ISSN 0101-8205 www.scielo.br/cam
On the eigenvalues of Euclidean distance matrices
A.Y. ALFAKIH∗
Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada
E-mail: alfakih@uwindsor.ca
Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues
of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic
poly-nomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents
methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues.
Mathematical subject classification: 51K05, 15A18, 05C50.
Key words:Euclidean distance matrices, eigenvalues, equitable partitions, characteristic
poly-nomial.
1 Introduction
Ann×nnonzero matrixD=(di j)is called aEuclidean distance matrix (EDM) if there exist pointsp1,p2, . . . ,pnin some Euclidean spaceℜr such that
di j = ||pi −pj||2 for all i, j =1, . . . ,n, where|| ||denotes the Euclidean norm.
Let pi, i
∈ N = {1,2, . . . ,n}, be the set of points that generate an EDM
D. An m-partition π of Dis an ordered sequence π = (N1,N2, . . . ,Nm) of nonempty disjoint subsets ofN whose union isN. The subsetsN1, . . . ,Nmare called thecellsof the partition. Then-partition of Dwhere each cell consists
#760/08. Received: 07/IV/08. Accepted: 17/VI/08.
∗Research supported by the Natural Sciences and Engineering Research Council of Canada
of a single point is called thediscrete partition, while the 1-partition of Dwith only one cell is called thesingle-cell partition.
Anm-partitionπ =(N1,N2, . . . ,Nm)of an EDM Dis said to beequitable if for alli,j =1, . . . ,m(casei = j included), there exist non-negative scalars
αi j such that for eachk ∈ Ni, the sum of the squared Euclidean distances from
pk to all points pl,linN
j, is equal toαi j. i.e.,
∀k ∈ Ni, X
l∈Nj
dkl =αi j, for alli, j =1, . . . ,m. (1)
The notion of equitable partitions for graphs, which was introduced by Sachs [13], is related to, among others, automorphism groups of graphs and distance-regular graphs [4]. Schwenk [15] used equitable partitions to find the eigenvalues of the adjacency matrix of a graph. Hayden et al. [7] also used equitable parti-tions, albeit under the nameblock structure, to investigate EDMs generated by points lying on a collection of concentric spheres. In particular, they devised an algorithm for finding the least number of concentric spheres containing the points that generate a given EDM. Their investigation was based on the block structure of EDMs and the corresponding eigenvectors.
Many of the results on the spectra of graphs obtained using equitable parti-tions have analogous counterparts in the case of EDMs. In particular, we show (see Theorem 3.1) that the characteristic polynomial of an EDM can be written as the product of the characteristic polynomials of two matrices associated with partitions. Theorem 3.1 is then used to determine the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. We also present methods for constructing cospectral EDMs and non-regular EDMs with exactly three distinct eigenvalues.
Recently, EDMs have received a great deal of attention for their many impor-tant applications. These applications include, among others, molecular confor-mation problems in chemistry [2], multidimensional scaling in statistics [9], and wireless sensor network localization problems [16].
denotes the vector consisting of the diagonal entries of A. Finally, the spec-trum of a matrix A, denoted byσ (A), is the multiset of the eigenvalues of A. If A has eigenvalues λ1, . . . , λk with multiplicities m1, . . . ,mk respectively, thenσ (D)= {λm1
1 , . . . , λ
mk
k }.
2 Preliminaries
LetDbe ann×nEDM, the dimension of the affine span of the pointsp1, . . . ,pn
that generateDis called theembedding dimensionofD. LetJn := In−eneTn/n denote the orthogonal projection on subspace
M:=e⊥=
x ∈ ℜn:eTx =0 . (2)
It is well known [14, 18] that a symmetric matrix D with zero diagonal is an EDM if and only ifDis negative semidefinite onM. Hence, EDMs have exactly one positive eigenvalue. LetSH denote the subset ofn×nsymmetric matrices with zero diagonal; and letSCdenote the subset ofn×nsymmetric matrices A
satisfyingAe=0. Following [3], letT :SH →SC andK:SC →SH be the
two linear maps defined by
T(D):= −1
2J D J, (3)
and
K(B):=diag(B)eT +e(diag(B))T −2B. (4)
It, then, immediately follows [3] thatT andKare mutually inverse, and that DinSH is an EDM of embedding dimensionr if and only ifT(D)is positive semidefinite of rankr.
