International Mathematical Forum, Vol. 11, 2016, no. 13, 599 - 613
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/imf.2016.6448
A Family of Nonnegative Matrices with
Prescribed Spectrum and Elementary Divisors1
Ricardo L. Soto2
Dpto. Matemáticas, Universidad Católica del Norte
1270709, Antofagasta, Chile
Elvis Valero
Dpto. Matemáticas, Universidad de Tarapacá
1010069 Arica, Chile
c 2016 Ricardo L. Soto and Elvis Valero. This article is distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
A perturbation result, due to Rado, shows how to modify r eigenvalues of a matrix of order n, via a perturbation of rank r ≤ n, without
changing any of the n − r remaining eigenvalues. This result extended
a previous one, due to Brauer, on perturbations of rank 1. Both results
have been exploited in connection with the nonnegative inverse eigenvalue problem and the nonnegative inverse elementary divisors problem.
In this paper, we use the Rado result from a more general point of view,
constructing a family of matrices with prescribed spectrum and elementary divisors, generalizing previous results. We also apply our results
to the nonnegative pole assigment problem.
Mathematics Subject Classification: 15A18, 15A29
Keywords: nonnegative inverse eigenvalue problem, nonnegative inverse
elementary divisors problem
1
2
Work supported by Fondecyt 1120180.
Corresponding author
600
1
Ricardo L. Soto and Elvis Valero
Introduction
The origin of the present paper is a perturbation result due to Brauer [1],
which shows how to modify one single eigenvalue of a matrix, via a rank-one
perturbation, without changing any of the remaining eigenvalues. This result
was first used in connection with the nonnegative inverse eigenvalue problem
(NIEP) by Perfect [8], and later in [10], giving rise to a number of realizability
criteria (see [14] and the references therein). Closer to our approach in this
paper is Rado’s extension of Brauer’s result, Theorem 2.2 below, which was
first used by Perfect in [9] and later in [11] to derive sufficient conditions for the
NIEP to have a solution. Both, Brauer and Rado results, have been also used
to derive efficient sufficient conditions for the real and symmetric nonnegative
inverse eigenvalue problem [11, 12], as well as for the nonnegative inverse
elementary divisors problem (NIEDP) [3, 15, 16]. In this paper we use the Rado
result (Theorem 2.1 below) in a more general context, to construct in Section
2, a family of nonnegative matrices with prescribed spectrum. In Section 3 we
construct a family of nonnegative matrices with prescribed elementary divisors.
In Section 4 we apply our results to the pole assigment problem. We denote
the spectrum of a matrix A as σ(A).
2
Rank-r perturbations
The following result, due to R. Rado and introduced by Perfect [9] in 1955, is
an extension of the above mentioned Brauer result. It shows how to change r
eigenvalues of an n×n matrix A via a perturbation of rank r, without changing
any of the remaining n − r eigenvalues.
Theorem 2.1 (Rado) Assume that for r ≤ n all following are given:
i) let A ∈ Cn×n be an arbitrary matrix with σ(A) = {λ1 , ..., λn };
ii) let U = [u1 | · · · | ur ] ∈ Cn×r be a matrix of rank r such that Aui = λi ui for
each i = 1, ..., r;
iii) let V = [v1 | · · · | vr ] ∈ Cn×r be an arbitrary matrix;
iv) let Ω = diag(λ1 , ..., λr ) ∈ Cr×r ;
Then σ(A + U V T ) = σ(Ω + V T U ) ∪ {λr+1 , ..., λn }.
We state an interesting consequence of Theorem 2.1
Corollary 2.1 Assume that for r ≤ n all following are given:
i) let A ∈ Cn×n be an arbitrary matrix with σ(A) = {λ1 , ..., λn };
ii) let U = [u1 | · · · | ur ] ∈ Cn×r be a matrix of rank r such that Aui = λi ui for
each i = 1, ..., r;
iii) let W = [w1 | · · · | wr ] ∈ Cn×r be an arbitrary matrix with W T U = Ir ;
A family of nonnegative matrices with prescribed spectrum and ...
601
iv) let C ∈ Cr×r be an arbitrary matrix and let Ω = diag(λ1 , ..., λr ) ∈ Cr×r ;
Then σ(A + U CW T ) = σ(Ω + C) ∪ {λr+1 , ..., λn }.
Proof. If we take V = W C T in Theorem 2.1, then Ω + V T U = Ω + C and
A + U V T = A + U CW T .
The advantage of this result with respect to Theorem 2.1 is that Corollary
2.1 permits to construct a family of matrices with a prescribed spectrum since
the set of eigenvalues {µ1 , µ2 , ..., µr } of Ω + C does not depend on the matrix
W : namely, for any W such that W T U = Ir we can construct a new matrix
A + U CW T with the desired spectrum {µ1 , ..., µr , λr+1 , ..., λn }.
Corollary 2.2 Assume all the following are given:
i) let S = [sij ]ri,j=1 ∈ Cr×r be an arbitrary matrix with σ(S) = {ρ1 , ..., ρr };
ii) for each i = 1, ..., r let Ai ∈ Cni ×ni be an arbitrary matrix with an eigenvalue
sii ;
iii) for each i = 1, ..., r let ui ∈ Cni be an eigenvector of Ai associated with the
eigenvalue sii ;
iv) for each i = 1, ..., r let wi ∈ Cni be a vector such that wiT ui = 1.
Then the matrix


A1
s12 u1 w2T · · · s1r u1 wrT
 s21 u2 wT A2
· · · s2r u2 wrT 
1


(1)

 ..
..
.
.
. . ..

 .
.
sr1 ur w1T sr2 ur w2T · · · Ar
has spectrum {σ(A1 ) − {s11 }} ∪ ... ∪ {σ(Ar ) − {srr }} ∪ {ρ1 , ..., ρr }.
Proof. We are going to
Let

A1
 0

A =  ..
 .
0
apply Corollary 2.1. Therefore:

0 ... 0
A2 . . . 0 

∈ C(n1 +...+nr )×(n1 +...+nr ) ;
..
. . . .. 

.
.
0 . . . Ar
Let



U =

u1
0
..
.
0
0 ... 0
u2 . . . 0
..
. . ..
. .
.
0 . . . ur



 ∈ C(n1 +...+nr )×r

and observe that for each i = 1, ..., r, the ith column of U is an eigenvector of
A with eigenvalue sii ;
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Ricardo L. Soto and Elvis Valero
Let



W =

w1
0
..
.
0
0 ...
w2 . . .
..
..
.
.
0 ...
and observe that W T U = Ir .
Let



C=

0
s21
..
.
sr1

0
0
..
.


 ∈ C(n1 +...+nr )×r

wr
s12 . . .
0
...
..
...
.
sr2 . . .
s1r
s2r
..
.



.

0
Then A+U CW T , as given in (1), has spectrum {ρ1 , ..., ρr }∪{σ(A)−{s11 , ..., srr }}.
Remark 2.1 Assume in Corollary 2.2 that the matrices Ai ∈ Cni ×ni are all
nonnegative with constant row sums equal to sii (the Perron eigenvalue). Then
ui = e = (1, 1, ..., 1)T , i = 1, ..., r. Let C an r×r nonnegative matrix. Moreover,
assume that ωi is nonnegative, i = 1, ..., r. The matrix A+U CW T in (1) is nonnegative with the spectrum {σ(A1 )−{s11 }}∪· · ·∪{σ(Ar )−{srr }}∪{ρ1 , ..., ρr }.
Hence, under these conditions we may construct a family of nonnegative matrices with a prescribed spectrum.
Example 2.1 Let Λ = {6, 3, 3, −5, −5}. We look for a family of nonnegative
matrices with spectrum Λ. Then we take the partition
Λ1 = {6, −5}, Λ2 = {3, −5}, Λ3 = {3} with
Γ1 = {5. − 5}, Γ2 = {5. − 5}, Γ3 = {2} being
realizable by
A1 = A2 =
0 5
5 0
, A3 = [2] .
Then the matrix


A=

A1
A2


=


A3
has eigenvalue 5, 5, 2, −5, −5. Moreover
0
5
0
0
0
5
0
0
0
0
0
0
0
5
0
0
0
5
0
0
0
0
0
0
2






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A family of nonnegative matrices with prescribed spectrum and ...



