Some equalities for estimations of variance components in a general linear model and its restricted and transformed models

For the unknown positive parameter σ² in a general linear model , the two commonly used estimations are the simple estimator (SE) and the minimum norm quadratic unbiased estimator (MINQUE). In this paper, we derive necessary and sufficient conditions for the equivalence of the SEs and MINQUEs of the variance component σ² in the original model ?, the restricted model , the transformed model , and the misspecified model .     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁ +A₂X₂B₂ +A₃X₃B₃ =C,(ii) A₁XB₁ =C₁ A₂XB₂ =C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A ₁ X ₁ B ₁+A ₂ X ₂ B ₂+A ₃ X ₃ B ₃+A ₄ X ₄ B ₄=C, (iv) A ₁ XB ₁=C ₁ A ₂ XB ₂=C ₂ A ₃ XB ₃=C ₃ A ₄ XB ₄=C ₄, (v) AXB+CXD=E are also considered.     

On some matrix equalities for generalized inverses with applications

Necessary and sufficient conditions are derived for the matrix equality A^-=PN^-Q to hold, where A^- and N^- are generalized inverses of matrices. Some consequences and applications are also given. In particular, necessary and sufficient conditions are derived for the additive decompositions C^-=A^-+B^- and to hold.     

Hermitian‐type generalized singular value decomposition with applications

For a pair of given Hermitian matrix A and rectangular matrix B with the same row number, we reformulate a well‐known simultaneous Hermitian‐type generalized singular value decomposition (HGSVD) with more precise structure and parameters and use it to derive some algebraic properties of the linear Hermitian matrix function A?BXB* and Hermitian solution of the matrix equation BXB* = A, and the canonical form of a partitioned Hermitian matrix and some optimization problems on its inertia and rank. Copyright © 2012 John Wiley & Sons, Ltd.     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.     

Similarity and consimilarity of elements in the real Cayley-Dickson algebras

The similarity and consimilarity of elements in the real quaternion, octonion and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equationsax=xb andax = [`(x)]bax = \bar xb in these algebras. Some consequences are also presented.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁+A₂X₂B₂+A₃X₃B₃=C,(ii) A₁XB₁=C₁A₂XB₂=C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A₁X₁B₁+A₂X₂B₂+A₃X₃B₃+A₄X₄B₄=C, (iv) A₁XB₁=C₁A₂XB₂=C₂A₃XB₃=C₃A₄XB₄=C₄, (v) AXB+CXD=E are also considered.     

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA * = B

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations AX = B and AXA ^* = B. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations AX = B and AXA ^* = B, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.     

Some identities for Moore–Penrose inverses of matrix products

A group of identities are established for the Moore–Penrose inverses and the weighted Moore–Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.