Let Dbe ann×n EDM of embedding dimensionr generated by the points
p1, . . . ,pnin
ℜr. Then then×r matrix
P =
p1T .. . pn T
ofD, then P′ = P Qis also a realization of Dfor anyr ×r orthogonal matrix
Q. Obviously, P and P′ in this case are obtained from each other by a rigid motion such as a rotation or a translation.
An EDMDis said to besphericalif the points that generateDlie on a hyper-sphere, otherwise, it is said to benon-spherical. It is well known [6, 17] that an EDM Dof embedding dimensionr is spherical if and only if rank D =r +1, and thatDis non-spherical if and only if rankD=r+2. A spherical EDMDis said to beregularif the pointsp1, . . . ,pnthat generateDlie on a hyper-sphere whose center coincides with the centroid of p1, . . . ,pn. It is not difficult to show that [12, 8] an EDM Dis regular if and only ife is an eigenvector of D
corresponding to the eigenvalue 1neTDe.
3 Equitable partition for EDMs
It immediately follows from (1) that the discrete partition of Dis always equi-table withαi j =di j; while the single-cell partition of Dis equitable if and only ifDis regular. In the latter case,α11 = 1neTDe. It also follows from (1) that
∀i =1, . . . ,m and ∀k ∈ Ni we have (De)k = m X
j=1
αi j. (5)
Let π = (N1,N2, . . . ,Nm) be an m-partition of an n ×n EDM D where
|Ni| =ni fori =1, . . . ,m. Define then×mmatrixPπ =(pi j)such that
pi j = (
n−1/2
j if i ∈Nj
0 otherwise.
(6)
Pπ is called thenormalized characteristic matrix[5] ofπ since its jth column
is equal ton−j1/2times the characteristic vector of Nj, and since PπTPπ =Im. Next we present a lemma whose graph adjacency matrix counterpart was proved by Godsil and McKay [5].
Lemma 3.1. Letπbe an m-partition of an EDM D. Thenπis equitable if and only if there exists an m×m symmetric matrix S=(si j)such that
D Pπ = PπS. (7)
Proof. Assume thatπis equitable then for allk∈ Ni and for all j =1, . . . ,m we have
(D Pπ)k j =n−j1/2 X
l∈Nj
dkl =n−j1/2αi j =(ni)−1/2 n
i
nj 1/2
αi j =(PπS)k j,
wheresi j =(ni/nj)1/2αi j.
On the other hand, assume that (7) holds for some partition π. Then for all
k∈ Ni and all j =1, . . . ,m, we have
(D Pπ)k j =n−j1/2 X
l∈Nj
dkl =(PπS)k j =(ni)−1/2si j.
Therefore,P
l∈Nj dkl = (nj/ni) 1/2s
i j, which is independent of k. Hence π is
equitable.
Given anm-partitionπ of EDM Dwithm ≤ n−1, let Pˉπ be then×(n−
m) matrix such that [Pπ Pˉπ] is an orthogonal matrix. Let χA(λ) denote the characteristic polynomial of matrix A. Then we have the following two results:
Theorem 3.1. Letπ be an equitable m-partition of an n×n EDM D, where m≤n−1. Then
χD(λ)=χS(λ) χSˉ(λ), (8)
where S is defined in(7)andSˉ = ˉPπTDPˉπ.
Proof. It follows from (7) and the definition ofPˉπthatPπTDPˉπ =S PπTPˉπ =0.
Thus,
det λIn− "
PT
π
ˉ PT
π
#
D
Pπ Pˉπ
!
=det "
λIm−PπTD Pπ 0
0 λIn−m− ˉPπTDPˉπ
#
.
Hence,
det(λIn−D)=det(λIm−S)det(λIn−m− ˉS).
