Page 109 - Matrices theory and applications

5. Nonnegative Matrices
92
M n+1 (IR), whichwedecomposeblockwise as
T

a
ξ
n
ξ, η ∈ IR ,
B ∈ M n (IR).
a ∈ IR,
,
A =
η
B
(a) Applying the induction hypothesis to the matrix B + J,where
> 0and J> 0 is a matrix, then letting go to zero, show
that ρ(B) is an eigenvalue of B, associated to a nonnegative
eigenvector (this avoids the use of Theorem 5.2.1).
(b) Using the formula
∞
−k k−1
−1
(λI n − B) = λ B ,
k=1
valid for λ ∈ (ρ(B), +∞), deduce that the function h(λ):=
T
λ−a−ξ (λI n −B) −1 η is strictly increasing on this interval and
that on the same interval the vector x(λ):=(λI n − B) −1 η is
positive.
(c) Prove the relation P A (λ)= P B (λ)h(λ) between the characteris-
tic polynomials.
(d) Deduce that the matrix A has one and only one eigenvalue in
(ρ(B), +∞), and that it is a simple one, associated to a positive
eigenvector. One denotes this eigenvalue by λ 0 .
T
(e) Applying the previous results to A , show that there exists ∈
n T
IR such that > 0and (A − λ 0 I n )= 0.
(f) Let µ be an eigenvalue of A, associated to an eigenvector X.
T
Show that (λ 0 −|µ|) |X|≥ 0. Conclusion?
5. Let A ∈ M n (IR) be a matrix satisfying a ij ≥ 0 for every pair (i, j)of
distinct indices.
(a) Using the Exercise 3, show that
R(h; A):= (I n − hA) −1 ≥ 0,
for h> 0 small enough.
(b) Deduce that exp(tA) ≥ 0 for every t> 0 (the exponential of
matrices is presented in Chapter 7). Consider Trotter’s formula
m
exp tA = lim R(t/m; A) ,
m→+∞
where exp is the exponential of square matrices, defined in
Chapter 7. Trotter’s formula is justified by the convergence (see
Exercise 10 in Chapter 7) of the implicit Euler method for the
differential equation
dx
= Ax. (5.3)
dt
(c) Deduce that if x(0) ≥ 0, then the solution of (5.3) is nonnegative
for every nonnegative t.