Linear Algebra II

The following sections are included:

• About This Compilation
• The Purpose of This Publication
• Editorial Board
• Preface
• Contents
• List of Figures

In this chapter we present a method of representing the structure of matrices in terms of graphs. Here, by “structure” we mean a structure of combinatorial nature, no more than the distinction between zeros and nonzeros of the matrix entries, disregarding numerical or quantitative information. The methodology of focusing on the structure is an effective means for qualitative analysis of large-scale systems and efficient large-scale numerical computations. The method of this chapter will also be used in Chap. 2 for the theory of nonnegative matrices.

This chapter presents the Perron–Frobenius theorem, which is a fundamental theorem for eigenvalues and eigenvectors of nonnegative matrices. As a primary application of this theorem, the theory of stochastic matrices (transition probability matrices of Markov chains, in particular) is developed, where effective use is made of the graph representation of matrix structures introduced in Chap. 1. We also explain the basic properties of M-matrices and Birkhoff’s theorem for doubly stochastic matrices.

In this chapter we deal with systems of linear inequalities and the polyhedra described by them. The Fourier–Motzkin elimination method is employed to prove Farkas’ lemma, which is used as the pivotal tool for deriving major results, including theorems of the alternative and the structure of solutions to systems of linear inequalities. These results form the mathematical foundation of linear programming for optimization.

In this chapter we focus on matrices with integer entries, with particular emphasis on integer solutions to systems of linear equations and inequalities. Since division is not always possible between integers, it is necessary to consider divisibility conditions in addition to the rank condition. Fundamental facts about integer solutions to systems of linear equations and inequalities are discussed by using integer elementary transformations, the Hermite normal form, and the Smith normal form.

In this chapter we deal with matrices with entries of polynomials. Such matrices often appear in engineering as representations of linear systems in the frequency domain (Laplace transforms). Since division is not always possible between polynomials, consideration of divisibility is crucial. Fundamental facts on matrices with entries of polynomials, such as the Hermite normal form, the Smith normal form, and the Kronecker canonical form of matrix pencils, are discussed. Facts pertaining to elementary transformations, the Hermite normal form, and the Smith normal form are almost direct analogues of those for integer matrices in Chap. 4.

For nonsingular square matrices the inverse matrices exist and are uniquely determined, but this is not the case with singular square matrices or rectangular matrices. Extending the concept of inverse matrices to arbitrary matrices, including rectangular matrices, turns out to be useful in applications. In this chapter, fundamental facts about generalized inverses are presented. The derivation and characterization of several types of generalized inverses are shown, and an application to Newton’s method is illustrated.

In this chapter we explain fundamental facts in group representation theory, which offers a mathematical methodology for expressing and utilizing the symmetry possessed by systems. The abstract structure of a group is described by concrete mathematical objects called representation matrices, and the symmetry possessed by engineering systems leads to block-diagonal decompositions of the matrices that describe the systems.

The following sections are included:

• Bibliography
• Index