Note that in case of discrete partitions, i.e., in casem = n, we have S = D
Theorem 3.2. Letπ be an equitable m-partition of an n×n EDM D, where m≤n−1. Then
χSˉ(λ)dividesχ−2T(D)(λ), (9)
whereS is as defined in Theoremˉ 3.1.
Proof. It follows from (5) and the definition of Pˉπ that PˉπTDe = 0 and
ˉ PT
πe =0. Thus,
ˉ
PπTT(D)Pπ =0 and PˉT
πT(D)Pˉπ = −
1 2Pˉ
T
π DPˉπ = −
ˉ S
2. Hence,
det λIn+2 "
PT
π
ˉ PT
π
#
T(D)
Pπ Pˉπ
!
=det "
λIm+2PπTT(D)Pπ 0
0 λIn−m +2PˉπTT(D)Pˉπ
#
.
Therefore,
det λIn+2T(D)=det λIm+2PπTT(D)Pπ
det λIn−m − ˉS.
4 Applications of Theorem 3.1
In this section we show that Theorem 3.1 provides a new method for determin-ing the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. It also provides a method for constructing cospectral EDMs. Let Vn be the n ×(n −1) whose columns form an orthonormal basis for subspaceMdefined in (2), i.e.,Vnhas full column rank and satisfies
VnTen =0, VnTVn= In−1, VnVnT = Jn= In−enenT/n
. (10)
4.1 χD(λ)of regular EDMs
Corollary 4.1 ([8, 1]). Let D 6= 0be an n×n regular EDM of embedding dimension r, then
χD(λ)=λn−r−1
λ− 1 ne
TDe r Y
i=1
(λ+2µi),
whereµi, for i =1, . . . ,r, are the nonzero eigenvalues ofT(D).
Proof. LetDbe ann×nregular EDM. Then, as we remarked earlier, the one-cell partitionπ of D, in this case, is equitable. Since Pπ = √1neand Pˉπ = Vn we haveS = 1
neTDeandSˉ =VnTDVn=−2VnTT(D)Vn. Thus,
χS(λ)=λ− 1
ne
TDeandχ ˉ
S(λ)=λn−r−1 r Y
i=1
(λ+2µi)
since the nonzero eigenvalues of Sˉ are equal to the nonzero eigenvalues of the
matrix−2T(D).
4.2 χD(λ)of non-spherical centrally symmetric EDMs
Let D1 be the 2n ×2n EDM generated by the points p1, . . . ,pn, . . . ,p2n, wherepn+i
= −pi for alli
=1, . . . ,n. Assume thatD1is non-spherical. Then
D1is called anon-spherical centrally symmetricEDM. It is easy to see that
D1=
"
D A A D
#
,
whereDis then×nEDM generated by the pointsp1, . . . ,pn, and
A=diag(T(D))eT
n +endiag(T(D))T +2T(D). (11)
The characteristic polynomial of D1, which was obtained in [1] using a
char-acterization of the nullspace of an EDM in terms of its Gale subspace, is given by
χD1(λ)=λ
2n−r−2
λ2−λ 1
2ne T
2nD1e2n− 2n xT
1x1
r Y
i=1
wherer is the embedding dimension of D1,µi, i = 1, . . . ,r, are the nonzero eigenvalues ofT(D1), andx1∈ ℜ2nis the unique vector satisfying
D1x1=e2n, e2Tnx1=0, and x1⊥Nullspace ofD1. (13) Note that such x1 satisfying (13) exists since D1 is non-spherical. A simple method to computex1is given in [1]. Next, we establish (12) using Theorem 3.1.
Then-partitionπofD1corresponding to the normalized characteristic matrix
Pπ =
1
√
2 "
In
In #
is, obviously, equitable. Since
ˉ Pπ =
1
√
2 "
In
−In #
,
it immediately follows thatS = PπTD1Pπ = D+ A = 2(diag(T(D))eTn +
endiag(T(D))T) and Sˉ = ˉPπTD1Pˉπ = D− A = −4T(D). Hence, the
nonzero eigenvalues ofSˉ = the nonzero eigenvalues of−4T(D)= the nonzero eigenvalues of−2T(D1)since
T(D1)=
"
T(D) −T(D) −T(D) T(D)
#
=
"
1 −1
−1 1 #
⊗T(D), (14)
where⊗denotes the Kronecker product. Hence,χSˉ(λ)=λn−r Qri=1(λ+2µi), whereµi,i =1, . . . ,r, are the nonzero eigenvalues ofT(D1).
