View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 436 (2012) 1473–1481 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Majorization for the eigenvalues of Euclidean distance matrices ∗ Hiroshi Kurata a, ,PabloTarazagab a Graduate School of Arts and Sciences, University of Tokyo, Japan b Department of Mathematics and Statistics, Texas A&M University, Corpus Christi, TX 78412, United States ARTICLE INFO ABSTRACT Article history: This paper investigates the relation between the eigenvalues of a Received 1 March 2011 Euclidean distance matrix (EDM) and those of the corresponding Accepted 11 August 2011 positive semidefinite matrix. More precisely, let D1 and D2 be two Available online 7 September 2011 EDMs and let B1 and B2 be the corresponding positive semidefinite SubmittedbyR.A.Brualdi matrices such that Di = κ(Bi)(i = 1, 2). In this paper, when the eigenvalues of B majorizes those of B , a condition under which the AMS classification: 1 2 15A57 eigenvalues of D1 majorizes those of D2 is derived. © 2011 Elsevier Inc. All rights reserved. Keywords: Majorization Euclidean distance matrix Positive semidefinite matrix Eigenvalues Orthogonal group 1. Introduction A Euclidean distance matrix (EDM) has a one-to-one correspondence with a centered positive semidefinite matrix. This paper investigates the relation between the eigenvalues of an EDM and those of the corresponding positive semidefinite matrix. A symmetric and nonnegative matrix with zero diagonal elements is called a predistance matrix. An n × n predistance matrix D = (dij) is said to be an EDM, if there exist a positive integer r ( n − 1) r and n points p1,...,pn ∈ such that 2 dij =pi − pj (i, j = 1,...,n), ∗ Corresponding author. E-mail addresses: [email protected] (H. Kurata), [email protected] (P. Tarazaga). 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2011.08.022 1474 H. Kurata, P. Tarazaga / Linear Algebra and its Applications 436 (2012) 1473–1481 where ·denotes the usual Euclidean norm on r . As is well-known, a necessary and sufficient condition for a predistance matrix D to be an EDM is that 1 1 T τ(D) =− PDP with P = In − ee (1.1) 2 n is positive semidefinite (Schoenberg [9]), where In is the n × n identity matrix and e is the vector of all ones. To be more specific, let n be the set of n × n EDMs, and denote by n(e) the set of n × n positive semidefinite matrices B such that Be = 0: n(e) ={B ∈ n | Be = 0}, where n is the set of n × n positive semidefinite matrices. Then the linear mapping τ : n → n(e) defined by (1.1) is one-to-one, and its inverse mapping, say κ : n(e) → n,isgivenby κ(B) = beT + ebT − 2B with b = diag(B), (1.2) where diag(B) is the vector consisting of the diagonal elements of B.LetSn be the linear space of n × n symmetric matrices, and consider its two subspaces SH and SC , where SH ={X = (xij) ∈ Sn | x11 =···= xnn = 0} is the space of hollow symmetric matrices and SC ={X ∈ Sn | Xe = 0} is the space of centered symmetric matrices. The sets n and n(e) are subsets (more specifically, closed convex cones) of SH and SC , respectively. Furthermore, as is shown by Johnson and Tarazaga [5], the two mapping τ and κ are mutually inverse when they are viewed as τ : SH → SC and κ : SC → SH. The structure of n and n(e) are fully studied by Hayden et al. [3], Tarazaga, Hayden and Wells [10] and Tarazaga [11] from geometric points of view. This paper investigates the relation between the eigenvalues of an EDM and those of the corre- sponding positive semidefinite matrix. To state the problem precisely, for a symmetric matrix X,let λ(X) be the vector of the decreasingly ordered eigenvalues of X.LetBi (i = 1, 2) be two arbitrary pos- itive semidefinite matrices in n(e), and let Di = κ(Bi). Suppose that the eigenvalues of B1 majorizes those of B2: λ(B1) λ(B2). (1.3) T T Here, for two vectors x = (x1,...,xn) and y = (y1,...,yn) , we say that x majorizes y and denote it by x y,if k k n n x[i] y[i] (k = 1,...,n − 1) and xi = yi, i=1 i=1 i=1 i=1 where x[i] and y[i] are the ith largest elements of x and y, respectively (see [8] for detail). Some properties of the eigenvalues of EDMs are studied by Hayden and Tarazaga [4] and Alfakih [1]. However, little is known about the relation between the eigenvalues of an EDM and those of