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.
Inhaltsverzeichnis
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.
