The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society. Volume 8, pp. 110-114, June 2001.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
A NOTE ON THE CONVEXITY OF THE REALIZABLE SET OF
EIGENVALUES FOR NONNEGATIVE SYMMETRIC MATRICES∗
C. KNUDSEN† AND J.J. MCDONALD†
Abstract. Geometric properties of the setRnofn–tuples of realizable spectra of nonnegative
symmetric matrices, and the Soules setSnintroduced by McDonald and Neumann, are examined.
It is established thatS5 is properly contained inR5. Two in terestin g examples are presen ted which
show that neitherRn norSn need be convex. It is proved that Rn andSn are star convex and
centered at (1,1, . . . ,1).
Key words. Symmetric matrices, inverse eigenvalue problem, realizable set, Soules set.
AMS subject classifications.15A18,15A29,15A57
1. Introduction, Definitions. The inverse eigenvalue problem forn×n sym-metric nonnegative matrices can be stated as follows:
Find necessary and sufficient conditions for a set of real numbersλ1, . . . , λn to be the
eigenvalues of ann×nsymmetric nonnegative matrix.
If there is a symmetric nonnegative matrix A with eigenvalues λ1, . . . , λn, then we
shall say that the set of numbersλ1, . . . , λn isrealizable.
Many necessary conditions for ann–tuple of numbers to be realizable are known, and there are also several known sufficient conditions. Nevertheless,for n >4,the realizablen–tuples have not yet been fully characterized. Sources of information on the inverse eigenvalue problem for symmetric nonnegative matrices include [7],[4], [2],[1,Ch. 4 and Supplement Sec. 2],and [5].
Without loss of generality we can assume throughout that 1 =λ1 ≥λ2≥. . .≥
λn ≥ −1. As in [6],we discuss three important sets in connection with the inverse
eigenvalue problem for symmetric matrices. The set of all points which satisfyλ2+
. . .+λn≥ −1 is referred to as thetrace nonnegative polytopeand denoted byTn. The
set of all points inTnwhich are realizable is referred to as therealizable setand denoted
byRn.In [8],Soules provides an algorithm for constructing orthogonal matrices. This
method is generalized by Elsner,Nabben,and Neumann [3] and they call the resulting matrices Soules matrices. In [6],the topological closure of the Soules set of matrices is used to construct nonnegative symmetric matrices. We explain how to construct a Soules matrix below. IfRis an orthogonal matrix,we say that (1, λ2, . . . , λn) is in the
feasible region forRifRΛRT ≥0,where Λ is the matrix whose diagonal entries are
1, λ2, . . . , λn.The set of alln–tuples which are in the feasible region for some Soules
matrix is referred to as theSoules setand denoted bySn.
In [6] it is shown that for n ≤ 4, Sn = Rn = Tn, and for n ≥5, that Rn is a
proper subset of Tn. The question as to whether or not S5 is equal to R5 is posed.
∗Received by the editors on20 November 2000. Final manuscript accepted for publicationon12
June 2001. Handling editor: Daniel Hershkowitz.
†Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada
S4S 0A2 (cinknudsen@hotmail.com, jmcdonald@math.wsu.edu). The work of the second author was supported by anNSERC Research Grant.
ELA
A Note onthe Convexity of the Realizable Set 111
Our examples in the present note show that S5 is properly contained in R5. This proves that forn= 5 there are nonnegative symmetric matrices which are not similar to nonnegative matrices constructed from Soules matrices.
The set Tn is always convex. For n ≤ 4, the set Rn is also convex,but as we
show, R5 and R6 are not convex. It is shown in [6] that forn ≤ 5, the set Sn is
convex. We show here thatS6 is not convex.
A set isstar convex centered atpif the line from any point in the set topis also contained in the set. We conclude our paper by showing that Sn and Rn are star
convex centered at (1,1, . . . ,1).
2. Examples, Results, and Convexity Arguments. Following Elsner, Nabben,and Neumann [3],we can describe the construction of a Soules matrix as
follows. Start with a positive vector n×1 unit vectorw. Partition w into
u v
.
