Page 110 - Matrices theory and applications

(d) Deduce also that
is an eigenvalue of A.
6. Let A ∈ M n (IR) be a matrix satisfying a ij ≥ 0 for every pair (i, j)of
distinct indices. σ := sup{ λ; λ ∈ Sp A} 5.6. Exercises 93
(a) Let us define
σ := sup{ λ; λ ∈ Sp A}.
Among the eigenvalues of A whose real parts equal σ,let us
denote by µ the one with the largest imaginary part. Show that
for every positive large enough real number τ, ρ(A + τI n )=
|µ + τ|.
(b) Deduce that µ = σ = ρ(A) (apply Theorem 5.2.1).
7. Let B ∈ M n (IR) be a matrix whose off-diagonal entries are positive
and such that the eigenvalues have strictly negative real parts. Show
that there exists a nonnegative diagonal matrix D such that B :=

D −1 BD is strictly diagonally dominant, namely,

b < − b .
ii
ij
j =i
8. Let B ∈ M n(IR) be a nonnegative matrix and

B 0 m
A := .
I m B
(a) If an eigenvalue λ of A is associated to a positive eigenvector,
show that there exists µ> λ and Z> 0 such that BZ ≥ µZ.
Deduce that λ< ρ(B).
(b) Deduce that A admits no strictly positive eigenvector (first of
T
all, apply Theorem 5.2.1 to the matrix A ).
9. (a) Let B ∈ M n (IR) be given, with ρ(B) = 1. Assume that the
eigenvalues of B of modulus one are (algebraically) simple. Show
m
that the sequence (B ) m≥1 is bounded.
(b) Let M ∈ M n (IR) be a nonnegative irreducible matrix, with
T
ρ(M) = 1. We denote by x and y the left and right eigenvectors
T
T
for the eigenvalue 1 (Mx = x and y M = y ), normalized by
T
y x =1. We define L := xy T and B = M − L.
i. Verify that B−I n is invertible. Determine the spectrum and
the invariant subspaces of B by means of those of M.
m
ii. Show that the sequence (B ) m≥1 is bounded. Express M m
m
in terms of B .