\contentsline {chapter}{\tocchapter {Chapter}{}{Glossary of notation}}{vii}
\contentsline {chapter}{\tocchapter {Chapter}{}{Introduction}}{1}
\contentsline {section}{\tocsection {}{1}{The inverse of a nonsingular matrix}}{1}
\contentsline {section}{\tocsection {}{2}{Generalized inverses of matrices}}{1}
\contentsline {section}{\tocsection {}{3}{Illustration: Solvability of linear systems}}{2}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{2}
\contentsline {section}{\tocsection {}{4}{Diversity of generalized inverses}}{3}
\contentsline {section}{\tocsection {}{5}{Preparation expected of the reader}}{3}
\contentsline {section}{\tocsection {}{6}{Historical note}}{3}
\contentsline {section}{\tocsection {}{7}{Remarks on notation}}{4}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{4}
\contentsline {chapter}{\tocchapter {Chapter}{0}{Preliminaries}}{5}
\contentsline {section}{\tocsection {}{1}{Scalars and vectors}}{5}
\contentsline {subsection}{\tocsubsection {}{1.1}{}}{5}
\contentsline {subsection}{\tocsubsection {}{1.2}{}}{5}
\contentsline {subsection}{\tocsubsection {}{1.3}{}}{5}
\contentsline {subsection}{\tocsubsection {}{1.4}{}}{5}
\contentsline {subsection}{\tocsubsection {}{1.5}{}}{6}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{6}
\contentsline {section}{\tocsection {}{2}{Linear transformations and matrices}}{8}
\contentsline {subsection}{\tocsubsection {}{2.1}{}}{8}
\contentsline {subsection}{\tocsubsection {}{2.2}{}}{9}
\contentsline {subsection}{\tocsubsection {}{2.3}{}}{9}
\contentsline {subsection}{\tocsubsection {}{2.4}{}}{9}
\contentsline {subsection}{\tocsubsection {}{2.5}{}}{9}
\contentsline {subsection}{\tocsubsection {}{2.6}{}}{10}
\contentsline {subsection}{\tocsubsection {}{2.7}{}}{10}
\contentsline {subsection}{\tocsubsection {}{2.8}{}}{10}
\contentsline {subsection}{\tocsubsection {}{2.9}{}}{11}
\contentsline {subsection}{\tocsubsection {}{2.10}{}}{11}
\contentsline {subsection}{\tocsubsection {}{2.11}{}}{11}
\contentsline {subsection}{\tocsubsection {}{2.12}{}}{11}
\contentsline {subsection}{\tocsubsection {}{2.13}{}}{12}
\contentsline {subsection}{\tocsubsection {}{2.14}{}}{12}
\contentsline {subsection}{\tocsubsection {}{2.15}{}}{12}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{13}
\contentsline {section}{\tocsection {}{3}{Elementary operations and permutations}}{18}
\contentsline {subsection}{\tocsubsection {}{3.1}{}}{18}
\contentsline {subsection}{\tocsubsection {}{3.2}{}}{19}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{20}
\contentsline {section}{\tocsection {}{4}{The Hermite normal form and related items}}{20}
\contentsline {subsection}{\tocsubsection {}{4.1}{}}{20}
\contentsline {subsection}{\tocsubsection {}{4.2}{}}{20}
\contentsline {subsection}{\tocsubsection {}{4.3}{}}{21}
\contentsline {subsection}{\tocsubsection {}{4.4}{}}{22}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{22}
\contentsline {section}{\tocsection {}{5}{Determinants and volume}}{24}
\contentsline {subsection}{\tocsubsection {}{5.1}{}}{24}
\contentsline {subsection}{\tocsubsection {}{5.2}{}}{25}
\contentsline {subsection}{\tocsubsection {}{5.3}{}}{25}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{25}
\contentsline {section}{\tocsection {}{6}{Some multilinear algebra}}{28}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{28}
\contentsline {section}{\tocsection {}{7}{The Jordan normal form}}{29}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{31}
\contentsline {section}{\tocsection {}{8}{The Smith normal form}}{33}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{33}
\contentsline {section}{\tocsection {}{9}{Nonnegative matrices}}{33}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{34}
\contentsline {chapter}{\tocchapter {Chapter}{1}{Existence and Construction of Generalized Inverses}}{35}
\contentsline {section}{\tocsection {}{1}{The Penrose equations}}{35}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{35}
\contentsline {section}{\tocsection {}{2}{Existence and construction of $\{1\}$--inverses}}{36}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{36}
\contentsline {section}{\tocsection {}{3}{Properties of $\{1\}$--inverses}}{37}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{38}
\contentsline {section}{\tocsection {}{4}{Existence and construction of $\{1,2\}$--inverses}}{39}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{40}
