Schneider, Hans

2019-02-15T14:20:14Z

2019-02-15T14:20:14Z

1952

Chapter 1 is a short introductory chapter dealing with some definitions and basic properties of matrices and vectors.

In Chapter 2 we introduce a partial order between matrices with real elements. For matrices with non-negative elements our notation' and terminology differ from the usual ones. The casual reader is advised to read section 2.3 before glancing further. The term "Positive Matrix" will be used from now on in the sense of 2.3.

In Chapter 3 we consider the "normal form" of a reducible matrix, and some associated sets. We define the "R-functions", a principal tool of investigation in later chapters.

Chapter 4 contains some consequences of the partial order "between matrices. Some analogues of the properties of positive matrices and positive numbers are developed.

In Chapter 5 we consider "chains of elements" and powers of positive matrices.

Chapter 6 contains a resumé of the chief algebraic properties of a matrix that are required in the rest of the thesis. These concern latent roots and latent vectors, sets of "generalized latent vectors", classical canonical submatrices, and principal idempotent and nilpotent elements.

In Chapter 7 we describe a method of proving the fundamental properties of positive matrices, which is based on some work by Probenius.

In Chapter 8 we review a method due to Wielandt (1950) of proving the basic results for irreducible positive matrices. We give a variant of our own. Lower and upper bounds are found in terms of the elements of the matrix, for the ratios of elements of the strictly positive latent column vector associated with the largest positive latent root of an irreducible positive matrix.

In Chapter 9 we deal with "P-matrices". We deduce a large number of algebraic results purely by inspection of positive elements, finally we examine "sets" of latent row vectors.

Chapter 10 is the longest chapter. In It we consider the singular matrix A = ρ I - P , where P is a positive matrix and ρ its largest positive latent root. We examine the number of linearly independent latent vectors associated with the latent root 0 (10. C1 ), sets of positive generalized latent vectors (10. 16), and, when the multiplicity of 0 does not exceed three, the classical canonical submatrices associated with 0. Provided we know which are singular when A is in normal form, these questions may be answered by an inspection of the positions of non-zero elements. In general the orders of the classical canonical submatrices associated with 0 can not be settled in this way, though such methods suffice to determine whether they are all equal to 1, (10. 31 ). Finally we consider the principal idempotent and nilpotent elements of A associated with 0, in some special cases.

The Bibliography then follows. In the text we give as reference the author's name and the date of publication, thus: Probenius (1912).

The Appendix consists of a paper accepted by the Journal of the London Mathematical Society, "An inequality for latent roots applied to determinants with dominant principal diagonal". The theory of matrices with dominant principal diagonal is closely connected with that for positive matrices. In this paper we "rejected" notation and terminology of 2.3.

http://hdl.handle.net/1842/33839

The University of Edinburgh

Annexe Thesis Digitisation Project 2019 Block 22

Matrices with non-negative elements

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy