On the Eigenvalues of Euclidean Distance Matrices

Home , Distance matrix, Identity matrix, Orthogonal matrix, Symmetric matrix

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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 polynomials 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 polynomial.

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 partitions, 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 partitions 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 conformation 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

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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

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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 .

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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. Hence is equitable. P

π π ( Given an m-partition of EDM D with m ≤ n − 1, let Pˉ be the n × n − ) π π χ (λ) m matrix such that [P Pˉ ] is an orthogonal matrix. Let A denote the characteristic polynomial of matrix A. Then we have the following two results:

π Theorem 3.1. Let be an equitable m-partition of an n × n EDM D, where m ≤ n − 1. Then χ (λ) χ (λ) χ (λ), D = S Sˉ (8) ( ) T π where S is defined in 7 and Sˉ = Pˉπ DPˉ .

π T π T π Proof. It follows from (7) and the definition of Pˉ that Pπ DPˉ = SPπ Pˉ = 0. Thus,

T λ T π λ Pπ Im − Pπ DP 0 . det In D Pπ Pπ det − T ˉ = λ T "Pπ # ! " 0 In m Pπ D Pπ # ˉ − − ˉ ˉ Hence, (λ ) (λ ) (λ ). det In − D = det Im − S det In−m − Sˉ

Note that in case of discrete partitions, i.e., in case m = n, we have S = D χ (λ) χ (λ) thus D = S follows trivially. An analogous result for graphs, namely that the characteristic polynomial of S divides the characteristic polynomial of a graph was obtained by Mowshowitz [10], and by Schwenk [15].

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242 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES

π Theorem 3.2. Let be an equitable m-partition of an n × n EDM D, where m ≤ n − 1. Then χ (λ) χ ( )(λ), Sˉ divides −2T D (9) . where Sˉ is as defined in Theorem 3 1.

π T Proof. It follows from (5) and the definition of Pˉ that Pˉπ De = 0 and T Pˉπ e = 0. Thus,

T ( ) T ( ) 1 T Sˉ . Pˉπ D Pπ 0 and Pˉπ D Pˉπ Pˉπ D Pˉπ T = T = −2 = − 2 Hence,

PπT λ ( ) π π det In + 2 T D P Pˉ "PπT # ! ˉ

λ T ( ) π Im + 2Pπ T D P 0 . = det " λ πT ( ) π # 0 In−m + 2Pˉ T D Pˉ Therefore,

T λ ( ) λ π ( ) π λ . det In + 2T D = det Im + 2P T D P det In−m − Sˉ 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 subspace M defined in (2), i.e., Vn has full column rank and satisfies T , T , T T / . Vn en = 0 Vn Vn = In−1 Vn Vn = Jn = In − enen n (10) χ (λ) 4.1 D of regular EDMs

The characteristic polynomial of a regular EDM was obtained in [8, 1]. The method used in [1] is based on a characterization of the nullspace of an EDM in terms of its Gale subspace. Next we determine the characteristic polynomial of regular EDMs as a corollary of Theorem 3.1.

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A.Y. ALFAKIH 243

Corollary 4.1 ([8, 1]). Let D 6= 0 be an n × n regular EDM of embedding dimension r, then

r χ (λ) λn−r−1 λ 1 T (λ μ ), D = − e De + 2 i n Yi=1 μ ,..., ( ) where i , for i = 1 r, are the nonzero eigenvalues of T D .

Proof. Let D be an n × n regular EDM. Then, as we remarked earlier, the one- π π 1 π cell partition of D, in this case, is equitable. Since P = √n e and Pˉ = Vn 1 T T T ( ) we have S = n e De and Sˉ = Vn DVn = −2 Vn T D Vn. Thus, r χ (λ) λ 1 T χ (λ) λn−r−1 (λ μ ) S e De and S 2 i = − n ˉ = + Yi=1 since the nonzero eigenvalues of Sˉ are equal to the nonzero eigenvalues of the ( ) matrix −2T D .

