On cell matrices: A class of Euclidean distance matrices

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Abstract

In this paper, we study the set of cell matrices and its relationship with the cone of positive semidefinite diagonal matrices. The set forms a convex polyhedral cone in the linear space of symmetric matrices. We describe the faces of the cone and its polar. We also provide a new linear inequality associated with cell matrices.

Introduction

A Euclidean distance matrix (EDM) is a matrix D=(dij) for which there exist n points x1,x2,,xn in some Euclidean space Rr such thatdij=xi-xj22,where ·2 is the usual Euclidean norm. An EDM D is called spherical if the points lie on a sphere in Rr. (See, for example, [9], [1].) During the last two decades, various kinds of subsets of EDMs with particular properties have been studied by several authors. The set of spherical EDMs is a typical example. The set of cell matrices, which is introduced lately by Jaklic̆ and Modic [6] is also an interesting one. As is stated in their paper, the notion of cell matrix is applied to many scientific areas. In this paper, we investigate the structure of the set of cell matrices as a polyhedral cone. We also identify its extreme directions and faces.

For this purpose, let us denote by Sn the linear space of symmetric matrices of order n. The Frobenius inner product in Sn will be denoted by A,BF=trace(AtB). The sets of positive semidefinite matrices and hollow symmetric matrices (i.e., symmetric matrices with only zero diagonal entries) are subsets of Sn, which will be denoted by Ωn and Hn, respectively. For a given vector x=(x1,x2,,xn)tRn,diag(x) stands for the diagonal matrix with diagonal entries equal to those of x:diag(x)=x100xn.We will denote by ei(i=1,2,,n) the canonical vectors, and by e the vector of all ones.

The set Λn of EDMs forms a closed convex cone in Sn. The set Λn is parametrized by the linear transformation κ on Ωn, that is,Λn=κ(Ωn),where κ(B) with B=(bij)Ωn is defined as,κ(B)=bet+ebt-2Bwithb=(b11,b22,,bnn)tRn.See Gower [4], Critchley [2] and Johnson and Tarazaga [5]. If κ is restricted to a maximal face of Ωn, given byΩn(s)={XΩn|Xs=0}withste=1,then the function κ:Ωn(s)Λn is one to one, and its inverse function τs:ΛnΩn(s) is given byτs(D)=-12I-estDI-set.

Here in general, a face F of a cone C is any subset of C such that for each aF, every decomposition a=b+c with b,cC implies b,cF. If the dimension dim(F) of F is dim(C)-1, then F is called a maximal face (or a facet). On the other hand, one-dimensional face is called an extreme ray. (See, for example, Chapter 2 of [3].) We call the direction of an extreme ray an extreme direction. Every face Ωn(s) with ste=1 corresponds to a different location of the origin of coordinates (for more details see Section 2 of [5]). The case in which s=e/n is of particular importance. In this case we write τ instead of τe/n. Needless to say, the two functions τ:ΛnΩn(e) and κ:Ωn(e)Λn are mutually inverse. A matrix in Ωn(e) is called centered, since the centroid of the corresponding configuration coincides with the origin.

Given an EDM D, we define its embedding dimension as the minimal dimension for which a configuration of the points that generate D can lie. It is well-known that the embedding dimension is the same as the rank of the matrix τs(D) as long as ste=1.

Let R+n be the set of n-dimensional vectors whose entries are nonnegative. For aR+n, a cell matrix D(a) is defined by(D(a))ij=0ifj=i,ai+ajifji.We will denote the set of cell matrices by Γn:Γn={D(a)|aR+n}.

The organization of the paper is as follows. Section 2 gives a characterization of cell matrix via the transformation κ. It is shown that the set Γn forms a convex polyhedral cone. Sections 3 Faces of the cone of cell matrices, 4 The polar cone of are devoted to describing the faces and the polar cone of Γn, respectively. In Section 5, we derive an interesting implication of cell matrix through a linear inequality.

Section snippets

Cell matrices structure

In this section we show a natural way to generate the cell matrices, by which a number of interesting properties can be derived. Let Δn be the set of nonnegative diagonal matrices:Δn=B=diag(b)|b=(b1,,bn)tR+n.We begin with the following basic result.

Theorem 2.1

A matrix D is a cell matrix if and only if D can be written as D=κ(B) for some BΔn. That is,Γn=κ(Δn).

Proof

By the definition of cell matrix, DΓn if and only if there exists a nonnegative vector aR+n such that D=D(a), where the function D(·) is defined

Faces of the cone of cell matrices

The structure of the convex cone Δn in (2.1) is very simple. In fact, its extreme rays are generated by the matrices Ei(i=1,2,,n). Maximal faces of this cone are given byFk={B=(bij)Δn|bkk=0}(k=1,2,,n)and the other faces can be written as the intersections of some of these maximal faces. The (relative) interior of the face is given byint(Fk)={BΔn|bkk=0andrank(B)=n-1},where rank denotes the rank of matrix. Most of the nice structure of Δn is carried out by κ to the cell matrices. However,

The polar cone of Γn

In this section, we describe the polar cone of the cone Γn. Here, for a convex cone C in the linear space Sn of symmetric matrices, the polar cone C of C is defined asC=XSn|X,YF0for allYC.

First let us consider the cone in the subspace Hn of hollow matrices, whereHn=H=(hij)Sn|h11=h22==hnn=0.

Theorem 4.1

A hollow matrix X=(xij)Hn belongs to the polar cone Γn if and only ifj=1nxij0fori=1,2,,n.

Proof

Suppose that XHn is in Γn. Then by the definition of polar cone, it is necessary for X to satisfyX,κ(Ei

Linear inequality for cell matrices

Given an EDM DΛn, it is not trivial to determine if it is a cell matrix or not. This problem is the one of finding a nonnegative vector aR+n such that D=D(a). For matrices in the faces of the cone Γn of cell matrices, the problem is solved by Theorem 3.1, Theorem 3.3. If D=(dij) is in the relative interior of Γn, the vector a=(a1,a2,,an)tR+n satisfying D=D(a) must be a positive solution (i.e., ai>0,i=1,2,,n) of the following systemai+aj=dij(1i<jn),which can be written in matrix form asMa=

Acknowledgments

The authors express their sincere gratitude to three anonymous reviewers for their careful reading and constructive comments, which greatly improved the quality of this paper.

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