* Glossary of notation * Introduction * Preliminaries * Existence and Construction of Generalized Inverses * Linear Systems and Characterization of Generalized Inverses * Minimal Properties of Generalized Inverses * Spectral Generalized Inverses * Generalized Inverses of Partitioned Matrices * A Spectral Theory for Rectangular Matrices * Computational Aspects of Generalized Inverses * Miscellaneous Applications * Generalized Inverses of Linear Operators between Hilbert Spaces * Appendix A: The Moore of the Moore-Penrose Inverse * Bibliography * Subject Index * Author Index

Generalized inverses: theory and applications

Preface MATRICES: Basic Properties of Vectors and Matrices Kronecker Products, the Vec-Operator and the Moore- Penrose Inverse Miscellaneous Matrix Results DIFFERENTIALS: THE THEORY: Mathematical Preliminaries Differentials and Differentiability The Second Differential Static Optimization DIFFERENTIALS: THE PRACTICE: Some Important Differentials First- Order Differentials and Jacobian Matrices Second-Order Differentials and Hessian Matrices INEQUALITIES: Inequalities THE LINEAR MODEL: Statistical Preliminaries The Linear Regression Model Further Topics in the Linear Model APPLICATIONS TO MAXIMUM LIKELIHOOD ESTIMATION: Maximum Likelihood Estimation Simultaneous Equations Topics in Psychometrics Subject Index Bibliography.

Matrix Differential Calculus with Applications in Statistics and Econometrics

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

For given data ($t_i\ , y_i), i=1, \ldots ,m$ , we consider the least squares fit of nonlinear models of the form F($\underset ~\to a\ , \underset ~\to \alpha\ ; t) = \sum_{j=1}^{n}\ g_j (\underset ~\to a ) \varphi_j (\underset ~\to \alpha\ ; t) , \underset ~\to a\ \epsilon R^s\ , \underset ~\to \alpha\ \epsilon R^k\ $ For this purpose we study the minimization of the nonlinear functional r($\underset ~\to a\ , \underset ~\to \alpha ) = \sum_{i=1}^{m} {(y_i - F(\underset ~\to a , \underset ~\to \alpha , t_i))}^2$ It is shown that by defining the matrix ${ \{\Phi (\underset ~\to \alpha\} }_{i,j} = \varphi_j (\underset ~\to \alpha ; t_i)$ , and the modified functional $r_2(\underset ~\to \alpha ) = \l\ \underset ~\to y\ - \Phi (\underset ~\to \alpha )\Phi^+(\underset ~\to \alpha ) \underset ~\to y \l_2^2$, it is possible to optimize first with respect to the parameters $\underset ~\to \alpha$ , and then to obtain, a posteriori, the optimal parameters $\overset ^\to {\underset ~\to a}$ The matrix $\Phi^+(\underset ~\to \alpha$) is the Moore-Penrose generalized inverse of $\Phi (\underset ~\to \alpha$), and we develop formulas for its Frechet derivative under the hypothesis that $\Phi (\underset ~\to \alpha$) is of constant (though not necessarily full) rank From these formulas we readily obtain the derivatives of the orthogonal projectors associated with $\Phi (\underset ~\to \alpha$), and also that of the functional $r_2(\underset ~\to \alpha$) Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given These results are generalizations of those of H D Scolnik [1971]

The differentiation of pseudoinverses and nonlinear least squares problems whose variables separate.

(1974). Equalities and Inequalities for Ranks of Matrices. Linear and Multilinear Algebra: Vol. 2, No. 3, pp. 269-292.

Equalities and Inequalities for Ranks of Matrices

Hadamard products and multivariate statistical analysis

https://www.math.uwaterloo.ca/~hwolkowi/henry/reports/bndseigs80.pdf

Bounds for eigenvalues using traces

It is well known that the ordinary least squares estimator of Xβ in the general linear model E y = Xβ, cov y = σ2 V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.

George P. H. Styan

Papers

Equalities and Inequalities for Ranks of Matrices

Hadamard products and multivariate statistical analysis

Bounds for eigenvalues using traces

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

Some properties of matrix partial orderings