U =


1
1
0
0
0
0
0
1
1
0
0
0
0
0
1






, W = 





α
γ
η
1 − α −γ −η 

β
δ
θ 
,
−β 1 − δ −θ 
0
0
1
with W T U = I. Next we compute




5 0 1
0 0 1
B =  1 5 0  , with C = B − diagB =  1 0 0  .
0 4 2
0 4 0
B has spectrum {6, 3, 3} and diagonal entries {5, 5, 2}.
Thus, for η = θ = γ = β = 0; 0 ≤ α, δ ≤ 1, the matrix

η
5−η
θ
−θ
 5+η
−η
θ
−θ

T

−α + 1
β
−β + 5
A + U CW =  α
 α
−α + 1 β + 5
−β
4γ
−4γ
4δ
−4δ + 4
1
1
0
0
2






gives a family of nonnegative matrices with spectrum Λ = {6, 3, 3 − 5, −5}.
We can also obtain a family of persymmetric nonnegative matrices with prescribed spectrum. Persymmetric matrices are common in physical sciences
and engineering. They arise, for instance, in the control of mechanical and
electric vibrations, where the eigenvalues of the Gram matrix, which is symmetric and persymmetric, play an important role. As the superscript T, in
AT , denotes the transpose of A, the superscript F, in AF , denotes the fliptranspose of A, which flips A across its skew-diagonal. If A = (aij )mn , then
AF = (an−j+1,m−i+1 )nm . A matrix A is said to be persymmetric if AF = A,
that is, if it is symmetric across its lower-left to upper-right diagonal. In [4],
the authors give a persymmetric version of Theorem 2.1, which we use in the
following
Corollary 2.3 Assume that for r ≤ n all following are given:
i) let A ∈ Cn×n be a persymmetric nonnegative matrix with σ(A) = {λ1 , ..., λn };
ii) let U = [u1 | · · · | ur ] ∈ Cn×r be a matrix of rank r such that Aui = λi ui for
each i = 1, ..., r;
iii) let C ∈ Cr×r be persymmetric nonnegative matrix and let Ω = diag(λ1 , ..., λr ) ∈
Cr×r ;
Then A + U CU F is persymmetric nonnegative matrix with spectrum σ(A +
U CU F ) = {µ1 , . . . , µr } ∪ {λr+1 , ..., λn }, where µ1 , . . . , µr are the eigenvalues of
Ω + CU F U (Ω + C if the columns of U are taken as orthonormal).
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Ricardo L. Soto and Elvis Valero
Proof. It is clear that U CU F is persymmetric nonnegative and then A +
U CU F es also persymmetric nonnegative. From Theorem 2.1 and Remark
2.1, A + U CU F has the spectrum σ(A).
Example 2.2 We construct a family of persymmetric nonnegative matrices
with spectrum Λ = {6, 4, −2, −2, −3, −3}. We take the partition Λ = Λ1 ∪ Λ2
with
Λ1 = {6, −2, −3}, Λ2 = {4, −2, −3}
and the associated realizable lists
Γ1 = {5, −2. − 3}, Γ2 = {5, −2, −3}.
Let

A=
A1 0
0 AF1




=



√0
√5
5
0
0
0

√ √
5
5 0
0
0
0
3
0
0
0 


3
0
0
0 √0 
,
0
0
0
3 √5 

0
0 √3 √0
5 
5
5 0
0
0
where A1 and AF1 realizan Γ1 and Γ2 respectively. Let U be the matrix of the
normalized eigenvectors of A corresponding to the eigenvalues 5, 5 :

