Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem

0.3 MB·English

by  Richard Ellard

Connecting sufficient conditions for the Symmetric
Nonnegative Inverse Eigenvalue Problem
Richard Ellard , Helena Sˇmigoc
5 ∗ ∗
1
School of Mathematical Sciences,
0
University College Dublin,
2
Belfield, Dublin 4, Ireland

n
a
J

6
2

Abstract
]
P
S We say that a list of real numbers is “symmetrically realisable” if it is the
. spectrum of some (entrywise) nonnegative symmetric matrix. The Sym-
h
t metric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of
a
m characterising all symmetrically realisable lists.
In this paper, we present a recursive method for constructing symmetri-
[
cally realisable lists. The properties of the realisable family we obtain allow
1
us to make several novel connections between a number of sufficient condi-
v
2 tions developed over forty years, starting with the work of Fiedler in 1974.
6
Weshowthatessentially allpreviouslyknownsufficientconditionsareeither
4
contained in or equivalent to the family we are introducing.
6
0
Keywords: Nonnegative matrices, Symmetric Nonnegative Inverse
.
1
Eigenvalue Problem, Soules matrix
0
5 2010 MSC: 15A18, 15A29
1
:
v
i 1. Introduction
X
r
This paper explores the spectral properties of symmetric nonnegative
a
matrices. Nonnegative matrices were a topic of special interest of Hans
Schneider: he had over fifty papers in the area, the most relevant of these
to our present paper being [1, 2].
∗The authors’ work was supported by Science Foundation Ireland under Grant
11/RFP.1/MTH/3157.
Email addresses: [email protected] (Richard Ellard),
[email protected] (Helena Sˇmigoc)
Let σ := (λ ,λ ,...,λ ) be a list of n real numbers. If there exists a
1 2 n
nonnegative symmetric matrix A with spectrum σ, then we say σ is sym-
metrically realisable and that A realises σ. The Symmetric Nonnegative
Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all
symmetrically realisable lists.
Since the spectrum of a symmetric matrix is necessarily real, the re-
striction that σ consist only of real numbers is a natural one; however, if we
allow Ato benot-necessarily-symmetric, butconsider only lists of real num-
bers, then the resulting problem is known as the Real Nonnegative Inverse
Eigenvalue Problem (RNIEP).
In this paper, we describe a recursive method of constructing symmet-
rically realisable lists, using a construction of Sˇmigoc [3]. The properties of
the realisable lists obtainable in this way allow us to show that essentially
all known sufficient conditions to date are either contained in or equivalent
to the realisability we are introducing. This includes the method of Soules
[4] (later generalised by Elsner, Nabben and Neumann [5]), one of the most
influential methodsof constructingsymmetrically realisable lists. Moreover,
since wealso show thatthe realising matrices we obtain by ourmethod have
the same form as the ones obtained by the method of Soules, our approach
gives a new insight into Soules realisability.
We also consider a sufficient condition for the RNIEP due to Borobia,
Moro and Soto [6] called “C-realisability” and a family of sufficient condi-
tions for the SNIEP due to Soto [7]. We show that C-realisability is also
sufficient for the SNIEP and that σ is C-realisable if and only if it satisfies
one of Soto’s conditions. Such σ are precisely those which may be obtained
by ourmethod orthemethod of Soules. Theequivalence of all fourmethods
is proved in Section 4.
In Section 2, we outline the background to and terminology used in this
paper. In Section 3, we describe our recursive approach and prove several
properties of the realisable lists which may be obtained in this manner.
Section 5 can be seen as a survey of sufficient conditions for the SNIEP
given in the literature, including Suleimanova [8], Perfect [9], Ciarlet [10],
