Linear Algebra, Theory And Applications, 2012a

Positive Matrices
Earlier theorems about Markov matrices were presented. These were matrices in which all
the entries were nonnegative and either the columns or the rows added to 1. It turns out
that many of the theorems presented can be generalized to positive matrices. When this is
done, the resulting theory is mainly due to Perron and Frobenius. I will give an introduction
to this theory here following Karlin and Taylor [18].
Definition A.0.1 For A a matrix or vector, the notation, A>>0 will mean every entry
of A is positive. By A>0 is meant that every entry is nonnegative and at least one is
positive. By A ≥ 0 is meant that every entry is nonnegative. Thus the matrix or vector
consisting only of zeros is ≥ 0. An expression like A>>Bwill mean A − B>>0 with
similar modifications for > and ≥.
For the sake of this section only, define the following for x =(x 1 , ··· ,x n ) T , avector.
|x| ≡(|x 1 | , ··· , |x n |) T .
Thus |x| is the vector which results by replacing each entry of x with its absolute value 1 .
Also define for x ∈ C n ,
||x|| 1
≡ ∑ |x k | .
k
Lemma A.0.2 Let A>>0 and let x > 0. ThenAx >> 0.
Proof: (Ax) i
= ∑ j A ijx j > 0 because all the A ij > 0 and at least one x j > 0.
Lemma A.0.3 Let A>>0. Define
S ≡{λ : Ax >λx for some x >> 0} ,
and let
Now define
Then
K ≡{x ≥ 0 such that ||x|| 1
=1} .
S 1 ≡{λ : Ax ≥ λx for some x ∈ K} .
sup (S) =sup(S 1 ) .
1 This notation is just about the most abominable thing imaginable. However, it saves space in the
presentation of this theory of positive matrices and avoids the use of new symbols. Please forget about it
when you leave this section.
403