On the eigenvalues of Euclidean distance matrices

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1 Volume 27, N. 3, pp , 2008 Copyright 2008 SBMAC ISSN On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada 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: 5K05, 5A8, 05C50. Key words: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic polynomial. Introduction An n n nonzero matrix D = (d i j ) is called a Euclidean distance matrix (EDM) if there exist points p, p 2,..., p n in some Euclidean space R r such that d i j = p i p j 2 for all i, j =,..., n, where denotes the Euclidean norm. Let p i, i N = {, 2,..., n}, be the set of points that generate an EDM D. An m-partition π of D is an ordered sequence π = (N, N 2,..., N m ) of nonempty disjoint subsets of N whose union is N. The subsets N,..., N m are called the cells of the partition. The n-partition of D where each cell consists #760/08. Received: 07/IV/08. Accepted: 7/VI/08. Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS.

2 238 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES of a single point is called the discrete partition, while the -partition of D with only one cell is called the single-cell partition. An m-partition π = (N, N 2,..., N m ) of an EDM D is said to be equitable if for all i, j =,..., m (case i = j included), there exist non-negative scalars α i j such that for each k N i, the sum of the squared Euclidean distances from p k to all points p l, l in N j, is equal to α i j. i.e., k N i, d kl = α i j, for all i, j =,..., m. () l N j The notion of equitable partitions for graphs, which was introduced by Sachs 3, is related to, among others, automorphism groups of graphs and distanceregular graphs 4. Schwenk 5 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.) 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. 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 important applications. These applications include, among others, molecular conformation problems in chemistry 2, multidimensional scaling in statistics 9, and wireless sensor network localization problems 6. We denote the identity matrix of order n by I n and the n-vector of all s by e n. E n,m = e n e T m denotes the n m matrix of all s, and E n = e n e T n denotes the square matrix of order n of all 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

3 A.Y. ALFAKIH 239 denotes the vector consisting of the diagonal entries of A. Finally, the spectrum of a matrix A, denoted by σ (A), is the multiset of the eigenvalues of A. If A has eigenvalues λ,..., λ k with multiplicities m,..., m k respectively, then σ (D) = {λ m,..., λm k k }. 2 Preliminaries Let D be an n n EDM, the dimension of the affine span of the points p,..., p n that generate D is called the embedding dimension of D. Let J n := I n e n e T n /n denote the orthogonal projection on subspace M := e = { x R n : e T x = 0 }. (2) It is well known 4, 8 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 S H denote the subset of n n symmetric matrices with zero diagonal; and let S C denote the subset of n n symmetric matrices A satisfying Ae = 0. Following 3, let T : S H S C and K : S C S H be the two linear maps defined by T (D) := J D J, (3) 2 and K(B) := diag (B)e T + e(diag (B)) T 2B. (4) It, then, immediately follows 3 that T and K are mutually inverse, and that D in S H 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 p,..., p n in R r. Then the n r matrix P = p T. p n T is called a realization of D. Given an EDM D, a realization P of D can be obtained by factorizing T (D) into T (D) = P P T. Note that if P is a realization

4 240 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES of D, then P = P Q is also a realization of D for any r 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 EDM D is said to be spherical if the points that generate D lie on a hypersphere, otherwise, it is said to be non-spherical. It is well known 6, 7 that an EDM D of embedding dimension r is spherical if and only if rank D = r +, 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 p,..., p n that generate D lie on a hyper-sphere whose center coincides with the centroid of p,..., p n. It is not difficult to show that 2, 8 an EDM D is regular if and only if e is an eigenvector of D corresponding to the eigenvalue n et De. 3 Equitable partition for EDMs It immediately follows from () that the discrete partition of D is always equitable with α i j = d i j ; while the single-cell partition of D is equitable if and only if D is regular. In the latter case, α = n et De. It also follows from () that i =,..., m and k N i we have (De) k = m α i j. (5) Let π = (N, N 2,..., N m ) be an m-partition of an n n EDM D where N i = n i for i =,..., m. Define the n m matrix P π = (p i j ) such that { /2 n j if i N j p i j = (6) 0 otherwise. P π is called the normalized characteristic matrix 5 of π since its jth column is equal to n /2 j times the characteristic vector of N j, and since P T π P π = I m. Next we present a lemma whose graph adjacency matrix counterpart was proved by Godsil and McKay 5. Lemma 3.. 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 = (s i j ) such that j= D P π = P π S. (7) Furthermore, if π is an equitable partition then s i j = (n i /n j ) /2 α i j.