Kellog [11], Salzmann [12], Fiedler [13], Borobia [14] and Soto [15]. We
show that if σ obeys any of these sufficient conditions, then σ may also be
obtained by our method.
2. Preliminaries and notation
To denote that σ is symmetrically realisable, we may sometimes write
σ . In this paper, the diagonal elements of the realising matrix will
n
∈ R
2
also be important; hence, if there exists a nonnegative symmetric matrix A
with diagonal elements (a ,a ,...,a ) and specrum σ, then we write
1 2 n
σ (a ,a ,...,a ).
n 1 2 n
∈ R
If we wish to specify that λ is the Perron eigenvalue of the realising matrix,
1
we will separate λ from the remaining entries in the list by a semicolon,
1
e.g. we may write
(λ ;λ ,...,λ )
1 2 n n
∈ R
or
(λ ;λ ,...,λ ) (a ,a ,...,a ).
1 2 n n 1 2 n
∈ R
The remaining eigenvalues λ ,λ ,...,λ will generally be considered un-
2 3 n
ordered. Thediagonalelementsa ,a ,...,a willalsogenerallybecosidered
1 2 n
unordered and they may appear in any order on the diagonal of A, i.e. we
do not assume that a is the (i,i) entry of A. Sometimes we will assume
i
that the λ or a are arranged in non-increasing order and if this is the case,
i i
we will say so explicitly. In this paper, will always be replaced by either
R
or , depending on whether we are considering realisability via Soules or
S H
our recursive method.
We begin by stating some necessary conditions (due to Fiedler [13]) for
σ to be the spectrum of a nonnegative symmetric matrix with specified
diagonal elements:
Theorem 2.1. [13] If λ λ λ , a a a 0
1 2 n 1 2 n
≥ ≥ ··· ≥ ≥ ≥ ··· ≥ ≥
and (λ ,λ ,...,λ ) is the spectrum of a nonnegative symmetric matrix with
1 2 n
diagonal elements (a ,a ,...,a ), then
1 2 n
λ a ,
1 1
≥
n n
λ = a
i i
i=1 i=1
X X
and
s s 1
−
λ +λ a +a +a
i k i k 1 k
≥ −
i=1 i=1
X X
for all 1 s < k n (with the convention that 0 a = 0).
≤ ≤ i=1 i
Fiedler also gave the following sufficient condPitions:
3
Theorem 2.2. [13] Let λ λ λ and a a a 0
1 2 n 1 2 n
≥ ≥ ··· ≥ ≥ ≥ ··· ≥ ≥
satisfy the following conditions:
k k
λ a : k = 1,2,...,n 1,
i i
≥ −
i=1 i=1
X X
n n
λ = a ,
i i
i=1 i=1
X X
λ a : k = 2,3,...,n 1. (1)
k k 1
≤ − −
Then (λ ,λ ,...,λ ) is the spectrum of a nonnegative symmetric matrix
1 2 n
with diagonal elements (a ,a ,...,a ).
1 2 n
For n 3, the question of whether σ (a ,a ,...,a ) is completely
n 1 2 n
≤ ∈ R
solved by Theorems 2.1 and 2.2. If n = 2, the matrix
a (λ a )(λ a )
1 1 1 1 2
− −
(λ a )(λ a ) a
(cid:20) 1− 1 1− 2 p 2 (cid:21)
has spectrum (λp,a +a λ ) and hence if λ λ and a a 0, then
1 1 2 1 1 2 1 2
− ≥ ≥ ≥
(λ ,λ ) is the spectrum of a nonnegative symmetric matrix with diagonal
1 2
elements (a ,a ) if and only if the following conditions are satisfied:
1 2
λ a ,
1 ≥ 1 (2)
λ +λ = a +a .
1 2 1 2
(cid:26)
Ifn = 3, thentheconditionsofTheorems2.1and2.2areidenticalandhence
if λ λ λ and a a a 0, then (λ ,λ ,λ ) is the spectrum of
1 2 3 1 2 3 1 2 3
≥ ≥ ≥ ≥ ≥
a nonnegative symmetric matrix with diagonal elements (a ,a ,a ) if and
1 2 3
only if the following conditions are satisfied:
λ a λ ,
2 1 1
≤ ≤
λ a , (3)
3 3
 ≤
λ +λ +λ = a +a +a .
 1 2 3 1 2 3
2.1. The Soules approach to the SNIEP
Soules’ approach to the SNIEP focuses on constructing the eigenvectors
of the realising matrix A. Starting from a positive vector x Rn, Soules [4]
∈
showed how to construct a real orthogonal n nmatrix R with first column
×
x such that for all λ λ λ 0, the matrix RΛRT—where
1 2 n
≥ ≥ ··· ≥ ≥
Λ := diag(λ ,λ ,...,λ )—is nonnegative. This motivated Elsner, Nabben
1 2 n
and Neumann [5] to make the following definition:
4
Definition 2.3. Let R Rn n be an orthogonal matrix with columns
×
∈
r ,r ,...,r . R is called a Soules matrix if r is positive and for every
1 2 n 1
diagonal matrix Λ := diag(λ ,λ ,...,λ ) with λ λ λ 0, the
1 2 n 1 2 n
≥ ≥ ··· ≥ ≥
matrix RΛRT is nonnegative.
With regard to the SNIEP, a key property of Soules matrices is the
following:
