Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem
We say that a list of real numbers is “symmetrically realisable” if it is the spectrum of some (entrywise) nonnegative symmetric matrix. The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all symmetrically realisable lists.
In this paper, we present a recursive method for constructing symmetrically realisable lists. The properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years, starting with the work of Fiedler in 1974. We show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing.
keywords:Nonnegative matrices, Symmetric Nonnegative Inverse Eigenvalue Problem, Soules matrix
Msc: 15A18, 15A29
This paper explores the spectral properties of symmetric nonnegative 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 MR3217406 (); MR1780191 ().
Let be a list of real numbers. If there exists a nonnegative symmetric matrix with spectrum , then we say is symmetrically realisable and that realises . 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 restriction that consist only of real numbers is a natural one; however, if we allow to be not-necessarily-symmetric, but consider only lists of real numbers, then the resulting problem is known as the Real Nonnegative Inverse Eigenvalue Problem (RNIEP).
In this paper, we describe a recursive method of constructing symmetrically realisable lists, using a construction of Šmigoc SmigocDiagonalElement (). 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 Soules () (later generalised by Elsner, Nabben and Neumann ElsnerNabbenNeumann ()), one of the most influential methods of constructing symmetrically realisable lists. Moreover, since we also show that the realising matrices we obtain by our method 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 UnifiedView () called “C-realisability” and a family of sufficient conditions for the SNIEP due to Soto Soto2013 (). We show that C-realisability is also sufficient for the SNIEP and that is C-realisable if and only if it satisfies one of Soto’s conditions. Such are precisely those which may be obtained by our method or the method of Soules. The equivalence of all four methods 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 Suleimanova1949 (), Perfect Perfect1953 (), Ciarlet Ciarlet1968 (), Kellog Kellog1971 (), Salzmann Salzmann1972 (), Fiedler Fiedler (), Borobia Borobia1995 () and Soto Soto2003 (). We show that if obeys any of these sufficient conditions, then may also be obtained by our method.
2 Preliminaries and notation
To denote that is symmetrically realisable, we may sometimes write . In this paper, the diagonal elements of the realising matrix will also be important; hence, if there exists a nonnegative symmetric matrix with diagonal elements and specrum , then we write
If we wish to specify that is the Perron eigenvalue of the realising matrix, we will separate from the remaining entries in the list by a semicolon, e.g. we may write
The remaining eigenvalues will generally be considered unordered. The diagonal elements will also generally be cosidered unordered and they may appear in any order on the diagonal of , i.e. we do not assume that is the entry of . Sometimes we will assume that the or are arranged in non-increasing order and if this is the case, we will say so explicitly. In this paper, will always be replaced by either or , depending on whether we are considering realisability via Soules or our recursive method.
We begin by stating some necessary conditions (due to Fiedler Fiedler ()) for to be the spectrum of a nonnegative symmetric matrix with specified diagonal elements:
Fiedler () If , and is the spectrum of a nonnegative symmetric matrix with diagonal elements , then
for all (with the convention that ).
Fiedler also gave the following sufficient conditions:
Fiedler () Let and satisfy the following conditions:
Then is the spectrum of a nonnegative symmetric matrix with diagonal elements .
has spectrum and hence if and , then is the spectrum of a nonnegative symmetric matrix with diagonal elements if and only if the following conditions are satisfied:
If , then the conditions of Theorems 2.1 and 2.2 are identical and hence if and , then is the spectrum of a nonnegative symmetric matrix with diagonal elements if and only if the following conditions are satisfied:
2.1 The Soules approach to the SNIEP
Soules’ approach to the SNIEP focuses on constructing the eigenvectors of the realising matrix . Starting from a positive vector , Soules Soules () showed how to construct a real orthogonal matrix with first column such that for all , the matrix —where —is nonnegative. This motivated Elsner, Nabben and Neumann ElsnerNabbenNeumann () to make the following definition:
Let be an orthogonal matrix with columns . is called a Soules matrix if is positive and for every diagonal matrix with , the matrix is nonnegative.
With regard to the SNIEP, a key property of Soules matrices is the following:
ElsnerNabbenNeumann () Let be a Soules matrix and let , where . Then the off-diagonal entries of the matrix are nonnegative.
Therefore, if is an Soules matrix and , where , then is the spectrum of a nonnegative symmetric matrix if the diagonal elements of are nonnegative. This motivates the following definition:
Let and let . We write
if there exists an Soules matrix such that the matrix —where —has diagonal elements . We write
if there exist such that (4) holds and we call the Soules set.
Elsner, Nabben and Neumann generalised the work of Soules by characterising all Soules matrices. In order to state their characterisation, we require two definitions:
Let be a sequence of partitions of . We say that is Soules-type if has the following properties:
for each , the partition consists of precisely subsets, say ;
for each , there exist indices with and , such that and , i.e. is constructed from by splitting one of the sets into two subsets.
If is a Soules-type sequence of partitions of , then we label the sets and in (ii) as and , i.e. for , we define and to be those sets in which do not coincide with any of the sets in .
