BX ± (BX)∗ with respect to a variable matrix X. As applications, we derive the extremal values of the ranks/inertias of the matrices X and X ± X∗, where X is a (Hermitian) solution to the matrix equation AXB = C, respectively, and give necessary and s...
متن کاملTo appear in Operator Theory: Advances and Applications INERTIA CONDITIONS FOR THE MINIMIZATION OF QUADRATIC FORMS IN INDEFINITE METRIC SPACES
We study the relation between the solutions of two minimization problems with inde nite quadratic forms We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions While these inertia conditions are automatically satis ed in a standard Hilbert space setting which is the case of classical least squares problems in both the determinis...
متن کاملOn the Stability of Some Hierarchical Rank Structured Matrix Algorithms
In this paper, we investigate the numerical error propagation and provide systematic backward stability analysis for some hierarchical rank structured matrix algorithms. We prove the backward stability of various important hierarchically semiseparable (HSS) methods, such as HSS matrix-vector multiplications, HSS ULV linear system solutions, HSS linear least squares solutions, HSS inversions, an...
متن کامل