χ (λ) 4.2 D of non-spherical centrally symmetric EDMs

1,..., n,..., 2n Let D1 be the 2n × 2n EDM generated by the points p p p , n+i i ,..., where p = −p for all i = 1 n. Assume that D1 is non-spherical. Then D1 is called a non-spherical centrally symmetric EDM. It is easy to see that

DA , D1 = " AD #

1,..., n where D is the n × n EDM generated by the points p p , and ( ( )) T ( ( ))T ( ). A = diag T D en + en diag T D + 2 T D (11)

The characteristic polynomial of D1, which was obtained in [1] using a characterization of the nullspace of an EDM in terms of its Gale subspace, is given by

r χ (λ) λ2n−r−2 λ2 λ 1 T 2n (λ μ ), D1 = − e2n D1e2n − T + 2 i (12) 2n x1 x1 Yi=1

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244 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES

μ ,..., where r is the embedding dimension of D1, i , i = 1 r, are the nonzero ( ) 2n eigenvalues of T D1 , and x1 ∈ < is the unique vector satisfying , T , . D1x1 = e2n e2n x1 = 0 and x1 ⊥ Nullspace of D1 (13)

Note that such x1 satisfying (13) exists since D1 is non-spherical. A simple method to compute x1 is given in [1]. Next, we establish (12) using Theorem 3.1. π The n-partition of D1 corresponding to the normalized characteristic matrix

1 In π P = √ 2 " In # is, obviously, equitable. Since

1 In π , Pˉ = √ 2 " −In #

T π ( ( ( )) T it immediately follows that S = Pπ D1 P = D + A = 2 diag T D en + ( ( ))T ) T π ( ) en diag T D and Sˉ = Pˉπ D1 Pˉ = D − A = −4 T D . Hence, the ( ) nonzero eigenvalues of Sˉ = the nonzero eigenvalues of −4T D = the nonzero ( ) eigenvalues of −2T D1 since ( ) ( ) ( ) T D −T D 1 −1 ( ), T D1 = ( ) ( ) = ⊗ T D (14) " −T D T D # " −1 1 # χ (λ) λn r r (λ μ ) − i where ⊗ denotes the Kronecker product. Hence, Sˉ = i=1 + 2 , μ ,..., ( ) where i , i = 1 r, are the nonzero eigenvalues of T D1 .Q χ (λ) In order to establish (12), we still need to determine S . To this end, note ( ) ( ) ( ( )) that T 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 D1 be an 2n × 2n non-spherical EDM satisfying ( ) . T D1 D1e2n = 0 Then T e2n D1 e2n 2n , D1e2n = e2n + T x1 (15) 2n x1 x1 ( ) where x1 satisfies 13 .

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A.Y. ALFAKIH 245

2n Proof. Let h y i denote the subspace generated by y ∈ < . Then it is shown in [1] that ( ) . Nullspace of T D1 = h e2n i ⊕ h x1 i ⊕ Nullspace of D1 Thus, β γ , D1e2n = e2n + x1 (16) for some scalars β and γ . The result follows by multiplying equation (16) from T T the left with e2n and x1 respectively. T ( T T ) n Therefore, it follows from (15) that x1 = x x for some vector x ∈ < and T ( ) e2n D1 e2n 2n . Sen = D + A en = en + T x (17) 2n x1 x1 T Now, from the definition of Vn in (10) we have Vn SVn = 0. Hence, T Vn χ (λ) λ en S det  In  eT  S Vn  = − n √n √   n       √ λ 2 n T In−1 − T Vn x  x1 x1  det . = √  2 n T λ 1 T   − T x Vn − e2n D1e2n   x x1 2n   1  Thus,

χ (λ) λn−2 λ λ 1 T 4n T T S = − e2n D1e2n − ( T )2 x Vn Vn x 2n x1 x1

λn−2 λ λ 1 T 2n , = − e2n D1 e2n − T 2n x1 x1 T T T T / since x Vn Vn x = x x = x1 x1 2. Hence, (12) is established.

4.3 Constructing cospectral EDMs

Two EDMs are said to be cospectral if they have the same eigenvalues, i.e., if they have the same characteristic polynomial. In this subsection, we present a method for constructing two cospectral EDMs.

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246 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES

μ > Given a scalar 1 and an n × n regular EDM D generated by the points 1,..., n p p , let D1 and D2 be the two 2n × 2n EDMs constructed from D as 1, 2,..., n μ 1, μ 2,... follows. D1 is generated by the 2n points p p p , − p − p , μ n 1, 2,..., n, μ 1, μ 2,... − p ; and D2 is generated by the 2n points p p p p p , μ n p . Next we show that D1 and D2 are cospectral. T ( ( )) en Den Since D is regular, we have diag T D = 2n2 en. Thus it follows from (3) and (4) that

DA1 , DA2 , D1 = μ2 and D2 = μ2 " A1 D # " A2 D # where 1 T μ2 T μ ( ), A1 e Den 1 ene 2 D (18) = 2n2 n + n + T and 1 T μ2 T μ ( ). A2 e Den 1 ene 2 D (19) = 2n2 n + n − T 1 T (μ2 ) π Note that A1en = A2en = 2n en Den + 1 en. Therefore, the 2-partition of D1 and D2 corresponding to the normalized characteristic matrix