Let be a positive vector and let be a Soules-type sequence of partitions of . For each , we define to be the vector in whose component is:
and we define to be the vector in whose component is:
We are now ready to state the characterisation of Soules matrices due to Elsner, Nabben and Neumann:
ElsnerNabbenNeumann () Let be a positive vector and let be a Soules matrix with columns , where . Then there exists a Soules-type sequence of partitions of such that is given (up to a factor of ) by
Conversely, if is a positive vector with and is a Soules-type sequence of partitions of , then the matrix —with and given by (5)—is a Soules matrix.
Note that, by (5), the entry of is nonzero if and only if .
Let us show that . To see this, conside the vector
and the partition sequence (illustrated in Figure 1), where
Using (5), we construct the Soules matrix
and the realising matrix
Note that if is irreducible, then any realising matrix for has a positive Perron eigenvector; however, the condition that Soules matrices have positive first column means that certain reducible lists (which are trivially symmetrically realisable) are not contained in ; for example, is symmetrically realisable, but . In order to complete the equivalence we prove in Section 4, we would like to include these reducible spectra in the Soules set. Hence we make the following definition:
Let and let . We write
if there exist two partitions
if there exist such that (7) holds.
The Soules set and its role in the SNIEP has been extensively studied, for example by McDonald and Neumann McDonaldNeumann () and Loewy and McDonald LoewyMcDonald (). Soules matrices and the associated orthonormal bases have also been considered elsewhere in the literature, for example in CHN2006 (); CNS2007 (); Nabben2007 (); CNS2008 (); EubanksMcDonald2009 (). In addition, Soules matrices have been applied to other areas of linear algebra, including nonnegative matrix factorisation CHNP2004 (), the cp-rank problem Naomi2004 () and describing the relationships between various classes of matrices ElsnerNabbenNeumann (); Nabben2007 ().
2.2 A constructive lemma
In (SmigocDiagonalElement, , Lemma 5), given a nonnegative matrix with Perron eigenvalue and specrtum and a nonnegative matrix with spectrum and a diagonal element , Šmigoc shows how to construct a nonnegative matrix with spectrum . For applications of this construction, see SmigocDiagonalElement (); SmigocSubmatrixConstruction (); NewListsFromOld (). Furthermore, if and are symmetric, then will be symmetric also. We state the symmetric case below.
SmigocDiagonalElement () Let be an nonnegative symmetric matrix with Perron eigenvalue and spectrum and let be an orthogonal matrix such that
Let be partitioned as
where and .
where is an nonnegative symmetric matrix and is nonnegative and let be an orthogonal matrix such that
Let be partitioned as
where and .
Then for matrices
2.3 C-realisability and the RNIEP
In UnifiedView (), 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 , provided we also increase the Perron eigenvalue by :
Guo () If is realisable, where is the Perron eigenvalue and is real, then
is realisable for all .
Note also the following well-known result, also proved by Guo Guo ():
Guo () If is the spectrum of a nonnegative matrix with Perron eigenvalue , then for all , 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:
If and are realisable, then is realisable.
Borobia, Moro and Soto make the following definition:
2.4 A family of realisability criteria in the SNIEP
Based on a theorem of Brauer, Soto Soto2013 () gives a family of realisability criteria denoted (not to be confused with ), such that if a list of real numbers satisfies the criterion for some , then is realisable. Soto also shows in Soto2013 () that the criteria are sufficient for symmetric realisability. In order to state , we will require some terminology and notation from Soto2013 (): Let , where , and let be a realisability criterion. Then we write
if satisfies the criterion . The Brauer -negativity of is defined to be the nonnegative number
and if , then the Brauer -realisability margin of is defined to be
The criteria are now defined recursively: We say satisfies the criterion if
and for odd , .
For , we say that satisfies the criterion if there exists a partition of into sublists , where
such that and
Note that if we allow above, then we have:
If satisfies , then satisfies .
Soto2013 () If satisfies for any , then is symmetrically realisable.
3 A recursive approach to the SNIEP
Here, we describe a method of recursively constructing symmetrically realisable lists, starting with lists of length 2 and repeatedly applying Lemma 2.11. Formally, we define the set in the following way:
For , we write if . For , we write if and . For , we write
if there exist two partitions
and a nonnegative number such that
if there exist such that (13) holds.
Note that by Lemma 2.11, if , then is the spectrum of a nonnegative symmetric matrix with diagonal elements .
Note that the conditions given in (2) coincide with the definition of and hence is the spectrum of a nonnegative symmetric matrix with Perron eigenvalue and diagonal elements if and only if . In the following lemma, we show that the same holds for :
Let and . If is the spectrum of a nonnegative symmetric matrix with diagonal elements , then
where . In particular, .
The result follows easily from (3) and the definition of . ∎
Suppose that for all and , the sets are known and we wish to determine whether . Our next result shows that depends only on and :
Let , let and let . Then if and only if there exist , , such that
We prove our claim by induction on . If , then the claim follows from Lemma 3.22. Now assume the claim holds for all , , and suppose . Then there exist a partition of into two subsets and , a partition of into two subsets and