CHAPTER 1 - Background on Linear Algebra and Related Topics

Abstract

This chapter discusses linear algebra and related topics and presents the properties and principles of basic matrix. A real nonsymmetric matrix A can have complex eigenvalues. As the coefficients of the characteristic polynomial are real, any complex eigenvalues of A must occur in complex conjugate pairs. For an N × N nonsymmetric matrix A, it is not always possible to find a basis for EN from the set of eigenvectors of A. However, it is always possible to form a basis from the independent eigenvectors of A supplemented by other vectors that are associated with the eigenvalues and eigenvectors of A. Such a basis can best be described in terms of the Jordan canonical form associated with A. A matrix whose set of eigenvectors does not include a basis is said to have an eigenvector deficiency. The chapter describes the matrix problem, which is obtained from a simple discretization of the generalized Dirichlet problem and illustrates the formulations of matrix problems arising from the discretization of boundary-value problems.