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).