Universal factorization equalities over generalized quaternion algebras.
Special forms of generalized inverses of row block matrices.
Ranks and independence of solutions of the matrix equation .
Completing block Hermitian matrices with maximal and minimal ranks and inertias.
Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix.
On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product.
The maximal and minimal ranks of with applications.
On a group of mixed-type reverse-order laws for generalized inverses of a triple matrix product with applications.
Schur complements and Banachiewicz-Schur forms.
Some algebraic and statistical properties of WLSEs under a general growth curve model.
How to establish universal block-matrix factorizations.
Some inequalities for sums of nonnegative definite matrices in quaternions.
Equalities for orthogonal projectors and their operations
A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we...
Matrix rank and inertia formulas in the analysis of general linear models
Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs),...
The extremal ranks of subject to a pair of matrix equations
Matrix rank/inertia formulas for least-squares solutions with statistical applications
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 − ZPZ0 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)...
On decompositions of estimators under a general linear model with partial parameter restrictions
A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the...
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