Yongge Tian

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 Au2009−u2009BXu2009±u2009X*B* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BXu2009±u2009X*B*u2009=u2009A, 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

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A−BXB*−CYC* and then solve two conjectures on the maximal and minimal possible ranks of A−BXB*−CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright

On an additive decomposition of the BLUE in a multiple-partitioned linear model

Necessary and sufficient conditions are derived for the BLUE in a general multiple-partitioned linear model {y,X[emailxa0protected]1+...+X[emailxa0protected]k,@s^[emailxa0protected]} to be the sum of the BLUEs under the k small models {y,X[emailxa0protected]1,@s^[emailxa0protected]}, ..., {y,X[emailxa0protected]k,@s^[emailxa0protected]}. Some consequences and further research topics are also given.

On Sum Decompositions of Weighted Least-Squares Estimators for the Partitioned Linear Model

For the partitioned linear model ℳ = {y, X 1 β 1 + X 2 β 2,σ2 Σ}, this article investigates decompositions of weighted least-squares estimator (WLSE) of X 1 β 1 + X 2 β 2 under ℳ as sums of WLSEs under the two small models {y, X 1 β 1,σ2 Σ} and {y, X 2 β 2,σ2 Σ}. Some consequences on the sum decomposition of the unique best unbiased linear estimator (BLUE) of X 1 β 1 + X 2 β 2 under ℳ are also given.

SCHUR COMPLEMENTS AND BANACHIEWICZ-SCHUR FORMS ∗

Through the matrix rank method, this paper gives necessary and sufficient conditions for a partitioned matrix to have generalized inverses with Banachiewicz-Schur forms. In addition, this paper investigates the idempotency of generalized Schur complements in a partitioned idempotent matrix.

The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems

A formula for the inverse of any nonsingular matrix partitioned into two-by-two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix. The formula is then applied to three matrix inverse completion problems to obtain their complete solutions.

On relations between BLUEs under two transformed linear models

For a given general linear model ℳ={y,Xβ,Σ}, we investigate relationships between the best linear unbiased estimations (BLUEs) under its two transformed models ℳ1={Ay,AXβ,AΣA′} and ℳ2={By,BXβ,BΣB′}. We first establish some expansion formulas for calculating the ranks and inertias of the covariance matrices of BLUEs and their operations under ℳ1 and ℳ2. We then derive from the rank and inertia formulas necessary and sufficient conditions for equalities and inequalities of BLUEs’ covariance matrices to hold. We also give applications of the rank and inertia formulas to two sub-sample models of ℳ.

Extremal ranks of a quadratic matrix expression with applications

In this article, we give some closed-form formulae for the maximal and minimal ranks of the quadratic matrix expression q(X 1,u2009X 2)u2009=u2009Au2009−u2009(A 1u2009−u2009B 1 X 1 C 1)D(A 2u2009−u2009B 2 X 2 C 2) with respect to the two variable matrices X 1 and X 2. As an application, we give the maximal and minimal ranks of the Schur complement with respect to the reflexive generalized inverse of A. In addition, we derive necessary and sufficient conditions for the solutions of two linear matrix equations to be orthogonal and proportional, respectively.

Two universal similarity factorization equalities for commutative involutory and idempotent matrices and their applications

A square matrix A of order n is said to be involutory if A 2u2009=u2009I n , and to be idempotent if A 2u2009=u2009A. In this article, we give two universal similarity factorization equalities for linear combinations of two commutative involutory and two idempotent matrices and their products. As applications, we derive some disjoint decompositions for these linear combinations, and use the disjoint decompositions to derive a variety of results on the determinants, ranks, traces, inverses, generalized inverses and similarity decompositions of these linear combinations. In particular, we present some collections of involutory, idempotent and tripotent matrices generated from these linear combinations.

An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices

Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices.