The theory of generalized inverses has its genetic roots essentially in the context of so-called "ill-posed" linear problems. These include problems in which one either specifies
too much information, or too little. These problems cannot be solved in the sense of a
solution of a nonsingular problem. However, there is a sense in which there is still a solution
(and in fact a unique "solution") if one adopts, for example, the notion of "least-squares
solution" (of minimal norm).
It is well known that if A is a (square) nonsingular matrix, then there exists a unique
matrix B, which is called the inverse of A, such that AB = BA = I, where / is the identity
matrix. If A is a singular or a rectangular matrix, no such matrix Я exists. Now if A'1 exists,
then the system of linear equations Ax = b has a unique solution x = A~lb. On the other
hand, in many cases, solutions of a system of linear equations exist even when the inverse of
the matrix defining these equations does not. Also, in the case when the equations are
inconsistent, one is often interested in least-squares solutions, i.e., vectors that minimize the
sum of the squares of the residuals. These problems, along with many others in numerical
linear algebra, optimization and control, statistics, and other areas of analysis and applied
mathematics, are readily handled via the concept of a generalized inverse (or pseudoinverse)
of a matrix or of a linear operator. Roughly speaking, a generalized inverse must possess
some of the following properties to be useful: (i) it must reduce \oA~l if A is nonsingular;
(ii) it must always exist (or in the case of linear operators, it must exist for a larger class than
the class of invertible operators); (Ш) it must have some of the properties of the inverse (or
appropriate modifications thereof), such as (A'1) = A, (A'1)* = D*), etc; (iv)when
used in place of the inverse (when the latter fails to exist), it should provide sensible answers
to important questions such as consistency of the equations, or least-squares solutions.