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.