Systems of Linear Algebraic Equations

Abstract

The solution of a system of n first-order linear algebraic equations with constant coefficients requires knowledge of certain properties of an n × n coefficient matrix and the nonhomogeneous matrix vector b belonging to the system. So the main purpose of this chapter is to provide an introduction to the solution of systems of m nonhomogeneous linear algebraic equations in the n unknown real variables x ₁, x ₂,…, x _n . Associated with this is the solution of a special type of homogeneous algebraic problem involving n homogeneous linear algebraic equations in n unknowns and a parameter λ, that leads to the study of the eigenvalues and eigenvectors of an n × n matrix. It will be recalled that an eigenvalue was introduced briefly at the end of Chapter 2, and encountered again in Exercises 22 and 23 at the end of Chapter 3. The formal definition of the eigenvalues and the associated eigenvectors of square matrices will be given in this chapter, though the properties and use of eigenvectors will be studied in greater detail in Chapter 5.