Page 111 - Matrices theory and applications

5. Nonnegative Matrices
94
iii. Deduce that
N−1
1

lim
N→+∞ N
m=0
iv. Under what additional assumption do we have the stronger
convergence
lim M N M m = L.
= L?
N→+∞
10. Let B ∈ M n (IR) be a nonnegative irreducible matrix and let C ∈
M n (IR) be a nonzero nonnegative matrix. For t> 0, we define r t :=
ρ(B + tC)and we let X t denote the nonnegative unitary eigenvector
associated to the eigenvalue r t .
(a) Show that t → r t is strictly increasing.
Define r := lim t→+∞ r t . We wish to show that r =+∞.Let X
be a cluster point of the sequence X t .We may assume,upto a
permutation of the indices, that

Y
X = , Y > 0.
0
(b) Suppose that in fact, r< +∞. Show that BX ≤ rX. Deduce

that B Y =0, where B is a matrix extracted from B.
(c) Deduce that X = Y ;that is, X> 0.
(d) Show, finally, that CX = 0. Conclude that r =+∞.
(e) Assume, moreover, that ρ(B) < 1. Show that there exists one
and only one t ∈ IR such that ρ(B + tC)= 1.
11. Show that ∆ is stable under multiplication. In particular, if M is
m
bistochastic, the sequence (M ) m≥1 is bounded.
12. Let M ∈ M n (IR) be a bistochastic irreducible matrix. Show that
1 ... 1
 
N−1
1 m 1  .
lim M =  . . 
.  =: J n
N→+∞ N n . .
m=0
1 ... 1
m
(use Exercise 9). Show by an example that the sequence (M ) m≥1
may or may not converge.
13. Show directly that for every p ∈ [1, ∞], J n p =1, where J n was
defined in the previous exercise.
14. Let P ∈ GL n (IR)begiven such that P, P −1 ∈ ∆ n . Show that P is a
permutation matrix.
15. If M ∈ ∆ n is given, we define an equivalence relation between in-

dices in the following way: i Ri if there exists a sequence i 1 =

i ,j 1 ,i 2,j 2 ,... ,i p = i such that m ij > 0each timethat (i, j)is