√2
√14
√5
√14
√5
14




U =
 0


 0
0
0

0 


0 
 and let C = 0 y .
√
x 0
√5 

√14 
√5 
14
√2
14
Then for all x, y ≥ 0,





A + U CU F = 



√0
√5
5
√
1
x 5
7 √
1
x 5
7
2
x
7
√
5
0
3
5
x
14
5
x
14√
1
x 5
7
√
5
3
0
5
x
14
5
x
14√
1
x 5
7
√
√
1
y 5
7
5
y
14
5
y
14
1
y 5
7
5
y
14
5
y
14
0
√3
5
3
√0
5
2
y
7√

1
y 5
7 √
1
y 5
7√





5
√ 

5 
0
is persymmetric nonnegative. In particular for x = y = 1, A + U CU F has the
spectrum Λ.
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A family of nonnegative matrices with prescribed spectrum and ...
3
Inverse elementary divisors problem
In this section we show how to construct a family of nonnegative matrices
with prescribed elementary divisors. Here we exploit Corollary 2.2 for the
nonnegative case (as in Remark 2.1).
Let A ∈ Cn×n and let


Jn1 (λ1 )
Jn2 (λ2 )


J(A) = S −1 AS = 

..




.
Jnk (λk )
be the Jordan canonical form of A.

λi 1

λi

Jni (λi ) = 

The ni × ni submatrices

..
.


 , i = 1, 2, . . . , k
..
. 1 
λi
are the Jordan blocks of J(A). The elementary divisors of A are the polynomials (λ − λi )ni , that is, the characteristic polynomials of Jni (λi ), i = 1, . . . , k.
The nonnegative inverse elementary divisor problem (NIEDP ) is the problem
of determining necessary and sufficient conditions under which the polynomials
(λ − λ1 )n1 , (λ − λ2 )n2 , . . . , (λ − λk )nk , n1 + · · · + nk = n, are the elementary
divisors of an n × n nonnegative matrix A [5, 6, 7]. The NIEDP contains
the NIEP and both problems remain unsolved (they have been solved only for
n ≤ 4).
The following result gives a sufficient condition for the existence and construction of a nonnegative matrix with prescribed spectrum and elementary
divisors.
Corollary 3.1 Let Λ = {λ1 , λ2 , ..., λn } be a list of complex numbers, with
n
P
λi ≥ 0. Assume that there exists a
Λ = Λ, λ1 ≥ maxi |λi | , i = 2, . . . , n, and
i=1
partition Λ = Λ0 ∪ Λ1 ∪ · · · ∪ Λr , where some of the lists Λi , i = 1, . . . , r, can
be empty, such that:
i) let S = [sij ]ri,j=1 is an r × r nonnegative matrix with spectrum σ(S) = Λ0 =
{λ1 , ..., λr };
ii) for each i = 1, ..., r, there exists a (pi + 1) × (pi + 1) nonnegative matrix Ai ,
with constant row sums equal to sii , spectrum Γi = {sii , λi1 , λi2 , . . . , λipi }, and
prescribed elementary divisors associated with Γi .
606
Ricardo L. Soto and Elvis Valero
iii) for each i = 1, ..., r let ui = e be an eigenvector of Ai corresponding to the
eigenvalue sii .
iv) Let A be the block diagonal matrix A = diag{A1 , . . . , Ar } and let U =
b r ] such that Ab
bi.
[b
u1 | · · · | u
ui = sii u
b i = 1, and
v) for each i = 1, ..., r let wi be a nonnegative vector such that wiT u
let W = [w1 | · · · | wr ] .
Then the nonnegative matrix A+U CW T , where C = S−diag{s11 , s22 , . . . , srr },
has spectrum Λ and the prescribed elementary divisors associated with Λi , i =
1, ..., r.
Proof. From ii) − iv) A = diag{A1 , . . . , Ar } is nonnegative with spectrum
b i , i = 1, . . . , r. Then from i) and Remark 2.1 the
Γ1 ∪ · · · ∪ Γr and Ab
ui = sii u
result follows.
Example 3.1 Let Λ = {7, 1, −2, −2, −1+3i, −1−3i}. We want to construct a
family of nonnegative matrices with spectrum Λ, and with elementary divisors
(λ − 7), (λ − 1), (λ + 2)2 , λ2 + 2λ + 10. Then we take the partition
Λ0 = {7, −1 + 3i, −1 − 3i}, Λ1 = {−2, −2}, Λ2 = {1}, Λ3 = ∅ with
Γ1 = {4, −2, −2}, Γ2 = {1, 1}, Γ3 = {0}.
We compute the