5 A.Y. ALFAKIH 24 Proof. Assume that π is equitable then for all k N i and for all j =,..., m we have ( ) /2 (D P π ) kj = n /2 j d kl = n /2 j α i j = (n i ) /2 ni α i j = (P π S) kj, n l N j j where s i j = (n i /n j ) /2 α i j. On the other hand, assume that (7) holds for some partition π. Then for all k N i and all j =,..., m, we have (D P π ) kj = n /2 j l N j d kl = (P π S) kj = (n i ) /2 s i j. Therefore, l N j d kl = (n j /n i ) /2 s i j, which is independent of k. Hence π is equitable. Given an m-partition π of EDM D with m n, let ˉP π be the n (n m) matrix such that P π ˉP π is an orthogonal matrix. Let χ A (λ) denote the characteristic polynomial of matrix A. Then we have the following two results: Theorem 3.. Let π be an equitable m-partition of an n n EDM D, where m n. Then χ D (λ) = χ S (λ) χ ˉS(λ), (8) where S is defined in (7) and ˉS = ˉP T π D ˉP π. Proof. It follows from (7) and the definition of ˉP π that Pπ T D ˉP π = S Pπ T ˉP π = 0. Thus, ( P T π det λ I n D ) λim Pπ T P π ˉP π = det D P π 0 0 λi n m ˉP π T D. ˉP π ˉP T π Hence, det(λi n D) = det(λi m S) det(λi n m ˉS). Note that in case of discrete partitions, i.e., in case m = n, we have S = D thus χ D (λ) = χ S (λ) 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 0, and by Schwenk 5.

6 242 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES Theorem 3.2. Let π be an equitable m-partition of an n n EDM D, where m n. Then χ ˉS(λ) divides χ 2T (D) (λ), (9) where ˉS is as defined in Theorem 3.. Proof. It follows from (5) and the definition of ˉP π that ˉP π T De = 0 and e = 0. Thus, ˉP T π Hence, ˉP T π T (D)P π = 0 and = det det ( λ I n + 2 ˉP T π T (D) ˉP π = 2 ˉP T π D ˉP π = ˉS 2. P T π T (D) ) P π ˉP π λim + 2P T π T (D) P π 0 ˉP T π 0 λi n m + 2 ˉP T π T (D) ˉP π. Therefore, det ( λi n + 2T (D) ) = det ( λi m + 2P T π T (D)P π) det ( λin m ˉS ). 4 Applications of Theorem 3. In this section we show that Theorem 3. provides a new method for determining the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. It also provides a method for constructing cospectral EDMs. Let V n be the n (n ) whose columns form an orthonormal basis for subspace M defined in (2), i.e., V n has full column rank and satisfies V T n e n = 0, V T n V n = I n, V n V T n = J n = ( I n e n e T n /n). (0) 4. χ D (λ) of regular EDMs The characteristic polynomial of a regular EDM was obtained in 8,. The method used in 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..

7 A.Y. ALFAKIH 243 Corollary 4. (8, ). dimension r, then Let D = 0 be an n n regular EDM of embedding χ D (λ) = λ (λ n r ) r n et De (λ + 2μ i ), where μ i, for i =,..., r, are the nonzero eigenvalues of T (D). Proof. Let D be an n n regular EDM. Then, as we remarked earlier, the onecell partition π of D, in this case, is equitable. Since P π = n e and ˉP π = V n we have S = n et De and ˉS = Vn T DV n = 2 Vn T T (D)V n. Thus, i= χ S (λ) = λ n et De and χ ˉS(λ) = λ n r r (λ + 2μ i ) since the nonzero eigenvalues of ˉS are equal to the nonzero eigenvalues of the matrix 2T (D). i= 4.2 χ D (λ) of non-spherical centrally symmetric EDMs Let D be the 2n 2n EDM generated by the points p,..., p n,..., p 2n, where p n+i = p i for all i =,..., n. Assume that D is non-spherical. Then D is called a non-spherical centrally symmetric EDM. It is easy to see that D A D =, A D where D is the n n EDM generated by the points p,..., p n, and A = diag (T (D)) e T n + e n diag (T (D)) T + 2 T (D). () The characteristic polynomial of D, which was obtained in using a characterization of the nullspace of an EDM in terms of its Gale subspace, is given by χ D (λ) = λ 2n r 2 ( λ 2 λ 2n et 2n D e 2n 2n ) r x T x (λ + 2μ i ), (2) i=

