On the eigenvalues of Euclidean distance matrices

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: [e-mail]

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.

Keywords: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic polynomial.

1 Introduction

An n × n nonzero matrix D = (d_ij) is called a Euclidean distance matrix (EDM) if there exist points p¹, p²,..., pⁿ in some Euclidean space ℜ^r such that

d_ij = ||pⁱ - p^j||² for all i,j = 1,...,n,

where || || denotes the Euclidean norm.

Let pⁱ, i Є N = {1,2,...,n}, be the set of points that generate an EDM D. An m-partition π of D is an ordered sequence π = (N₁, N₂, ..., N_m) of nonempty disjoint subsets of N whose union is N. The subsets N₁,...,N_m are called the cells of the partition. The n-partition of D where each cell consists 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.

An m-partition π = (N₁, N₂, ..., N_m) 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_ij such that for each k Є 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_ij. i.e.,

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 important 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 I_n and the n-vector of all 1's by e_n. E_n,m = e_n denotes the n × m matrix of all 1's, and E_n = e_n 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 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 λ₁,...,λ_k with multiplicities m₁, ..., m_k respectively, then σ(D) = {,..., }.

2 Preliminaries

Let D be an n ×n EDM, the dimension of the affine span of the points p¹,..., pⁿ that generate D is called the embedding dimension of D. Let J_n: = I_n - e_n/n denote the orthogonal projection on subspace

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 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 : S_H → S_C and : S_C → S_H be the two linear maps defined by

and

It, then, immediately follows [3] that and are mutually inverse, and that D in S_H is an EDM of embedding dimension r if and only if (D) is positive semidefinite of rank r.

Let D be an n × n EDM of embedding dimension r generated by the points p¹,...,pⁿ in ℜ^r. Then the n × r matrix

is called a realization of D. Given an EDM D, a realization P of D can be obtained by factorizing (D) into (D) = PP^T. Note that if P is a realization of D, then P^' = P is also a realization of D for any r × r orthogonal matrix . 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 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 corresponding to the eigenvalue e^T De.

3 Equitable partition for EDMs

It immediately follows from (1) that the discrete partition of D is always equitable with α_ij = d_ij; while the single-cell partition of D is equitable if and only if D is regular. In the latter case, α₁₁ = e^T De. It also follows from (1) that

Let π = (N₁, N₂, ..., N_m) be an m-partition of an n × n EDM D where |N_i| = n_i for i = 1,...,m. Define the n × m matrix P_π = (p_ij) such that

P_π is called the normalized characteristic matrix [5] of π since its jth column is equal to times the characteristic vector of N_j, and since P_π = I_m.

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. Thenπ is equitable if and only if there exists an m × m symmetric matrix S = (s_ij) such that

Furthermore, ifπis an equitable partition then s_ij = (n_i/n_j)^1/2α_ij.

Proof. Assume that π is equitable then for all k Є N_i and for all j = 1,...,m we have

where s_ij = ( n_i/n_j)^1/2α_ij.

On the other hand, assume that (7) holds for some partition π. Then for all k Є N_i and all j = 1,...,m, we have

Therefore, Σ_lЄNjd_kl = (n_j/n_i)^1/2s_ij, which is independent of k. Hence π is equitable.

□

Given an m-partition π of EDM D with m < n-1, let _π be the n × (n-m) matrix such that [P_{π p} ] is an orthogonal matrix. Let χ_A(λ) denote the characteristic polynomial of matrix A. Then we have the following two results:

Theorem 3.1. Letπbe an equitable m-partition of an n × n EDM D, where m < n-1. Then

where S is defined in (7) and = D_π.

Proof. It follows from (7) and the definition of _p that D _p = S

_p = 0. Thus,

Hence,

det(λ In – D) = det(λ Im – S) det(λ I_n–m).

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 [10], and by Schwenk [15].

Theorem 3.2.Letπbe an equitable m-partition of an n × n EDM D, where m < n-1. Then

where is as defined in Theorem 3.1.

Proof. It follows from (5) and the definition of _p that De = 0 and e = 0. Thus,

Hence,

Therefore,

det (λ I_n + 2(D)) = det (λ I_m + 2(D) P_π ) det (λ I_n–m –

□

4 Applications of Theorem 3.1

In this section we show that Theorem 3.1 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-1) whose columns form an orthonormal basis for subspace M defined in (2), i.e., V_n has full column rank and satisfies

4.1 χ_D(λ) 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.

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

whereµ_i, for i = 1,...,r, are the nonzero eigenvalues of (D).

Proof. Let D be an n × n regular EDM. Then, as we remarked earlier, the one-cell partition π of D, in this case, is equitable. Since P_π = e and _π = V_n we have S = e^T De and = DV_n = -2 (D) V_n. Thus,

since the nonzero eigenvalues of are equal to the nonzero eigenvalues of the matrix - 2 (D).