Then it is straightforward to see that the vector
˜
w =
(v2/u2)u −(u2/v2)v
satisfies that w˜2 = 1, and ˜wTw = 0. We can then create additional orthogonal vectors by further re–partitioning the vectors (v2/u2)u and (u2/v2)v to obtain a setw1, w2, . . . , wk, k≤n,of n×1 vectors which are mutually orthogonal.
Aftern−1 steps we can construct an orthogonal matrixR with columnsw1, . . . , wn.
The matrices which can be constructed in this way are referred to as Soules matri-ces. By letting a subvector of wapproach zero and taking the limit of the matrices constructed,we generate additional matrices in the topological closure of the original Soules set which we also consider to be Soules matrices.
It is observed in [6] thatT5 is the convex hull of the following points:
a = (1,1,1,1,1), b = (1,1,1,1,−1), c = (1,1,1,−1,−1), d = (1,1,0,−1,−1), e = (1,0,0,0,−1), f = (1,1,−12,−12,−1),
g = (1,−14,−14,−14,−14), h = (1,1,−23,−23,−23),
i = (1,1
2, 1
2,−1,−1),
and that the Soules setS5 is the convex hull ofa, b, c, d, e, f, g, j, k, lwhere
ELA
112 C. Knudsen and J. J. McDonald
k = (1,1
We introduce one additional point:
m= (1,−1 +
This point is interesting for a variety of reasons. Foremost,we will show that it is a realizable point on the line froml toi.In [6] it was shown thatlis the only point on this line which is contained inS5.Secondly,the level curves 1 +λ2+λ3+λ4+λ5= 0 line froml toiwhich can be realizable since the remaining points on the line fail the necessary condition that 1+λq2+λ
mconsists of 1,a multiple of the golden mean,and a multiple of its reciprocal.
Observation 2.1. The line froml tomis contained in R5, and this is the only portion of the line froml toi which is realizable.
Proof. Consider the matrix
l andmrespectively. Then
ELA
A Note onthe Convexity of the Realizable Set 113
satisfied for any 0 ≤ t ≤ 1. Hence the line from l to m is realizable. It is easy to check that the remaining points on the line froml to i fail the necessary condition that 1 +λq2+λ
Corollary 2.2. The set S5 is properly contained in R5.
Proof. In Observation 2.1 we see thatm∈ R5.By [6,Lemma 4.3],m∈ S5. Observation 2.3. The setR5 is not convex.
Proof. The point
is on the line frommto the realizable pointc,but 1 + 2(1 + 3
thuspfails a necessary condition for realizability and hence is not in R5. We now turn our attention toS6 andR6.
Observation 2.4. The setsS6 andR6 are not convex.
Proof. Let
ThenR1 andR2 are Soules matrices. The point
r= (1,1,1,−1,−1,−1) is feasible forR1 and the point
s= (1,1,−1/2,−1/2,−1/2,−1/2)
ELA
114 C. Knudsen and J. J. McDonald
that the sum of the numbers in each set is nonnegative. ThusS6 and R6 cannot be convex.
We conclude with the following result.
Theorem 2.5. The sets Sn and Rn are star convex centered at the point
(1,1, . . . ,1).
Proof. Let p be a point in Sn or Rn. Let P = diag(p), the n×n diagonal
matrix formed from the elements ofp.If we are working with the setSn,then there
is a Soules matrixR such thatRP RT ≥0.If we are working withR
n, then we can
choose a nonnegative symmetric matrix A whose eigenvalues are listed in p. Since
A can be diagonalized using an orthogonal similarity,there is an orthogonal matrix
R such that A = RP RT ≥ 0. Let Λ = diag(1, λ
2, . . . , λn). Then RΛRT ≥ 0 is
a set of linear inequalities in λ2, λ3, . . . , λn, which combined with the inequalities
1 ≥λ2 ≥. . . ≥λn ≥ −1, generate a polytope. Since RIRT =I ≥0 we know that
(1,1, . . . ,1) is in the polytope,as is the line frompto (1,1, . . . ,1).
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[3] L. Elsner, R. Nabben, and M. Neumann. Orthogonal bases which lead to symmetric nonnegative bases. Linear Algebra and its Applications, 271:323–343, 1998.
[4] M. Fiedler. Eigenvalues of nonnegative symmetric matrices.Linear Algebra and its Applications, 9:119–142, 1974.
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