Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

Jiang, Bo; Tian, Yongge

doi:10.1016/j.amc.2017.07.079

Cited by 4 publications

(1 citation statement)

References 31 publications

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“…(4) subject to a consistent linear matrix equation XA = B by pure algebraic operations of matrices. Tian [14] and Tian and Jiang [15,16] established the explicit formulas for calculating the extremal ranks and inertias of matrix function XP X * − Q * subject to the general solution, the least-squares solutions and the weighted least-squares solutions of the matrix equation AX = B, respectively. Wang [17] considered the extreme inertias and ranks of a new quasi-quadratic Hermitian structure XX * − P subject to a consistent system of Eq.(1.1).…”

Section: Introductionmentioning

confidence: 99%

The Optimization Problems of a Quadratic Hermitian Matrix-valued Function with the Constraint of Matrix Equations

Lan,

Yuan

2023

Preprint

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Let Ω={X∈Cn×p| AX=B, XH=K, and AA+B=B, KH+H=K, AK=BH}, and let f(X)=(XC+D)M(XC+D)∗−G be a given quadratic Hermitian matrix-valued function. In this paper, we first establish a series of closed-form formulas for calculating the extremal ranks and inertias of f(X) subject to X∈Ω by applying the generalized inverses of matrices. Further, we present the solvability conditions for X∈Ω to satisfy the matrix equality (XC+D)M(XC+D)∗=G and matrix inequalities (XC+D)M(XC+D)∗> G(≥ G, < G, ≤ G)to hold, respectively. In addition, we provide closed-form solutions to two Löwner partial ordering optimization problems on f(X) subject to X∈Ω. Mathematics Subject Classification (2010). Primary 15A03; Secondary 15A09; 15A24; 15A63.

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Section: Introductionmentioning

confidence: 99%

The Optimization Problems of a Quadratic Hermitian Matrix-valued Function with the Constraint of Matrix Equations

Lan,

Yuan

2023

Preprint

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Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model

Tian

Wang

2018

Communications in Statistics - Theory and Methods

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On best linear unbiased estimation and prediction under a constrained linear random-effects model

Jiang

Tian

2023

JIMO

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<p style='text-indent:20px;'>This paper is concerned with solving some fundamental estimation, prediction, and inference problems on a linear random-effects model with its parameter vector satisfying certain exact linear restrictions. Our work includes deriving analytical formulas for calculating the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the model by way of solving certain constrained quadratic matrix optimization problems, characterizing various mathematical and statistical properties of the predictors and estimators, establishing various fundamental rank and inertia formulas associated with the covariance matrices of predictors and estimators, and presenting necessary and sufficient conditions for several equalities and inequalities of covariance matrices of the predictors and estimators to hold.</p>

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Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

Cited by 4 publications

References 31 publications

The Optimization Problems of a Quadratic Hermitian Matrix-valued Function with the Constraint of Matrix Equations

The Optimization Problems of a Quadratic Hermitian Matrix-valued Function with the Constraint of Matrix Equations

Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model

On best linear unbiased estimation and prediction under a constrained linear random-effects model

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