4
34

S=
7
0
nonnegative matrix

0 3
1 87  with spectrum Λ0 and diagonal entries 4, 1, 0,
7 0
and the realizing matrices

  


4 0
0
1 0
2
2
A1 =  7 −2 −1  +  1  −4 2 2 =  3 0 1  ,
6 0 −2
1
2 2 0
1 0
A2 =
and A3 = [0] ,
0 1
with spectra Γ1 , Γ2 , Γ3 , respectively. Observe that the matrix A1 has elementary
divisors (λ − 4), (λ + 2)2 . Let




α
γ
η
1 0 0
 1 − α −γ −η 
 1 0 0 




 0

 1 0 0 
0
0
 and U = 

W =
 β

 0 1 0 .
δ
θ




 −β 1 − δ −θ 
 0 1 0 
0
0
1
0 0 1
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A family of nonnegative matrices with prescribed spectrum and ...
Then W T U = I, and


A1
 + U CW T
A2
A=
A3

3η
2 − 3η
 3η + 3
−3η

 3η + 2
2 − 3η
=
 34 α + 8 η 34 − 8 η − 34 α
7
7
7
7
7
 34
 α + 8 η 34 − 8 η − 34 α
7
7
7
7
7
7γ
−7γ
3θ
−3θ
3θ
−3θ
3θ
−3θ
8
34
8
θ
+
β
+
1
−
θ
− 34
β
7
7
7
7
8
34
34
θ+ 7β
1 − 7 β − 78 θ
7
7δ
7 − 7δ
2
1
0
0
0
0
3
3
3
8
7
8
7








0
is a family of matrices with spectrum Λ and the desired elementary divisors.
In particular for
η = γ = θ = β = 0, and 0 ≤ α, δ ≤ 1
we have the nonnegative family

0
 3

 2
B=
 34 α 34
7
7
 34
 α 34
7
7
0
2
0
2
− 34
α
7
34
− 7α
0
2 0
0
1 0
0
0 0
0
0 1
0
0 0
1
0 7δ 7 − 7δ
3
3
3
8
7
8
7




,



0
with spectrum Λ and elementary divisors
(λ − 7), (λ − 1), (λ + 2)2 , λ2 + 2λ + 10.
We can also solve the inverse elementary divisors problem for a family of
nonnegative matrices with constant row sums and spectrum Λ = {λ1 , ..., λn }
in the following cases, which have been solved for a single nonnegative matrix
in [13, 15, 3]:
i) λ1 > λ2 ≥ · · · ≥ λn ≥ 0
ii) λ1 > 0 ≥ λ2 ≥ · · · ≥ λn
iii) Reλi ≤ 0, |Reλ
√ i | ≥ |Imλi |
iv) Reλi ≤ 0, 3Reλi ≥ |Imλi | .
For each one of these cases, we may apply the same techniques used in [13, 15, 3]
for a single nonnegative matrix. For example, in the first case i) Λ = {5, 3, 3, 3},
we use from [8], the Perfect matrix


1 1
1
1
 1 1
1 −1 

P =
 1 1 −1 0  ,
1 −1 0
0
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Ricardo L. Soto and Elvis Valero
to obtain a positive matrix