the corresponding positive semidefinite matrix. In this paper, when the eigenvalues of B1 majorizes those of B1, a condition under which the eigenvalues of D1 majorizes those of D2 is derived. While this problem is primarily of theoretical interest, the results here will be helpful in some applied areas such as multidimensional scaling in statistics, where it is often required to compare several EDMs to detect possibly significant difference among them. In such cases, it is natural to compare the location, size, shape and spreadth of their configurations by using statistical measures, H. Kurata, P. Tarazaga / Linear Algebra and its Applications 436 (2012) 1473–1481 1475 some of which are functions of the eigenvalues of the corresponding positive semidefinite matrices. Toseethismoreprecisely,letD and B be the EDM and its corresponding positive semidefinite matrix in question, and let B = CCT be a rank decomposition of B, where C is an n × r matrix of full rank { ,..., } and the set of its row vectors c1 cn gives a configuration of D. A typical measure for the size n 2 = ( T ) = ( ) of the configuration is i=1 ci tr CC tr B , which is clearly the sum of the eigenvalues of B. When measuring the sphericity of the configuration, one may be interested in how close the = ( / ) n T × covariance matrix of the configuration, V 1 n i=1 cici , which is an r r positive definite matrix, to the identity matrix (up to a multiplicative constant). In such a case, majorization ordering of the eigenvalues of V plays an important role, where if the eigenvalues of V are majorized by those of another covariance matrix, say V ,thenV is understood to be more spherical than V . Here, it is easy to see that V = (1/n)CT C and hence the eigenvalues of nV coincides with the nonzero eigenvalues of B. Therefore, when comparing the sphericity of the configurations of D1 = κ(B1) and D2 = κ(B2) with the same size tr(B1) = tr(B2), the inequality λ(B1) λ(B2) suggests that the configuration of D2 is more spherical than that of D1. Hence it is interesting to ask if the eigenvalues of the EDMs Di (i = 1, 2) share this kind of monotonicity. We begin with a well-known matrix result, in which the majorization for the eigenvalues of sym- metric matrices is described as a convex combination of matrices. For proof, see, for example, Example 2.4 of Eaton [2]. In the sequel, in order to avoid double suffix, we write two arbitrary EDMs as D and D˜ instead of D ˜ 1 and D2. Correspondingly, B1 and B2 in n(e) will be written as B and B, respectively. Let O(n) be the group of n × n orthogonal matrices. Lemma 1. Let X and X˜ be two arbitrary n × n symmetric matrices. Then the inequality λ(X) λ(X˜) (1.4) holds if and only if X˜ is expressed as the following convex combination m ˜ = T X ciUiXUi (1.5) i=1 ,..., m = for some integer m, some positive reals c1 cm satisfying i=1 ci 1, and some orthogonal matrices U1,...,Um ∈ O(n). In the next section, we derive a matrix identity that is similar to (1.5) to characterize the majorization (1.3) for two positive semidefinite matrices in n(e). The identity is applied to describe the relation between the eigenvalues of EDMs and positive semidefinite matrices in Section 3. 2. A characterization of majorization for eigenvalues Let O(n; e) be the set of orthogonal matrices satisfying e = e: O(n; e) ={ ∈ O(n) | e = e}, (2.1) which forms a subgroup of O(n). It is clear that O(n; e) contains the group P(n) of n × n permutation matrices as its subgroup, since every permutation matrix satisfies e = e. To describe an explicit expression of ∈ O(n; e),letF be any n × (n − 1) matrix satisfying T T F e = 0, F F = In−1. (2.2) Then it is easy to derive the following expression of O(n; e) (see, for example, Lemma 2.1 of Kurata [6]): O(n; e) ={FAF T + (1/n)eeT | A ∈ O(n − 1)}. The n × n matrix − / G = (F, n 1 2e) (2.3) 1476 H. Kurata, P. Tarazaga / Linear Algebra and its Applications 436 (2012) 1473–1481 is orthogonal. It is often convenient to express ∈ O(n; e) as ⎛ ⎞ ⎛ ⎞ A 0 FT = ( , −1/2 ) ⎝ ⎠ ⎝ ⎠ = ( ⊕ ) T , F n e − / G A 1 G (2.4) 0 1 n 1 2eT where ⊕ denotes the direct sum of matrices, i.e., ⎛ ⎞ A 0 A ⊕ 1 = ⎝ ⎠ ∈ O(n). 0 1 ˜ Theorem 1. Let B and B be two arbitrary n × n positive semidefinite matrices in n(e),andletβ = λ(B) and β˜ = λ(B˜).