8 244 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES where r is the embedding dimension of D, μ i, i =,..., r, are the nonzero eigenvalues of T (D ), and x R 2n is the unique vector satisfying D x = e 2n, e T 2n x = 0, and x Nullspace of D. (3) Note that such x satisfying (3) exists since D is non-spherical. A simple method to compute x is given in. Next, we establish (2) using Theorem 3.. The n-partition π of D corresponding to the normalized characteristic matrix P π = I n 2 I n is, obviously, equitable. Since ˉP π = 2 I n I n, it immediately follows that S = Pπ T D P π = D + A = 2 (diag (T (D)) en T + e n diag (T (D)) T ) and ˉS = ˉP π T D ˉP π = D A = 4 T (D). Hence, the nonzero eigenvalues of ˉS = the nonzero eigenvalues of 4T (D) = the nonzero eigenvalues of 2T (D ) since T (D ) = T (D) T (D) T (D) = T (D) T (D), (4) where denotes the Kronecker product. Hence, χ ˉS(λ) = λ n r r i= (λ + 2μ i), where μ i, i =,..., r, are the nonzero eigenvalues of T (D ). In order to establish (2), we still need to determine χ S (λ). To this end, note that T (D )D e 2n = 2nT (D ) diag (T (D )) = 0, where the first equality follows from (4). Then we have the following technical lemma. Lemma 4.. Let D be an 2n 2n non-spherical EDM satisfying Then where x satisfies (3). T (D )D e 2n = 0. D e 2n = et 2n D e 2n e 2n + 2n 2n x T x x, (5)

9 A.Y. ALFAKIH 245 Proof. Let y denote the subspace generated by y R 2n. Then it is shown in that Thus, Nullspace of T (D ) = e 2n x Nullspace of D. D e 2n = βe 2n + γ x, (6) for some scalars β and γ. The result follows by multiplying equation (6) from the left with e2n T and x T respectively. Therefore, it follows from (5) that x T = (x T x T ) for some vector x R n and Se n = (D + A)e n = et 2n D e 2n e n + 2n 2n x T x x. (7) Now, from the definition of V n in (0) we have Vn T SV n = 0. Hence, Vn T e n χ S (λ) = det λi n en T S V n n n Thus, = det λi n 2 n x T x 2 n x T x V T n x x T V n λ 2n et 2n D e 2n. ( χ S (λ) = λ (λ n 2 λ ) 2n et 2n D e 2n 4n ) (x T x ) x T V 2 n Vn T x ( = λ (λ n 2 λ ) 2n et 2n D e 2n 2n x T x ), since x T V n V T n x = x T x = x T x /2. Hence, (2) 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.

10 246 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES Given a scalar μ > and an n n regular EDM D generated by the points p,..., p n, let D and D 2 be the two 2n 2n EDMs constructed from D as follows. D is generated by the 2n points p, p 2,..., p n, μp, μp 2,..., μp n ; and D 2 is generated by the 2n points p, p 2,..., p n, μp, μp 2,..., μp n. Next we show that D and D 2 are cospectral. Since D is regular, we have diag (T (D)) = et n De n 2n 2 (3) and (4) that D A D =, and D μ 2 2 = D A e n. Thus it follows from D A 2 μ 2 D A 2, where and A = 2n 2 et n De ( n μ 2 + ) e n en T A 2 = 2n 2 et n De ( n μ 2 + ) e n en T + 2 μt (D), (8) 2 μt (D). (9) Note that A e n = A 2 e n = 2n et n De n(μ 2 + )e n. Therefore, the 2-partition π of D and D 2 corresponding to the normalized characteristic matrix is equitable. Since it immediately follows that S = P T π D P π = n et n De n Furthermore, ˉS = ˉP T π D ˉP π = P π = n V T ˉP π = μv T e n 0 0 e n V n 0 0 V n, 2 (μ2 + ) 2 (μ2 + ) μ 2 μv T μ 2 V T = = P T π D 2 P π = S 2. μ μ μ 2 V T,

