4 1. SOME FUNDAMENTALS
Let A be an n × n positive matrix. Then the following properties hold:
PF1. λmax is a simple eigenvalue of A.
PF2. There is a positive eigenvector of A for the eigenvalue λmax:
Ax = λmaxx, x is a positive vector.
PF3. λmax is the only eigenvalue of A with a corresponding nonnegative
eigenvector.
PF4. If y is a positive eigenvector and α is a scalar such that
Ay αy (entrywise) but Ay = αy,
then λmax α.
PF5. If y is a positive vector and α is a scalar such that
Ay αy (entrywise) but Ay = αy,
then λmax α.
PF6. If B is an n × n matrix with B A (entrywise) but B = A, then
λmax(B) λmax(A).
PF7. Let rmin be the minimum row sum of A and let rmax be the maxi-
mum row sum of A, then
rmin λmax rmax,
with equality at either end if and only if rmin = rmax.
The basic properties here are PF1 and PF2, and there are many expo-
sitions of their proofs. We shall assume PF1 and PF2, and show how the
other properties follow.
Proofs of PF3 to PF7 from PF1 and PF2: Let A = [aij] and let x =
(x1,x2,...,xn)t
be a positive eigenvector of the transpose matrix
At
for
λmax. Then
xtA
=
λmaxxt,
that is,
n
j=1
xjaji = λmaxxi (1 i n).
We assume that y =
(y1,y2,...,yn)t
is an arbitrary positive vector, and we
let
μi =
1
yi
n
j=1
aijyj (1 i n).
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