In order to establish (12), we still need to determineχS(λ). To this end, note thatT(D1)D1e2n= 2nT(D1)diag(T(D1))= 0, where the first equality follows
from (4). Then we have the following technical lemma.
Lemma 4.1. Let D1be an2n×2n non-spherical EDM satisfying
T(D1)D1e2n=0.
Then
D1e2n=
eT
2nD1e2n 2n e2n+
2n
xT
1x1
x1, (15)
Proof. Lethyidenote the subspace generated by y ∈ ℜ2n. Then it is shown in [1] that
Nullspace of T(D1)= he2ni ⊕ hx1i ⊕ Nullspace of D1.
Thus,
D1e2n = βe2n + γx1, (16)
for some scalarsβandγ. The result follows by multiplying equation (16) from
the left withe2Tnandx1T respectively.
Therefore, it follows from (15) that xT
1 = (xT xT)for some vector x ∈ ℜn
and
Sen = (D+A)en =
eT
2nD1e2n
2n en +
2n xT
1x1
x. (17)
Now, from the definition ofVnin (10) we haveVnTSVn =0. Hence,
χS(λ) = det
λIn−
VT n eT n √n S Vn en √n = det
λIn−1 −
2√n xT 1x1 VT n x −2 √n xT
1x1 xTV
n λ− 1 2ne
T
2nD1e2n . Thus,
χS(λ) = λn−2
λ
λ− 1
2ne
T
2nD1e2n
−(xT4n
1x1)2
xTVnVnTx
= λn−2
λ
λ− 1
2ne
T
2nD1e2n
−x2Tn
1x1
,
sincexTV
nVnTx =xTx =x1Tx1/2. Hence, (12) is established. 4.3 Constructing cospectral EDMs
Given a scalarµ > 1 and ann×nregular EDM Dgenerated by the points
p1, . . . ,pn, let D1and D2 be the two 2n
×2n EDMs constructed from D as follows. D1 is generated by the 2n points p1,p2, . . . ,pn,
−µp1,
−µp2, . . ., −µpn; and D2 is generated by the 2n points p1,p2, . . . ,pn, µp1, µp2, . . ., µpn. Next we show thatD1andD2are cospectral.
Since Dis regular, we have diag(T(D)) = enTDen
2n2 en. Thus it follows from (3) and (4) that
D1=
"
D A1 A1 µ2D
#
, and D2=
"
D A2 A2 µ2D
#
,
where
A1 = 1
2n2 e
T
nDen µ2+1enenT +2µT(D), (18) and
A2 = 1
2n2 e
T
nDen µ2+1enenT −2µT(D). (19) Note that A1en = A2en = 21neTnDen(µ2+1)en. Therefore, the 2-partitionπ ofD1andD2corresponding to the normalized characteristic matrix
Pπ =
1
√n
"
en 0 0 en
#
is equitable. Since
ˉ Pπ =
"
Vn 0
0 Vn
#
,
it immediately follows that
S1= PπTD1Pπ =
1
ne
T nDen
1 1 2(µ 2 +1)
1 2(µ
2
+1) µ2
= P
T
πD2Pπ =S2.
Furthermore,
ˉ
S1= ˉPπTD1Pˉπ =
"
VT
n DVn −µVnTDVn
−µVnTDVn µ2VnTDVn #
=
"
1 −µ
−µ µ2
#
and
ˉ
S2= ˉPπTD2Pˉπ =
"
VT
n DVn µVnTDVn
µVT
n DVn µ2VnTDVn #
=
"
1 µ
µ µ2
#
⊗VnTDVn.