\contentsline {section}{\tocsection {}{5}{Existence and construction of $\{1,2,3\}$--, $\{1,2,4\}$-- and $\{1,2,3,4\}$--inverses}}{40}
\contentsline {section}{\tocsection {}{6}{Explicit formula for $A^{\dag }$}}{42}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{42}
\contentsline {section}{\tocsection {}{7}{Construction of $\{2\}$--inverses of prescribed rank}}{43}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{44}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{44}
\contentsline {chapter}{\tocchapter {Chapter}{2}{Linear Systems and Characterization of Generalized Inverses}}{45}
\contentsline {section}{\tocsection {}{1}{Solutions of linear systems}}{45}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{46}
\contentsline {section}{\tocsection {}{2}{Characterization of $A\{1,3\}$ and $A\{1,4\}$}}{48}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{48}
\contentsline {section}{\tocsection {}{3}{Characterization of $A\{2\}$, $A\{1,2\}$ and other subsets of $A\{2\}$.}}{49}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{50}
\contentsline {section}{\tocsection {}{4}{Idempotent matrices and projectors}}{50}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{54}
\contentsline {section}{\tocsection {}{5}{Matrix functions}}{56}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{59}
\contentsline {section}{\tocsection {}{6}{Generalized inverses with prescribed range and null space}}{61}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{63}
\contentsline {section}{\tocsection {}{7}{Orthogonal projections and orthogonal projectors}}{64}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{66}
\contentsline {section}{\tocsection {}{8}{Efficient characterization of classes of generalized inverses}}{73}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{75}
\contentsline {section}{\tocsection {}{9}{Restricted generalized inverses}}{76}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{78}
\contentsline {section}{\tocsection {}{10}{The Bott--Duffin inverse}}{79}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{81}
\contentsline {section}{\tocsection {}{11}{An application of $\{1\}$--inverses in interval linear programming}}{82}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{83}
\contentsline {section}{\tocsection {}{12}{A $\{1,2\}$--inverse for the integral solution of linear equations}}{84}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{85}
\contentsline {section}{\tocsection {}{13}{An application of the Bott--Duffin inverse to electrical networks}}{85}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{87}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{89}
\contentsline {chapter}{\tocchapter {Chapter}{3}{Minimal Properties of Generalized Inverses}}{91}
\contentsline {section}{\tocsection {}{1}{Least--squares solutions of inconsistent linear systems}}{91}
\contentsline {subsection}{\tocsubsection {}{}{Exercises, examples and supplementary notes}}{92}
\contentsline {section}{\tocsection {}{2}{Solutions of minimum norm}}{95}
\contentsline {subsection}{\tocsubsection {}{}{Exercises, examples and supplementary notes}}{96}
\contentsline {section}{\tocsection {}{3}{Tihonov regularization}}{99}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{101}
\contentsline {section}{\tocsection {}{4}{Weighted generalized inverses}}{102}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{104}
\contentsline {section}{\tocsection {}{5}{Least--squares solutions and basic solutions}}{107}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{111}
\contentsline {section}{\tocsection {}{6}{Minors of the Moore--Penrose inverse}}{111}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{114}
\contentsline {section}{\tocsection {}{7}{Essentially strictly convex norms and the associated projectors and generalized inverses${}^*$}}{114}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{120}
\contentsline {section}{\tocsection {}{8}{An extremal property of the Bott--Duffin inverse with application to electrical networks}}{131}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{133}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{134}
\contentsline {chapter}{\tocchapter {Chapter}{4}{Spectral Generalized Inverses}}{135}
\contentsline {section}{\tocsection {}{1}{Introduction}}{135}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{135}
