Positivity 8: 327–338, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
On Matrices with Perron–Frobenius Properties and
Some Negative Entries
CHARLES R. JOHNSON
and PABLO TARAZAGA
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA;
Department of Computing and Mathematical Sciences, Texas A&M University-Corpus Christi,
6300 Ocean Dr., Corpus Christi, TX 78412, USA. E-mail: firstname.lastname@example.org
Received 21 December 2001; accepted 2 March 2003
Abstract. We consider those n-by-n matrices with a strictly dominant positive eigenvalue of multi-
plicity 1 and associated positive left and right eigenvectors. Such matrices may have negative entries
and generalize the primitive matrices in important ways. Several ways of constructing such matrices,
including a very geometric one, are discussed. This paper grew out of a recent survey talk about
nonnegative matrices by the ﬁrst author and a joint paper, with others, by the second author about the
symmetric case [Tarazaga et al. (2001) Linear Algebra Appl. 328: 57].
In the early part of the twentieth century Perron and Frobenius described the ba-
sic spectral properties of the (entry-wise) positive matrices and their closure, the
nonnegative matrices. A description may be found in  or . Ever since, such
matrices have only grown in applied importance, and naturally, mathematicians
have tried to broaden the matrices to which the key conclusion apply.
Most notable has been the ‘cone-theoretic approach’, summarized in , whose
primary intent is to characterize n-by-n matrices whose dominant eigenvalue is
positive. Here we study matrices of which even more of the Perron–Frobenius
properties are asked, precisely the properties most important in applications. In-
terestingly, such matrices may have some negative entries so that a proper general-
ization is provided.
2 Matrices with the Strict Perron–Frobenius Property
denote those n-by-n real matrices for which there is a positive eigenvalue
(of multiplicity 1) that strictly dominates all other eivenvalues and for which there
is both an entry-wise positive left and right eigenvector. Note that PF
under positive scalar multiplication and under addition of positive multiples of the
identity. Note also that A∈PF
must have a positive entry in every row and
column because of the eigenvector equation.