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“main” — 2008/10/13 — 23:12 — page 237 — #1 Volume 27, N. 3, pp. 237–250, 2008 Copyright © 2008 SBMAC 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: [email protected] 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 ( ) An n ×n nonzero matrix D = di j is called a Euclidean distance matrix (EDM) 1, 2,..., n r if there exist points p p p in some Euclidean space < such that i j 2 , ,..., , di j = ||p − p || for all i j = 1 n where || || denotes the Euclidean norm. i , ,..., Let p , i ∈ N = {1 2 n}, be the set of points that generate an EDM π π ( , ,..., ) D. An m-partition of D is an ordered sequence = N1 N2 Nm of ,..., nonempty disjoint subsets of N whose union is N. The subsets N1 Nm are called the cells of the partition. The n-partition of D where 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 and MITACS. “main” — 2008/10/13 — 23:12 — page 238 — #2 238 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES of a single point is called the discrete partition, while the 1-partition of D with only one cell is called the single-cell partition. π ( , ,..., ) An m-partition = N1 N2 Nm of an EDM D is said to be equitable , ,..., if for all i j = 1 m (case i = j included), there exist non-negative scalars α i j such that for each k ∈ Ni , the sum of the squared Euclidean distances from k l α p to all points p , l in N j , is equal to i j . i.e., , α , , ,..., . ∀ k ∈ Ni dkl = i j for all i j = 1 m (1) lX∈N j 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 name block 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]. We denote the identity matrix of order n by In and the n-vector of all 1’s , T T by en. En m = enem denotes the n × m matrix of all 1’s, and En = enen denotes the square matrix of order n of all 1’s. The subscripts of I , e and E will be deleted if the order is clear from the context. For a matrix A, diag A Comp. Appl. Math., Vol. 27, N. 3, 2008 “main” — 2008/10/13 — 23:12 — page 239 — #3 A.Y. ALFAKIH 239 denotes the vector consisting of the diagonal entries of A. Finally, the spec- trum of a matrix A, denoted by σ (A), is the multiset of the eigenvalues of A. λ , . , λ ,..., If A has eigenvalues 1 k with multiplicities m1 mk respectively, σ( ) λm1 , . , λmk then D = { 1 k }. 2 Preliminaries 1,..., n Let D be an n×n EDM, the dimension of the affine span of the points p p T / that generate D is called the embedding dimension of D. Let Jn := In − enen n denote the orthogonal projection on subspace ⊥ n T . M := e = x ∈ < : e x = 0 (2) It is well known [14, 18] that a symmetric matrix D with zero diagonal is an EDM if and only if D is negative semidefinite on M. Hence, EDMs have exactly one positive eigenvalue. Let SH denote the subset of n × n symmetric matrices with zero diagonal; and let SC denote the subset of n × n symmetric matrices A satisfying Ae = 0. Following [3], let T : SH → SC and K : SC → SH be the two linear maps defined by 1 (D) JDJ, (3) T := −2 and ( ) ( ) T ( ( ))T . K B := diag B e + e diag B − 2B (4) It, then, immediately follows [3] that T and K are mutually inverse, and that ( ) D in SH is an EDM of embedding dimension r if and only if T D is positive semidefinite of rank r. Let D be an n × n EDM of embedding dimension r generated by the points 1,..., n r p p in < . Then the n × r matrix T p1 . . P = n T p is called a realization of D. Given an EDM D, a realization P of D can be ob- ( ) ( ) T tained by factorizing T D into T D = PP . Note that if P is a realization Comp. Appl. Math., Vol. 27, N. 3, 2008 “main” — 2008/10/13 — 23:12 — page 240 — #4 240 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES 0 of D, then P = PQ is also a realization of D for any r × r orthogonal matrix Q. Obviously, P and P0 in this case are obtained from each other by a rigid motion such as a rotation or a translation. An EDM D is said to be spherical if the points that generate D lie on a hyper- sphere, otherwise, it is said to be non-spherical. It is well known [6, 17] that an EDM D of embedding dimension r is spherical if and only if rank D = r + 1, and that D is non-spherical if and only if rank D = r + 2. A spherical EDM D is said to be regular if the points p1,..., pn that generate D lie on a hyper-sphere whose center coincides with the centroid of p1,..., pn. It is not difficult to show that [12, 8] an EDM D is regular if and only if e is an eigenvector of D 1 T corresponding to the eigenvalue n e De. 3 Equitable partition for EDMs It immediately follows from (1) that the discrete partition of D is always equi- α table with i j = di j ; while the single-cell partition of D is equitable if and only α 1 T if D is regular. In the latter case, 11 = n e De. It also follows from (1) that m ,..., ( ) α . ∀ i = 1 m and ∀ k ∈ Ni we have De k = i j (5) Xj=1 π ( , ,..., ) Let = N1 N2 Nm be an m-partition of an n × n EDM D where ,..., π ( ) |Ni | = ni for i = 1 m. Define the n × m matrix P = pi j such that / −1 2 n j if i ∈ N j pi j = (6) ( 0 otherwise. Pπ is called the normalized characteristic matrix [5] of π since its jth column 1/2 − T π is equal to n j times the characteristic vector of N j , and since Pπ P = Im. Next we present a lemma whose graph adjacency matrix counterpart was proved by Godsil and McKay [5]. Lemma 3.1. Let π be an m-partition of an EDM D. Then π is equitable if and ( ) only if there exists an m × m symmetric matrix S = si j such that π π . DP = P S (7) π ( / )1/2 α Furthermore, if is an equitable partition then si j = ni n j i j . Comp. Appl. Math., Vol. 27, N. 3, 2008 “main” — 2008/10/13 — 23:12 — page 241 — #5 A.Y. ALFAKIH 241 π ,..., Proof. Assume that is equitable then for all k ∈ Ni and for all j = 1 m we have 1/2 1/2 1/2 / ni ( π ) − − α ( )−1 2 α ( π ) , DP kj = n j dkl = n j i j = ni i j = P S kj n j lX∈N j ( / )1/2α where si j = ni n j i 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 1/2 / ( π ) − ( π ) ( )−1 2 . DP kj = n j dkl = P S kj = ni si j lX∈N j ( / )1/2 π kl j i i j Therefore, l∈N j d = n n s , which is independent of k.
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