Generalized Inverses: Theory and Applications [Paperback]

1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.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.

From the reviews of the second edition:

The book under review which is the second edition of the 30 years ago published one provides a detailed survey of generalized inverses and their main properties & . An important feature of this book is the over 600 exercises & . Each chapter ends with the section Suggested further reading. These sections provide excellent additional references on topics treated & . it can be used profitably by gradualSs