„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 graduate or advanced undergraduate students of mathematics and computer science, and by PhD students … .“ (Róbert Rajkó, Acta Scientiarum Mathematicarum, Vol. 71, 2005)
„Each chapter is accompanied by suggestions for further reading, while the bibliography contains 901 references. … The book contains 450 exercises at different levels of difficulty, many of which are solved in detail. This feature makes it suitable either for reference and self-study or for use as a classroom text. It can be used profitably by graduate students or advanced undergraduate students … .“ (Nicholas Karampetakis, Zentralblatt MATH, Vol. 1026, 2004)
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.