\contentsline {section}{\tocsection {}{2}{The matrix index}}{136}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{137}
\contentsline {section}{\tocsection {}{3}{Spectral inverse of a diagonable matrix}}{138}
\contentsline {section}{\tocsection {}{4}{The group inverse}}{138}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{140}
\contentsline {section}{\tocsection {}{5}{Spectral properties of the group inverse}}{143}
\contentsline {section}{\tocsection {}{6}{The Drazin inverse}}{145}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{147}
\contentsline {section}{\tocsection {}{7}{Spectral properties of the Drazin inverse}}{149}
\contentsline {section}{\tocsection {}{8}{Index 1--nilpotent decomposition of a square matrix}}{149}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{151}
\contentsline {section}{\tocsection {}{9}{Quasi--commuting inverses}}{152}
\contentsline {section}{\tocsection {}{10}{Other spectral generalized inverses}}{152}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{153}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{154}
\contentsline {chapter}{\tocchapter {Chapter}{5}{Generalized Inverses of Partitioned Matrices}}{155}
\contentsline {section}{\tocsection {}{1}{Introduction}}{155}
\contentsline {section}{\tocsection {}{2}{Partitioned matrices and linear equations}}{155}
\contentsline {subsection}{\tocsubsection {}{}{Case 1. $A\in \@mathbb {C}^{r\times n}_{r}$, i.e. $A$ is of full row rank}}{156}
\contentsline {subsection}{\tocsubsection {}{}{Case 2. $A\in \@mathbb {C}^{m\times r}_{r}$, i.e. $A$ is of full column rank}}{157}
\contentsline {subsection}{\tocsubsection {}{}{Case 3. $A\in \@mathbb {C}^{m\times n}_{r}$, with $r\leq mnn$}}{157}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{159}
\contentsline {section}{\tocsection {}{3}{Intersection of manifolds}}{162}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{167}
\contentsline {section}{\tocsection {}{4}{Common solutions of linear equations and generalized inverses of partitioned matrices}}{168}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{173}
\contentsline {section}{\tocsection {}{5}{Generalized inverses of bordered matrices}}{175}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{176}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{178}
\contentsline {chapter}{\tocchapter {Chapter}{6}{A Spectral Theory for Rectangular Matrices}}{179}
\contentsline {section}{\tocsection {}{1}{Introduction}}{179}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{180}
\contentsline {section}{\tocsection {}{2}{The singular value decomposition}}{182}
\contentsline {section}{\tocsection {}{}{Historical notes}}{185}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{185}
\contentsline {section}{\tocsection {}{3}{The Schmidt approximation theorem}}{189}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{192}
\contentsline {section}{\tocsection {}{4}{Partial isometries and the polar decomposition theorem}}{194}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{198}
\contentsline {section}{\tocsection {}{5}{Principal angles between subspaces}}{205}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{210}
\contentsline {section}{\tocsection {}{6}{Perturbations}}{212}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{214}
\contentsline {section}{\tocsection {}{7}{A spectral theory for rectangular matrices}}{216}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{222}
\contentsline {section}{\tocsection {}{8}{Generalized singular value decompositions}}{224}
\contentsline {subsection}{\tocsubsection {}{8.1}{The $B$--SVD}}{224}
\contentsline {subsection}{\tocsubsection {}{8.2}{The $\{W,Q\}$--SVD}}{227}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{227}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{228}
\contentsline {chapter}{\tocchapter {Chapter}{7}{Computational Aspects of Generalized Inverses}}{229}
\contentsline {section}{\tocsection {}{1}{Introduction}}{229}
\contentsline {section}{\tocsection {}{2}{Computation of unrestricted $\{1\}$--inverses and $\{1,2\}$--inverses}}{229}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{230}
\contentsline {section}{\tocsection {}{3}{Computation of unrestricted $\{1,3\}$--inverses}}{231}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{232}
