Computer Science

Computation of Generalized Matrix Inverses and Applications
Ivan Stanimirović, PhD

Computation of Generalized Matrix Inverses and Applications

Published. Available now.
Pub Date: December 2017
Hardback Price: $149.95 US
Hard ISBN: 9781771886222
E-Book ISBN: 9781315115252
Pages: 292pp w/Index
Binding Type: hardback
Notes: 2 illustrations

Computation of General Matrix Inverses and Applications offers a gradual exposition to matrix theory as a subject of linear algebra. It presents both the theoretical results in generalized matrix inverses and the applications. The book is as self-contained as possible, assuming no prior knowledge of matrix theory and linear algebra.

The book first addresses the basic definitions and concepts of an arbitrary generalized matrix inverse with special reference to the calculation of {i,j,...,k} inverse and the Moore-Penrose inverse. Then, the results of LDL* decomposition of the full rank polynomial matrix are introduced, along with numerical examples. Methods for calculating the Moore-Penrose’s inverse of rational matrix are presented, which are based on LDL* and QDR decompositions of the matrix. A method for calculating the A(2)T;S inverse using LDL ∗ decomposition using methods is derived as well as the symbolic calculation of A(2)T;S inverses using QDR factorization.

The text then offers several ways on how the introduced theoretical concepts can be applied in restoring blurred images and linear regression methods, along with the well-known application in linear systems. The book also explains how the computation of generalized inverses of matrices with constant values is performed, and covers several methods, such as methods based on full-rank factorization, Leverrier-Faddeev method, method of Zhukovski, and variations of partitioning method.


  • Provides in-depth coverage of important topics in matrix theory and generalized inverses of matrices, highlighting the Moore-Penrose inverse
  • Requires no prior knowledge of linear algebra
  • Offers an extensive collection of numerical examples on numerical computations
  • Describes several computational techniques for evaluating matrix full-rank decompositions, which are then used to compute generalized inverses
  • Highlights popular algorithms for computing generalized inverses of both polynomial and constant matrices
  • Presents several comparisons of performances of methods for computing generalized inverses
  • Includes material relevant in theory of image processing, such as restoration of blurred images
  • Shows how multiple matrix decomposition techniques can be exploited to derive the same result


1. Introduction
1.1 Basic definitions and properties
1.2 Representation of generalized inverses using full-rank decomposition
1.3 LU decomposition
1.4 The structure and organization of the book

2. Computing Generalized Inverses of Matrices with Numerical Values
2.1 Preliminaries and notation
2.2 Methods for computing generalized inverses of constant matrices
2.3 Partitioning method
2.4 Matrix multiplication is as hard as generalized inversion

3. Generalized Inverses of Polynomial and Rational Matrices
3.1 Symbolic matrix computations
3.2 Calculation of {i, j, k}| inverse and generalized inverse matrix of rational
3.3 LDL* full rank decomposition of polynomial matrix
3.4 Calculating the Moore-Penrose’s inverse polynomial matrix
3.5 LDL* decomposition of the full range of rational matrix
3.6 Calculating the Moore-Penrose’s inverse rational matrix
3.7 Calculating A(2) T;S inverse LDL* with decomposition
3.8 Representations complete ranking based on the QR decomposition
3.9 Symbolic computations of A(2) T;S inverses using QDR factorization
3.10 Singular value-decomposition
3.11 Generalized inverses of block matrices
3.12 Block LDL* decomposition of the full range
3.13 Greville’s partitioning method
3.14 Leverrier-Faddeev method for computing Moore-Penrose inverse of polynomial matrices

4. Applications
4.1 The problem of feedback compensation
4.2 Linear regression method
4.3 Restoration of blurred images

5. Conclusion



About the Authors / Editors:
Ivan Stanimirović, PhD
Assistant Professor, Department of Computer Science, Faculty of Sciences and Mathematics, University of Niš, Serbia

Ivan Stanimirović, PhD, is currently with the Department of Computer Science, Faculty of Sciences and Mathematics at the University of Niš, Serbia, where he is an Assistant Professor. He formerly was with the Faculty of Management at Megatrend University, Belgrade, as a Lecturer. His work spans from multi-objective optimization methods to applications of generalized matrix inverses in areas such as image processing and restoration and computer graphics. His current research interests include computing generalized matrix inverses and its applications, applied multi-objective optimization and decision making, as well as deep learning neural networks. Dr. Stanimirović was the Chairman of a workshop held at 13th Serbian Mathematical Congress, Vrnjačka banja, Serbia, in 2014.

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