Theorem 2.4. [5] Let R be a Soules matrix and let Λ := diag(λ ,λ ,...,
1 2
λ ), where λ λ λ . Then the off-diagonal entries of the matrix
n 1 2 n
≥ ≥ ··· ≥
RΛRT are nonnegative.
Therefore, if R = (r ) is an n n Soules matrix and Λ := diag(λ ,
ij 1
×
λ ,...,λ ), where λ λ λ , then σ := (λ ,λ ,...,λ ) is the
2 n 1 2 n 1 2 n
≥ ≥ ··· ≥
spectrum of a nonnegative symmetric matrix if the diagonal elements of
RΛRT are nonnegative. This motivates the following definition:
Definition 2.5. Let λ λ λ and let a ,a ,...,a 0. We
1 2 n 1 2 n
≥ ≥ ··· ≥ ≥
write
(λ ;λ ,...,λ ) (a ,a ,...,a ) (4)
1 2 n n 1 2 n
∈ S
if there exists an n n Soules matrix R such that the matrix RΛRT—where
×
Λ := diag(λ ,λ ,...,λ )—has diagonal elements (a ,a ,..., a ). We write
1 2 n 1 2 n
(λ ,λ ,...,λ )
1 2 n n
∈ S
if there exist a ,a ,...,a 0 such that (4) holds and we call the Soules
1 2 n n
≥ S
set.
Elsner, Nabben and Neumann generalised the work of Soules by char-
acterising all Soules matrices. In order to state their characterisation, we
require two definitions:
Definition 2.6. Let = ( , ,..., ) be a sequence of partitions of
1 2 n
N N N N
1,2,...,n . Wesaythat isSoules-type if hasthefollowingproperties:
{ } N N
(i) foreachi 1,2,...,n ,thepartition consistsofpreciselyisubsets,
i
∈{ } N
say = , ,..., ;
i i,1 i,2 i,i
N {N N N }
(ii) for each i 2,3,...,n , there exist indices j,k,l with 1 j i 1
∈ { } ≤ ≤ −
and 1 k < l i, such that = , and
i 1 i 1,j i i,k i,l
≤ ≤ N− \ N− N \ {N N }
= , i.e. is constructed from by splitting one
i 1,j i,k i,l i i 1
N− N ∪N N N−
of the sets , ,..., into two subsets.
i 1,1 i 1,2 i 1,i 1
N− N− N− −
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If = ( , ,..., ) is a Soules-type sequence of partitions of 1,2,...,
1 2 n
N N N N {
n , then we label the sets and in (ii) as and , i.e. for
} Ni,k Ni,l Ni∗ Ni∗∗
i 2,3,...,n , we define and to be those sets in which do not
∈ { } Ni∗ Ni∗∗ Ni
coincide with any of the sets in .
i 1
N−
Definition 2.7. Let x Rn be a positive vector and let = ( , ,
1 2
∈ N N N
..., ) be a Soules-type sequence of partitions of 1,2,...,n . For each
n
N { }
i 2,3,...,n , we define x(i) to be the vector in Rn whose ith component
∈ { }
is: N
x : i
i ∈ Ni∗
0 : i /
(cid:26) ∈ Ni∗
and we define xˆ(i) to be the vector in Rn whose ith component is:
N
x : i
i ∈ Ni∗∗
0 : i / .
(cid:26) ∈ Ni∗∗
We are now ready to state the characterisation of Soules matrices due
to Elsner, Nabben and Neumann:
Theorem 2.8. [5] Let x Rn be a positive vector and let R be a Soules
∈
matrix with columns r ,r ,...,r , where r = x. Then there exists a Soules-
1 2 n 1
type sequence of partitions of 1,2,...,n such that r is given (up to a
i
N { }
factor of 1) by
±
(i) (i)
1 xˆ x
2 (i) 2 (i)
r = || || x || || xˆ , (5)
i N N
x(i) 2+ xˆ(i) 2 x(i) 2 N − xˆ(i) 2 N !
|| ||2 || ||2 || N || || N ||
N N
q
i = 2,3,...,n.
Conversely, if x Rn is a positive vector with x = 1 and is
2
∈ || || N
a Soules-type sequence of partitions of 1,2,...,n , then the matrix R =
{ }
r r r —with r = x and r ,r ,...,r givenby(5)—isa Soules
1 2 n 1 2 3 n
···
matrix.
(cid:2) (cid:3)
Remark. Note that, by (5), the jth entry of r is nonzero if and only if
i
j .
∈ Ni∗∪Ni∗∗
Example 2.9. Let us show that (7;5, 2, 4, 6) (0,0,0,0,0). To see
5
− − − ∈ S
this, conside the vector
T
x = 1 1 1 √3 √3
2 2 2√2 4 4
h i
6
and the partition sequence = ( , , , , ) (illustrated in Figure
1 2 3 4 5
N N N N N N
1), where
= 1,2,3,4,5 ,
1
N {{ }}
= 1,2 , 3,4,5 ,
2
N {{ } { }}
= 1,2 , 3 , 4,5 ,
3
N {{ } { } { }}
= 1,2 , 3 , 4 , 5 ,
4
N {{ } { } { } { }}
= 1 , 2 , 3 , 4 , 5 .
5
N {{ } { } { } { } { }}
Using (5), we construct the Soules matrix
1,2,3,4,5
{ }
1,2 3,4,5
{ } { }
1,2 3 4,5
{ } { } { }
1,2 3 4 5
{ } { } { } { }
1 2 3 4 5
{ } { } { } { } { }
Figure 1: Partition sequence N
1 1 0 0 1
2 2 √2
1 1 0 0 1
 2 2 −√2 
R = 1 1 √3 0 0
 2√2 −2√2 2 
 √3 √3 1 1 0 
 4 − 4 −2√2 √2 
 √3 √3 1 1 0 
 
