On cell matrices: A class of Euclidean distance matrices
Introduction
A Euclidean distance matrix (EDM) is a matrix for which there exist n points in some Euclidean space such thatwhere is the usual Euclidean norm. An EDM D is called spherical if the points lie on a sphere in . (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 the linear space of symmetric matrices of order n. The Frobenius inner product in will be denoted by . The sets of positive semidefinite matrices and hollow symmetric matrices (i.e., symmetric matrices with only zero diagonal entries) are subsets of , which will be denoted by and , respectively. For a given vector stands for the diagonal matrix with diagonal entries equal to those of x:We will denote by the canonical vectors, and by e the vector of all ones.
The set of EDMs forms a closed convex cone in . The set is parametrized by the linear transformation on , that is,where with is defined as,See Gower [4], Critchley [2] and Johnson and Tarazaga [5]. If is restricted to a maximal face of , given bythen the function is one to one, and its inverse function is given by
Here in general, a face F of a cone C is any subset of C such that for each , every decomposition with implies . If the dimension of F is , 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 with corresponds to a different location of the origin of coordinates (for more details see Section 2 of [5]). The case in which is of particular importance. In this case we write instead of . Needless to say, the two functions and are mutually inverse. A matrix in 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 as long as .
Let be the set of n-dimensional vectors whose entries are nonnegative. For , a cell matrix is defined byWe will denote the set of cell matrices by :
The organization of the paper is as follows. Section 2 gives a characterization of cell matrix via the transformation . It is shown that the set 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 , 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 be the set of nonnegative diagonal matrices: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 for some . That is, Proof By the definition of cell matrix, if and only if there exists a nonnegative vector such that , where the function is defined
Faces of the cone of cell matrices
The structure of the convex cone in (2.1) is very simple. In fact, its extreme rays are generated by the matrices . Maximal faces of this cone are given byand the other faces can be written as the intersections of some of these maximal faces. The (relative) interior of the face is given bywhere rank denotes the rank of matrix. Most of the nice structure of is carried out by to the cell matrices. However,
The polar cone of
In this section, we describe the polar cone of the cone . Here, for a convex cone C in the linear space of symmetric matrices, the polar cone of C is defined as
First let us consider the cone in the subspace of hollow matrices, where Theorem 4.1 A hollow matrix belongs to the polar cone if and only if Proof Suppose that is in . Then by the definition of polar cone, it is necessary for X to satisfy
Linear inequality for cell matrices
Given an EDM , it is not trivial to determine if it is a cell matrix or not. This problem is the one of finding a nonnegative vector such that . For matrices in the faces of the cone of cell matrices, the problem is solved by Theorem 3.1, Theorem 3.3. If is in the relative interior of , the vector satisfying must be a positive solution (i.e., ) of the following systemwhich can be written in matrix form as
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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