Formal Problems in Linear Algebra

A great many practical problems in the scientific and engineering world give rise to models or descriptions of reality which involve matrices. In consequence, a very large proportion of the literature of numerical mathematics is devoted to the solution of various matrix equations. In the following sections, the major formal problems in numerical linear algebra will be introduced. Some examples are included to show how these problems may arise directly in practice. However, the formal problems will in most cases occur as steps in larger, more difficult computations. In fact, the algorithms of numerical linear algebra are the keystones of numerical methods for solving real problems. Matrix computations have become a large area for mathematical and computational research. Textbooks on this subject, such as Stewart (1973) and Strang (1976), offer a foundation useful for understanding the uses and manipulations of matrices and vectors. More advanced works detail the theorems and algorithms for particular situations. An important collection of well-referenced material is Golub and Van Loan (1983). Kahaner, Moler and Nash (1989) contains a very readable treatment of numerical linear algebra.