11 A.Y. ALFAKIH 247 and ˉS 2 = ˉP T π D 2 ˉP π = V T μv T μv T μ 2 V T = μ μ μ 2 V T. But the eigenvalues of ˉS are equal to the eigenvalues of ˉS 2. In particular, the nonzero eigenvalues of ˉS = the nonzero eigenvalues of ˉS 2 = ( + μ 2 ) times the nonzero eigenvalues of V T = ( + μ 2 ) times the nonzero eigenvalues of 2T (D). Therefore, D and D 2 are two cospectral EDMs. Furthermore, D and D 2 are not regular since μ >. 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 D = γ (E I ) for some scalar γ > 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 introduced the notion of strength for distance matrices as a measure of their regularity. According to Neumaier regular EDMs are of strength 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. Theorem 5.. 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 + obtained from n n regular 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 that σ (D) = { e T De/n, λ r, λ n r } ( ) 2. Assume that either e T De/n + λ λ = n α 2 or ( e T De/n +

12 248 ON THE EIGENVALUES OF EUCLIDEAN DISTANCE MATRICES ) λ 2 λ2 = n α 2 for some α e T De, α = 2n 2 n(n ) et De. Let γ be some positive scalar. Then D α e n D = γ α en T 0 is a non-regular EDM of order n + with exactly 3 distinct eigenvalues. Proof. It is easy to see that D is a non-regular EDM since α = n(n ) et De. Now assume (e T De/n + λ )λ = nα 2. Let V be the n (n ) matrix whose columns form an orthonormal basis for subspace M defined in (2), thus V V T = J = I ee T /n. Let e n / n 0 V Q =. 0 0 Then Q T D Q = γ e T De/n α n /2 0 α n / V T DV But it is well known, 8 that σ ( 2T (D)) = σ (V V T DV V T ) = { λ r, λ n r 2, 0 }. Thus σ (V T DV ) = { λ r, λ n r } 2. Furthermore, the submatrix e T De/n α n /2 α n /2 0 has eigenvalues λ and n et De + λ. The result follows since Q is an orthogonal matrix. Note that, taking α = 2n 2 e T De in Theorem 5.2 is equivalent to placing point p n+ at the center of the hyper-sphere containing p,..., p n. Also, note that if α = n(n ) et De, then D becomes a regular EDM. The second class consists of non-regular EDMs of order n + obtained from EDMs of the form D = λ(e n I n ), λ > 0, by adding one row and one column of equal entries. The proof of the next theorem is similar to that of Theorem Theorem 5.3. Let γ be a positive scalar and α (n )/2n, α =. Then E n I n α e n D = γ α en T 0

13 A.Y. ALFAKIH 249 is a non-regular EDM with exactly 3 distinct eigenvalues, i.e., σ (D ) = { λ, λ 2, γ n } where λ = 2 γ (n + (n ) 2 + 4α 2 n) and λ 2 = 2 γ (n (n ) 2 + 4α 2 n). Acknowledgement. The author would like to thank an anonymous referee for his/her comments and suggestions which improved the presentation of this paper. REFERENCES A.Y. Alfakih, On the nullspace, the rangespace and the characteristic polynomial of Euclidean distance matrices. Linear Algebra Appl., 46 (2006), G.M. Crippen and T.F. Havel, Distance Geometry and Molecular Conformation. Wiley, New York, F. Critchley, On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 05 (988), C.D. Godsil, Compact graphs and equitable partitions. Linear Algebra Appl., 255 (997), C.D. Godsil and B.D. McKay, Feasibility conditions for the existence of walk-regular graphs. Linear Algebra Appl., 30 (980), J.C. Gower, Properties of Euclidean and non-euclidean distance matrices. Linear Algebra Appl., 67 (985), T.L. Hayden, J. Lee, J. Wells and P. Tarazaga, Block matrices and multispherical structure of distance matrices. Linear Algebra Appl., 247 (996), T.L. Hayden and P. Tarazaga, Distance matrices and regular figures. Linear Algebra Appl., 95 (993), J. De Leeuw and W. Heiser, Theory of multidimensional scaling. In: P.R. Krishnaiah and L.N. Kanal (Eds.), Handbook of Statistics, volume 2, pages North-Holland (982). 0 A. Mowshowitz, The adjacency matrix and the group of a graph. In: F. Harary (Ed.), New directions in the theory of graphs, pages Academic Press, New York (973). A. Neumaier, Distances, graphs and designs. Europ. J. Combinatorics, (980), A. Neumaier, Distance matrices, dimension and conference graphs. Nederl. Akad. Wetensch. Indag. Math., 43 (98),

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