But the eigenvalues ofS1ˉ are equal to the eigenvalues of S2ˉ . In particular, the nonzero eigenvalues ofS1ˉ = the nonzero eigenvalues of S2ˉ =(1+µ2)times the nonzero eigenvalues ofVT
n DVn = (1+µ2) times the nonzero eigenvalues of
−2T(D). Therefore, D1 and D2are two cospectral EDMs. Furthermore, D1
andD2are not regular sinceµ >1.
5 Constructing EDMs with three distinct eigenvalues
EDMs have two or more distinct eigenvalues since the eigenvalues of an EDM
Dcan not all be equal. Moreover, EDMs with exactly two distinct eigenvalues are precisely those generated by the standard simplex, i.e., EDMs of the form
D = γ (E −I)for some scalarγ > 0. The problem of obtaining a complete characterization of EDMs with exactly three distinct eigenvalues, just like its graph counterpart, seems to be difficult.
Neumaier [11] introduced the notion ofstrengthfor distance matrices as a mea-sure of their regularity. According to Neumaier regular EDMs are of strength 1 while regular EDMs with three distinct eigenvalues, where one of the eigen-values is a zero are of strength 2. The following theorem follows from a more general result due to Neumaier [11].
Theorem 5.1. Let D be a regular EDM with exactly three distinct eigenvalues such that its off-diagonal entries have exactly 2 or 3 distinct values. Then any two rows (columns) of D are obtained from each other by a permutation.
Next we present two classes of non-regular EDMs with exactly 3 distinct eigenvalues. The first class consists of non-regular EDMs of ordern+1 obtained fromn×nregular EDMs with 3 distinct eigenvalues by adding one row and one column of equal entries.
Theorem 5.2. Let D be an n×n regular EDM such thatσ (D)=eTDe/n, −λr1
1,−λ
n−1−r1
2 . Assume that either eTDe/n+λ1
λ2
λ2=nα2for someα ≥ 21n2eTDe, α6=
1
n(n−1)e
TDe. Letγ be some positive
scalar. Then
D1=γ
"
D αen
αeTn 0 #
is a non-regular EDM of order n+1with exactly 3 distinct eigenvalues.
Proof. It is easy to see that D1is a non-regular EDM sinceα 6= n(n1
−1)e
TDe.
Now assume(eTDe/n
+λ1)λ1=nα2. LetV be then×(n−1)matrix whose
columns form an orthonormal basis for subspaceMdefined in (2), thusV VT =
J =I −eeT/n. Let
Q=
"
en/√n 0 V
0 1 0
#
.
Then
QTD
1Q=γ
eTDe/n αn1/2 0 αn1/2 0 0
0 0 VTDV
.
But it is well known [1, 8] that σ (−2T(D)) = σ (V VTDV VT) =
−λr1
1, −λn−1−r1
2 ,0 . Thusσ (VTDV) =
−λr1
1,−λ
n−1−r1
2 . Furthermore, the
sub-matrix
"
eTDe/n αn1/2 αn1/2 0
#
has eigenvalues−λ1and1neTDe+λ1. The result follows sinceQis an
orthog-onal matrix.
Note that, takingα = 1
2n2eTDein Theorem 5.2 is equivalent to placing point
pn+1 at the center of the hyper-sphere containing p1, . . . ,pn. Also, note that ifα= n(n−11)eTDe, thenD1becomes a regular EDM.
The second class consists of non-regular EDMs of ordern+1 obtained from EDMs of the formD=λ(En−In),λ >0, by adding one row and one column of equal entries. The proof of the next theorem is similar to that of Theorem 5.2.
Theorem 5.3. Letγ be a positive scalar andα ≥(n−1)/2n,α 6=1. Then
D1=γ
"
En−In αen
is a non-regular EDM with exactly 3 distinct eigenvalues, i.e., σ (D1) =
λ1,−λ2,−γn−1 where λ1=
1
2γ (n−1+ p
(n−1)2
+4α2n) and
λ2= 1
2γ (n−1− p
(n−1)2
+4α2n).
Acknowledgement. The author would like to thank an anonymous referee for his/her comments and suggestions which improved the presentation of this paper.
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