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£

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Nonnegative matrices play a crucial role in the theory of matrices. They are important in

the study of convergence of iterative methods and arise in many applications including

economics, queuing theory, and chemical engineering.

A nonnegative matrix is simply a matrix whose entries are nonnegative. More gener-

ally, a partial order relation can be de¬ned on the set of matrices.

q¤ ¥

H RFP I#F D

Q FH ©U

T

C

˜

h˜

Let and be two matrices. Then

˜#

…p # …pz‚

# # # # ¡ ¡

˜ h ˜

if by de¬nition, for , . If denotes the zero matrix,

( ¡

then is nonnegative if , and positive if . Similar de¬nitions hold in which

“positive” is replaced by “negative”.

z“–

”’‘

The binary relation “# ” imposes only a partial order on since two arbitrary matrices

u“j

”’‘

in are not necessarily comparable by this relation. For the remainder of this section,

d “ ©¢ d¡ ¥ © ”¥ 7© ¥ ¥ ¨¥

e u § u§ ¨ £

u ¡

¡£ ¡£ ¢ ¡ ¥ 5 ¥

§¥

©

§ ¥

we now assume that only square matrices are involved. The next proposition lists a number

of rather trivial properties regarding the partial order relation just de¬ned.

V¤ G S£

£¦

SRI FcP` `

Q

bQ

H QF U

T

The following properties hold.

˜ # s# –# ˜ #

– The relation for matrices is re¬‚exive ( ), antisymmetric (if and

# # ˜

#

˜

˜ , then ), and transitive (if and , then ).

˜

˜(

˜™— If and are nonnegative, then so is their product and their sum .

'

'#

If is nonnegative, then so is .

¨

˜ # ¡

G # G

˜

If , then .

0

¡ §

¡

'©

˜s# s#

0˜#

˜# If , then and similarly .

The proof of these properties is left as Exercise 23.

A matrix is said to be reducible if there is a permutation matrix such that G

is block upper triangular. Otherwise, it is irreducible. An important result concerning non-

negative matrices is the following theorem known as the Perron-Frobenius theorem.

„ SG b¡¤

§¥¦ ¥ B ¡

Q ©T ˜ ˜

Let be a real nonnegative irreducible matrix. Then , @

C

Y

the spectral radius of , is a simple eigenvalue of . Moreover, there exists an eigenvector

with positive elements associated with this eigenvalue.

V

A relaxed version of this theorem allows the matrix to be reducible but the conclusion is

somewhat weakened in the sense that the elements of the eigenvectors are only guaranteed

to be nonnegative.

Next, a useful property is established.

V¤ G S£

£¦ – ˜

SRI FcP` `

Q

bQ

H QF ¡U

T

s#

˜

Let be nonnegative matrices, with . Then

x–s# x–

–}˜s# D– ˜

and

£

©¨¦

§ §¥ T

Consider the ¬rst inequality only, since the proof for the second is identical. The

result that is claimed translates into

# a` # … ' ' 0 ‘)(&' # … ' ' ‚ 1‘)(&'

˜ 0

which is clearly true by the assumptions.

A consequence of the proposition is the following corollary.

b£ £ G

¦¢ ¦ U ¡

`Q

Q T

˜ ˜#

Let and be two nonnegative matrices, with . Then

z( t

' # ' ¨ ¤ ¢ ¦§¥

˜ 7