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