A = P DP −1 = 

13
4
1
4
1
4
1
4
1
4
13
4
1
4
1
4
1
2
1
2
7
2
1
2

1
1 
,
1 
4
where D = diag{5, 3, 3, 3}. If Ei,j is the matrix with 1 in position (i, j) and
zeros elsewhere, then

 1
1
a + 13
− 21 a 12 1
2
4
4
 1 a + 1 13 − 1 a 1 1 
2
4
4
2
2

M = A + aP E3,4 P −1 = 
 1 − 1a 1a + 1 7 1  ,
4
2
2
4
2
1
1
1
4
4
4
2
with 0 ≤ a ≤ 12 , is a nonnegative matrix with constant row sums equal to 5,
and with elementary divisors J1 (5), J2 (3), J1 (3).
For J1 (5), J3 (3) we have with E = aE3,4 + bE2,3 ,


M = A + aP EP −1 = 

1
a + 14 b + 13
2
4
1
1
1
a
+
b
+
2
4
4
1
b − 12 a + 41
4
1
− 14 b
4
1
b − 21 a + 14
4
1
b − 12 a + 13
4
4
1
a + 14 b + 14
2
1
− 41 b
4
1
− 21 b
2
1
− 21 b
2
7
− 21 b
2
1
b + 12
2

1
1 
,
1 
4
which for 0 ≤ a, b ≤ 1, is nonnegative with the desired JCF.
For the second case ii) λ1 > 0 ≥ λ2 ≥ · · · ≥ λn , we apply Brauer result:

  
10
0
0
0
0
1
 12


−1 −1 0
0   1 


 
0 −1 0
0 
M =
 11
 +  1  −10 1 1 4 4
 15 − a a
0 −4 −1   1 
14 − a 0
a
0 −4
1



=


0
1
1
2
0
0
1
1
0
5−a a+1
1
4−a
1
a+1
4
4
4
0
4
4
4
4
3
0



,


which, for 0 ≤ a ≤ 4, is nonnegative, with constant row sums λ1 = 10, and
elementary divisors J1 (10), J2 (−1), J2 (−4). In the same way, for the list
Λ = {4, −1 + i, −1 − i, −1 + i. − 1 − i}
A family of nonnegative matrices with prescribed spectrum and ...
609
we have



M =





=


4 0
0
0
0
4 −1 1
0
0
7 −1 −1 −1 0
4 0
0 −1 1
6 0
0 −1 −1
1−a
1−a
4−a
1−a
3−a
1
0
0
1
1
1
2
0
1
1
1
a
1
a
0
a
0 a+1
0 a−1


 
 
+
 
 
1
1
1
1
1



 −3 − a 1 1 1 a





 , 0 ≤ a ≤ 1,


with JCF



J(M ) = 


4
4
0
0
0
0
0 −1 − i
0
0
0
0
1
−1 − i
0
0
0
0
0
−1 + i
0
0
0
0
1
−1 + i






Nonnegative pole assigment problem
In [2] the authors show applications of Theorem 2.1 (Rado Theorem), to deflation techniques and the pole assigment problem for multi-imput multi-output
(MIMO) systems defined by pairs of matrices (A, B). Corollary 2.1 may also
be applied to deflation techniques and the pole assignment problem to obtain
a nonnegative solution matrix. Let A be an n × n matrix, let B be an n × r
matrix with σ(A) = {λ1 , λ2 , ..., λn }. Let {µ1 , µ2 , ..., µr } be a list of numbers.
The pole assignment problem for MIMO systems asks conditions on the pair
(A, B) for the existence of a matrix F such that
σ(A + BF T ) = {µ1 , µ2 , ..., µr , λr+1 , λr+2 , ..., λn }.
For the nonnegative pole assigment problem we have the following result:
Corollary 4.1 Let (A, B) be a given pair of matrices with A an n × n nonnegative matrix and B an n × r matrix. Let σ(A) = {λ1 , λ2 , ..., λn } and let
µ = {µ1 , µ2 , ..., µr } be a list of numbers. Let X = [α1 x1 | ... | αr xr ] be an
n × r matrix such that rankX = r and AT xi = λi xi , i = 1, 2, ..., r. If there
610
Ricardo L. Soto and Elvis Valero
exists a column bj = [b1j , b2j , ..., brj ]T of the matrix B, with all its entries being nonnegative suchPthat bTj xi 6= 0, i = 1, 2, ..., r, σ(Ω + [bj | ... | bj ]T X) =
r
{µ1 , µ2 , ..., µr } and
i=1 αi xsi ≥ 0, s = 1, 2, ..., n, then there exists a nonnegative matrix F such that the nonnegative matrix A + BF T has spectrum
{µ1 , ..., µr , λr+1 , ..., λn }.
Proof. Let the n × r matrix
C T = [bj | ... | bj ] = B[ej | ... | ej ],
where ej is the j th column of the identity matrix of order r. Then
C T X T = B[e
... | ej ]X T
 j |P
b1j ri=1 αi x1i · · ·