 4 − 4 −2√2 −√2 
 
and the realising matrix
0 6 1 √3 √3
2√2 4 4
 6 0 1 √3 √3 
2√2 4 4
A =RΛRT =  1 1 0 √6 √6 , (6)
 2√2 2√2 
 √3 √3 √6 0 4 
 4 4 
 √3 √3 √6 4 0 
 4 4 
 
where Λ := diag(7,5, 2, 4, 6).
− − −
7
Note that if σ is irreducible, then any realising matrix for σ has a pos-
itive Perron eigenvector; however, the condition that Soules matrices have
positive first column means that certain reducible lists (which are trivially
symmetrically realisable) are notcontained in ; for example, (1,1) is sym-
n
S
metrically realisable, but (1,1) . In order to complete the equivalence
2
6∈ S
we prove in Section 4, we would like to include these reducible spectra in
the Soules set. Hence we make the following definition:
Definition 2.10. Let λ λ λ and let a ,a ,...,a 0. We
1 2 n 1 2 n
≥ ≥ ··· ≥ ≥
write
(λ ;λ ,...,λ ) (a ,a ,...,a ) (7)
1 2 n n 1 2 n
∈ S
if there exist two partitions
1,...,n = α(1),...,α(1) α(2),...,α(2) α(k),...,α(k) ,
{ } { 1 n1}∪{ 1 n2}∪···∪{ 1 nk}
1,...,n = β(1),...,β(1) β(2),...,β(2) β(k),...,β(k) ,
{ } { 1 n1 }∪{ 1 n2 }∪···∪{ 1 nk }
such that
λ ;λ ,...,λ a ,a ,...,a : i = 1,2,...,k.
α1(i) α2(i) αn(ii) ∈Sni β1(i) β2(i) βn(ii)
(cid:16) (cid:17) (cid:16) (cid:17)
We write
(λ ,λ ,...,λ )
1 2 n n
∈ S
if there exist a ,a ,...,a 0 such that (7) holds.
1 2 n
≥
The Soules set and its role in the SNIEP has been extensively studied,
for example by McDonald and Neumann [16] and Loewy and McDonald
[17]. Soules matrices and the associated orthonormal bases have also been
considered elsewhere in the literature, for example in [18, 19, 20, 21, 22]. In
addition, Soules matrices have been applied to other areas of linear algebra,
including nonnegative matrix factorisation [23], the cp-rank problem [24]
and describing the relationships between various classes of matrices [5, 20].
2.2. A constructive lemma
In [3, Lemma 5], given a nonnegative matrix B with Perron eigenvalue c
and specrtum (c,ν ,ν ,...,ν ) and a nonnegative matrix A with spectrum
2 3 l
(µ ,µ ,...,µ ) and a diagonal element c, Sˇmigoc shows how to construct
1 2 k
a nonnegative matrix C with spectrum (µ , µ ,...,µ ,ν ,ν ,...,ν ). For
1 2 k 2 3 l
applications of this construction, see [3, 25, 26]. Furthermore, if A and B
are symmetric, then C will be symmetric also. We state the symmetric case
below.
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Lemma 2.11. [3] Let B be an l l nonnegative symmetric matrix with
×
Perron eigenvalue c and spectrum (c,ν ,ν ,...,ν ) and let Y Rl l be an