\contentsline {section}{\tocsection {}{4}{Computation of $\{2\}$--inverses with prescribed range and null space}}{232}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{233}
\contentsline {section}{\tocsection {}{5}{Greville's method and related results}}{234}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{239}
\contentsline {section}{\tocsection {}{6}{Computation of least--squares solutions}}{240}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{241}
\contentsline {section}{\tocsection {}{7}{Iterative methods for computing $A^{\dag }$}}{241}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{245}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{251}
\contentsline {chapter}{\tocchapter {Chapter}{8}{Miscellaneous applications}}{253}
\contentsline {section}{\tocsection {}{1}{Introduction}}{253}
\contentsline {section}{\tocsection {}{2}{Parallel sums}}{253}
\contentsline {section}{\tocsection {}{3}{The linear statistical model}}{255}
\contentsline {subsection}{\tocsubsection {}{3.1}{$X$ of full column--rank, $V$ nonsingular}}{256}
\contentsline {subsection}{\tocsubsection {}{3.2}{The general case}}{257}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{259}
\contentsline {section}{\tocsection {}{4}{Ridge regression}}{261}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{263}
\contentsline {section}{\tocsection {}{5}{An application of $\{2\}$--inverses in iterative methods for solving nonlinear equations}}{264}
\contentsline {subsection}{\tocsubsection {}{}{Exercises}}{269}
\contentsline {section}{\tocsection {}{6}{Linear systems theory}}{270}
\contentsline {section}{\tocsection {}{7}{Application of the group inverse in finite Markov chains}}{271}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{275}
\contentsline {section}{\tocsection {}{8}{An application of the Drazin inverse to difference equations}}{277}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{279}
\contentsline {section}{\tocsection {}{9}{Matrix volume and the change-of-variables formula in integration}}{280}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{283}
\contentsline {section}{\tocsection {}{10}{An application of the matrix volume in probability}}{289}
\contentsline {subsection}{\tocsubsection {}{10.1}{Probability densities and surface integrals}}{289}
\contentsline {subsection}{\tocsubsection {}{10.2}{Hyperplanes}}{290}
\contentsline {subsection}{\tocsubsection {}{10.3}{Spheres}}{291}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{292}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{294}
\contentsline {chapter}{\tocchapter {Chapter}{9}{Generalized Inverses of Linear Operators between Hilbert Spaces}}{295}
\contentsline {section}{\tocsection {}{1}{Introduction}}{295}
\contentsline {section}{\tocsection {}{2}{Hilbert spaces and operators: Preliminaries and notation}}{295}
\contentsline {subsection}{\tocsubsection {}{2.1}{Exercises and examples}}{299}
\contentsline {section}{\tocsection {}{3}{Generalized inverses of linear operators between Hilbert spaces}}{300}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{306}
\contentsline {section}{\tocsection {}{4}{Generalized inverses of linear differential operators}}{311}
\contentsline {subsection}{\tocsubsection {}{}{Exercises and examples}}{312}
\contentsline {section}{\tocsection {}{5}{Minimal properties of generalized inverses}}{318}
\contentsline {subsection}{\tocsubsection {}{5.1}{Exercises and examples}}{321}
\contentsline {section}{\tocsection {}{6}{Series and integral representations and iterative computation of generalized inverses}}{324}
\contentsline {subsection}{\tocsubsection {}{6.1}{Exercises and examples}}{327}
\contentsline {section}{\tocsection {}{7}{Frames}}{327}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{329}
\contentsline {chapter}{\tocappendix {Appendix}{A}{The Moore of the Moore--Penrose Inverse}}{331}
\contentsline {section}{\tocsection {}{1}{Introduction}}{331}
\contentsline {section}{\tocsection {}{2}{The 1920 lecture to the AMS}}{331}
\contentsline {section}{\tocsection {}{3}{The general reciprocal in \textit {General Analysis}}}{332}
\contentsline {section}{\tocsection {}{}{Suggested further reading}}{335}
\contentsline {chapter}{\tocappendix {Appendix}{}{Bibliography}}{337}
\contentsline {chapter}{\tocappendix {Appendix}{}{Subject Index}}{361}
\contentsline {chapter}{\tocappendix {Appendix}{}{Author Index}}{367}