...
=  ...
Pr
bnj i=1 αi x1i · · ·
b1j
..
.
Pr
bnj
Pr
i=1 αi xni
i=1


 ≥ 0.
αi xni
If we take F T = [ej | ... | ej ]X T then
0 ≤ A + C T X T = A + B[ej | ... | ej ]X T = A + BF T
and for Theorem 2.1
σ(A + BF T ) = σ(A + C T X T ) = σ(Ω + [ej | ... | ej ]T B T X) ∪ {λr+1 , ..., λn }.
√
√
Example
4.1
Consider
the
pair
(A,
B)
and
µ
=
{8
+
19,
8
−
19} where




0 5 1 1 0
0 0
 0 0 
 5 0 0 0 0 







A=
 0 0 0 5 4  and B =  0 0  = [b1 | b2 ] ,
 0 0 5 0 0 
 0 4 
0 1
1 1 0 0 2
with σ(A) = {6, 3, 3, −5, −5}. We compute the eigenvector of AT :
1 1 1 1
xλ=6 = (1, 1, 1, 1, 1)T , xλ=3 = (− , − , , , 1)T
2 2 4 4
xλ=−05 = {(−5, 5, 0, 1, 0)T , (
35 179 7
,−
, − , 0, 1)T }.
4
20
4
A family of nonnegative matrices with prescribed spectrum and ...
611


0
 0 
  T
T

Since for b2 = 
 0  , b2 xλ=6 6= 0, and b2 xλ=3 6= 0, we may change the eigen 4 
1
√
√
values λ = 6 and λ = 3 to µ1 = 8 + 19 and µ2 = 8 − 19, respectively. To
do this, let
0 0
T
C = [b2 | b2 ] = B
= B [e2 | e2 ]
1 1
with
T
X =
α1
α1
α1 α1 α1
1
1
− 2 α2 − 2 α2 14 α2 14 α2 α2
.
Since σ(Ω + [b2 | b2 ]T X) = {µ1 , µ2 }, must be
√
√
6 + 5α1 + 3 + 2α2
= 8 + √19 + 8 − √ 19
.
(6 + 5α1 )(3 + 2α2 ) − (5α1 )(2α2 ) = (8 + 19)(8 − 19)
Then α1 = 1, α2 = 1
Taking F T = [e2 | e2 ] X T , we have that

0
 5

A + BF T = 
 0
 2
3
2
5
0
0
2
1
0
0
10
1
0
5
5
3
2
5
4
5
4
0
0
4
8
4






is nonnegative and
σ(A + BF T ) = σ(Ω + [e2 | e2 ]T B T X) ∪ {3, −5, −5}
where
T
T
Ω + [e2 | e2 ] B X =
has the spectrum 8 +
11 2
5 5
√
√
19, 8 − 19.
References
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http://dx.doi.org/10.1215/s0012-7094-52-01910-8
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Ricardo L. Soto and Elvis Valero
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Received: May 2, 2016; Published: June 16, 2016