Page 103 - Matrices theory and applications

5. Nonnegative Matrices
86
with nonzero sizes:
0

.
.
.
.
.
.


.


.


.
 .


0



.
.


0
 0 . . . . . . M 1 . . . 0 . . . . ··· . . . M p−1 

0 0
M p ··· ···
Remarks:
• The converse is true. For example, the spectrum of a cyclic matrix is
stable under multiplication by exp(2iπ/p).
• One may show that p divides n − n 0 ,where n 0 is the multiplicity of
the zero eigenvalue.
• The nonzero eigenvalues of A are the pth roots of those of the matrix
M 1 M 2 ··· M p , which is square, though its factors might not be square.
Proof
Let us denote by X the unique nonnegative eigenvector of A normalized
by X 1 =1. If Y is a unitary eigenvector, associated to an eigenvalue µ
of maximal modulus ρ(A), the inequality ρ(A)|Y | = |AY |≤ A|Y | im-
plies (Lemma 5.3.3) |Y | = X. Hence there is a diagonal matrix D =
diag(e iα 1 ,... ,e iα n )such that Y = DX. Let us define a unimodular com-
plex number e iγ = µ/ρ(A)and let B be the matrix e −iγ D −1 AD. One has
|B| = A and BX = X. For every j, one therefore has
n n

b jk x k = |b jk |x k .

k=1 k=1

Since X> 0, one deduces that B is real-valued and nonnegative; that is,
iγ
B = A. Hence D −1 AD = e A. The spectrum of A is thus invariant under
iγ
multiplication by e .
1
Let U = ρ(A) −1 R(A), which is included in S , the unit circle. The previ-
ous discussion shows that U is stable under multiplication. Since U is finite,
it follows that its elements are roots of unity. Since the inverse of a dth root
of unity is its own (d − 1)th power, U is stable under inversion. Hence it is
1
a finite subgroup of S ; that is, it is U p , for a suitable p.
Let P A be the characteristic polynomial and let ω =exp(2iπ/p). One
may apply the firstpartof the proofto µ = ωρ(A). One has thus D −1 AD =
n
ωA, and it follows that P A (X)= ω P A (X/ω). Therefore, multiplication by
ω sends eigenvalues to eigenvalues of the same multiplicities. In particular,
the eigenvalues of maximal modulus are simple.
p
Iterating the conjugation, one obtains D −p AD = A.Let us set
p
D = diag(d 1 ,... ,d n ).