□

4.2χ_D(λ) of non-spherical centrally symmetric EDMs

Let D₁ be the 2n × 2n EDM generated by the points p¹,...,pⁿ,...,p²ⁿ, where pⁿ⁺ⁱ = -pⁱ 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

where D is the n × n EDM generated by the points p¹,...,pⁿ, and

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

where r is the embedding dimension of D₁, µ_i, i = 1,...,r, are the nonzero eigenvalues of (D₁), and x₁Є ℜ²ⁿ is the unique vector satisfying

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. The n-partition π of D₁ corresponding to the normalized characteristic matrix

is, obviously, equitable. Since

it immediately follows that S = D₁P_π = D + A = 2 (diag ( (D)) + e_n diag ( (D))^T) and = D₁ P_π = D - A = - 4 (D). Hence, the nonzero eigenvalues of = the nonzero eigenvalues of - 4 (D) = the nonzero eigenvalues of - 2 (D₁) since

where ⊗ denotes the Kronecker product. Hence, where µ_i, i = 1,...,r, are the nonzero eigenvalues of (D₁).

In order to establish (12), we still need to determine χ_S(λ). To this end, note that (D₁) D₁e_2n = 2n (D₁) diag ( (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

(D₁) D₁e_2n = 0.

Then

where x₁ satisfies (13).

Proof. Let 〈y〉 denote the subspace generated by y Є ℜ²ⁿ. Then it is shown in [1] that

Nullspace of (D₁) = 〈e_2n〉 ⊕ 〈x₁〉 ⊕ Nullspace of D₁

Thus,

for some scalars β and γ. The result follows by multiplying equation (16) from the left with and respectively.

□

Therefore, it follows from (15) that = (x^T x^T) for some vector x Єℜⁿ and

Now, from the definition of V_n in (10) we have SV_n = 0. Hence,

Thus,

since x^T V_n x = x^T x = 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.

Given a scalar µ> 1 and an n × n regular EDM D generated by the points p¹,..., pⁿ, let D₁ and 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 that D₁ and D₂ are cospectral.

Since D is regular, we have diag ( (D)) = e_n. Thus it follows from (3) and (4) that

where

and

Note that A₁e_n = A₂e_n = De_n (µ²+1)e_n. Therefore, the 2-partition π of D₁ and D₂ corresponding to the normalized characteristic matrix

is equitable. Since

it immediately follows that

Furthermore,

and

But the eigenvalues of

₁ are equal to the eigenvalues of ₂. In particular, the nonzero eigenvalues of ₁ = the nonzero eigenvalues of ₂ = (1 +µ²) times the nonzero eigenvalues of DV_n = (1 +µ²) times the nonzero eigenvalues of - 2 (D). Therefore, D₁ and D₂ are two cospectral EDMs. Furthermore, D₁ and D₂ are not regular since µ> 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 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 [11] introduced the notion of strength for distance matrices as a measure 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.

Theorem 5.2. Let D be an n × n regular EDM such that σ (D) = {e^TDe/n, - , -^-1-r₁}. Assume that either (e^TDe/n+λ₁) λ₁ = n α² or (e^TDe/n+ λ₂)λ₂ = n α² for some α > e^TDe, α ≠ e^TDe. Let γ be some positive scalar. Then

is a non-regular EDM of order n + 1 with exactly 3 distinct eigenvalues.

Proof. It is easy to see that D₁ is a non-regular EDM since α ≠ e^TDe. Now assume (e^TDe/n + λ₁)λ₁ = nα². Let V be the n×(n-1) matrix whose columns form an orthonormal basis for subspace M defined in (2), thus VV^T = J = I - ee^T/n. Let

Then

But it is well known [1, 8] that σ (-2 (D)) = σ (VV^T DVV^T) = { - , -^-1-r₁, 0 }. Thus s(V^T DV) = {-, -^-1-r₁ }. Furthermore, the sub-matrix

has eigenvalues - λ₁ and e^TDe+ λ₁. The result follows since is an orthogonal matrix.

□

Note that, taking α = e^TDe in Theorem 5.2 is equivalent to placing point pⁿ⁺¹ at the center of the hyper-sphere containing p¹,...,pⁿ. Also, note that if α = e^TDe, then D₁ becomes a regular EDM.

The second class consists of non-regular EDMs of order n+1 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 5.2.

Theorem 5.3. Letγ be a positive scalar andα< (n-1)/2n, α ≠ 1. Then

is a non-regular EDM with exactly 3 distinct eigenvalues, i.e., σ (D₁) = {λ₁, - λ₂, -γ ^n-1} where

and

Acknowledgement. The author would like to thank an anonymous referee for his/her comments and suggestions which improved the presentation of this paper.

Received: 07/IV/08. Accepted: 17/VI/08.

#760/08.

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*

Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS.

Publication Dates

• Publication in this collection
  29 Oct 2008
• Date of issue
  2008

History

• Accepted
  17 June 2008
• Received
  07 Apr 2008