2 3 l ×
∈
orthogonal matrix such that
YTBY = diag(c,ν ,ν ,...,ν ).
2 3 l
Let Y be partitioned as
Y = v V
where v Rl and V Rl (l 1). (cid:2) (cid:3)
× −
∈ ∈
Let
A a
A := 1 ,
aT c
(cid:20) (cid:21)
where A is an (k 1) (k 1) nonnegative symmetric matrix and a Rk 1
1 −
− × − ∈
is nonnegative and let X Rk k be an orthogonal matrix such that
×
∈
XTAX = diag(µ ,µ ,...,µ ).
1 2 k
Let X be partitioned as
U
X = ,
uT
(cid:20) (cid:21)
where u Rk and U R(k 1) k.
− ×
∈ ∈
Then for matrices
A avT
C := 1
vaT B
(cid:20) (cid:21)
and
U 0
Z := ,
vuT V
(cid:20) (cid:21)
we have
ZTCZ = diag(µ ,µ ,...,µ ,ν ,ν ,...,ν ).
1 2 k 2 3 l
2.3. C-realisability and the RNIEP
In [6], Borobia, Moro and Soto construct realisable lists in the RNIEP,
starting from trivially realisable lists, using three well-known results.
Specifically, in 1997, Guo gave the following result, which states that we
may perturb a real eigenvalue of a realisable list by ǫ, provided we also
±
increase the Perron eigenvalue by ǫ:
9
Theorem2.12. [27]If(ρ,λ ,λ ,...,λ )isrealisable, where ρisthePerron
2 3 n
eigenvalue and λ is real, then
2
(ρ+ǫ,λ ǫ,λ ,λ ,...,λ )
2 3 4 n
±
is realisable for all ǫ 0.
≥
Note also the following well-known result, also proved by Guo [27]:
Theorem 2.13. [27] If (ρ,λ ,λ ,...,λ ) is the spectrum of a nonnegative
2 3 n
matrix with Perron eigenvalue ρ, then for all ǫ 0, (ρ+ǫ,λ ,λ ,..., λ )
2 3 n
≥
is the spectrum of a nonnegative matrix also.
Finally, recall that the spectrum of a block diagonal matrix is the union
of the spectra of the diagonal blocks, in other words:
Observation 2.14. If (λ ,λ ,...,λ ) and (µ ,µ ,...,µ ) are realisable,
1 2 m 1 2 n
then (λ ,λ ,...,λ ,µ ,µ ,...,µ ) is realisable.
1 2 m 1 2 n
Borobia, Moro and Soto make the following definition:
Definition 2.15. Alistofrealnumbers(λ ,λ ,...,λ )iscalledC-realisable
1 2 n
if it may be obtained by starting with the n trivially realisable lists (0),(0),
...,(0) and then using results 2.12, 2.13 and 2.14 any number of times in
any order.
Example 2.16. In Example 2.9, we showed that (7,5, 2, 4, 6) .
5
− − − ∈ S
To see that (7,5, 2, 4, 6) is C-realisable, consider the following series of
− − −
steps:
1. (0),(0),(0),(0),(0)
2. (0,0),(0,0),(0)
3. (6, 6),(4, 4),(0)
− −
4. (6, 6),(4,0, 4)
− −
5. (6, 6),(6, 2, 4)
− − −
6. (6,6, 2, 4, 6)
− − −
7. (7,5, 2, 4, 6)
− − −
We used Ovservation 2.14 at steps 1 2, 3 4 and 5 6. We used
→ → →
Theorem 2.12 at steps 2 3, 4 5 and 6 7.
→ → →
10