1 en 0 π P = √ n " 0 en # is equitable. Since Vn 0 π , Pˉ = " 0 Vn # it immediately follows that

1(μ2 ) 1 1 + 1 T π T 2 T π . S1 = Pπ D1 P = en Den   = Pπ D2 P = S2 n 1(μ2 ) μ2  + 1   2  Furthermore, T μ T μ Vn DVn Vn DVn 1 T π − − T , Sˉ1 Pˉπ D1 Pˉ Vn DVn = = " μ T μ2 T # = " μ μ2 # ⊗ − Vn DVn Vn DVn −

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A.Y. ALFAKIH 247 and T μ T μ Vn DVn Vn DVn 1 T π T . Sˉ2 Pˉπ D2 Pˉ Vn DVn = = " μ T μ2 T # = " μ μ2 # ⊗ Vn DVn Vn DVn

But the eigenvalues of Sˉ1 are equal to the eigenvalues of Sˉ2. In particular, the ( μ2) nonzero eigenvalues of Sˉ1 = the nonzero eigenvalues of Sˉ2 = 1 + times the T ( μ2) nonzero eigenvalues of Vn DVn = 1 + times the nonzero eigenvalues of ( ) −2T D . Therefore, D1 and D2 are two cospectral EDMs. Furthermore, D1 μ > and D2 are not regular since 1.

5 Constructing EDMs with three distinct eigenvalues

EDMs have two or more distinct eigenvalues since the eigenvalues of an EDM D can 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 γ ( ) γ > D = E − I for some scalar 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 of strength for 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 eigenvalues 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 order n+1 obtained from n × n regular EDMs with 3 distinct eigenvalues by adding one row and one column of equal entries.

σ( ) T / Theorem 5.2. Let D be an n × n regular EDM such that D = e De n, λr1 , λn−1−r1 T / λ λ α2 T / − 1 − 2 . Assume that either e De n + 1 1 = n or e De n + Comp. Appl. Math., Vol. 27, N. 3, 2008 “main” — 2008/10/13 — 23:12 — page 248 — #12

248 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES

λ λ α2 α 1 T , α 1 T γ 2 2 n for some 2 e De ( ) e De. Let be some positive = ≥ 2n 6= n n−1 scalar. Then α γ D en D1 = α T " en 0 # is a non-regular EDM of order n + 1 with exactly 3 distinct eigenvalues.

α 1 T Proof. It is easy to see that D1 is a non-regular EDM since ( ) e De. 6= n n−1 ( T / λ )λ α2 ( ) Now assume e De n + 1 1 = n . Let V be the n × n − 1 matrix whose T columns form an orthonormal basis for subspace M defined in (2), thus VV = T / J = I − ee n. Let /√ en n 0 V Q . = " 0 1 0 #

Then / eT De/n α n1 2 0 T γ α 1/2 . Q D1 Q =  n 0 0  T  0 0 V DV    σ ( ( )) σ( T T ) λr1 But it is well known [1, 8] that −2T D = VV DVV = − 1 , λn−1−r1 , σ ( T ) λr1 , λn−1−r1 − 2 0 . Thus V DV = − 1 − 2 . Furthermore, the sub- matrix / eT De/n α n1 2 / " α n1 2 0 # λ 1 T λ has eigenvalues − 1 and n e De + 1. The result follows since Q is an orthogonal matrix. α 1 T Note that, taking = 2n2 e De in Theorem 5.2 is equivalent to placing point pn+1 at the center of the hyper-sphere containing p1,..., pn. Also, note that α 1 T if ( ) e De, then D1 becomes a regular EDM. = n n−1 The second class consists of non-regular EDMs of order n + 1 obtained from λ( ) λ > EDMs of the form D = En − In , 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. γ α ( )/ α Theorem 5.3. Let be a positive scalar and ≥ n − 1 2n, 6= 1. Then α γ En − In en D1 = α T " en 0 #

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A.Y. ALFAKIH 249

σ ( ) is a non-regular EDM with exactly 3 distinct eigenvalues, i.e., D1 = λ , λ , γ n−1 1 − 2 − where λ 1γ ( ( )2 α2 ) 1 n 1 n 1 4 n = 2 − + − + p and λ 1γ ( ( )2 α2 ). 2 n 1 n 1 4 n = 2 − − − + p 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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