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Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2

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Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 1{4 Eigenvalues of Euclidean Distance Matrices and rs-majorization on R2 Asma Ilkhanizadeh Manesh∗ Department of Pure Mathematics, Vali-e-Asr University of Rafsanjan Alemeh Sheikh Hoseini Department of Pure Mathematics, Shahid Bahonar University of Kerman Abstract Let D1 and D2 be two Euclidean distance matrices (EDMs) with correspond- ing positive semidefinite matrices B1 and B2 respectively. Suppose that λ(A) = ((λ(A)) )n is the vector of eigenvalues of a matrix A such that (λ(A)) ... i i=1 1 ≥ ≥ (λ(A))n. In this paper, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs, on R will be investigated. ≺ Keywords: Euclidean distance matrices, Rs-majorization. Mathematics Subject Classification [2010]: 34B15, 76A10 1 Introduction An n n nonnegative and symmetric matrix D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean distance matrix (EDM) if there exist a positive integer r and a set of n points p1, . , pn r 2 2 { } such that p1, . , pn R and d = pi pj (i, j = 1, . , n), where . denotes the ∈ ij k − k k k usual Euclidean norm. The smallest value of r that satisfies the above condition is called the embedding dimension. As is well known, a predistance matrix D is Euclidean if and 1 1 t only if the matrix B = − P DP with P = I ee , where I is the n n identity matrix, 2 n − n n × and e is the vector of all ones, is positive semidefinite matrix. Let Λ be the set of n n n × EDMs, and Ωn(e) be the set of n n positive semidefinite matrices B such that Be = 0. × 1 Then the linear mapping τ :Λ Ω (e) defined by τ(D) = − P DP is invertible, and its n → n 2 inverse mapping, say κ :Ω (e) Λ is given by κ(B) = bet + ebt 2B with b = diag(B), n → n − where diag(B) is the vector consisting of the diagonal elements of B. For general refrence on this topic see, e.g. [1]. Majorization is one of the vital topics in mathematics and statistics. It plays a basic role in matrix theory. One can see some type of majorization in [2]-[13]. In this paper, the relation between the eigenvalues of EDMs and those of the corresponding positive ∗Speaker 408 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 2{4 2 semidefinite matrices respect to rs on R will be investigated. An nonnegative matrix R ≺ is called row stochastic if the sum of entries of each row of R is equal to one. The following notation will be fixed throughout the paper. m m Co(A) := λiai m N, λi 0, λi = 1, ai A, i Nm , { i=1 | ∈ ≥ i=1 ∈ ∈ } for a subset A Rn; P ⊂ P Sgn α be 1 if α > 0 and be 1 if α < 0, Sgn 0 can be 1 or 1; { } − { } − [T ] be the matrix representation of a linear function T : Rn Rn with respect to the → standard basis; ri be the sum of entries on the ith row of [T ]. A linear function T : Rn Rn is said to be a linear preserver (strong linear preserver) −→ of if T (x) T (y) whenever x y (T (x) T (y) if and only if x y). ∼ ∼ ∼ ∼ ∼ 1.1 Rs-majorization n We introduce the relation rs on R and we state some properties of rs-majorization on ≺ R2. Definition 1.1. A matrix R M with nonnegative entries is called row stochastic if the ∈ n sum of entries of each row of R is equal to one. Definition 1.2. For two real vector x and y, we say that x is rs-majorized by y (denoted by x y) if there exists an n-by-n row stochastic matrix R with all its column entries ≺rs equal such that x = Ry. In this paper, we consider this relation on R2. The following proposition gives an equivalent condition for rs-majorization on R2. t t 2 Proposition 1.3. Let x = (x1, x2) , y = (y1, y2) R . Then x rs y if and only if ∈ ≺ x = x y , y . 1 2 ∈ C{ 1 2} 2 Here we state all (resp. strong) linear preservers of rs on R . ≺ 2 2 a b Theorem 1.4. Let T : R R be a linear function, and let [T ] = c d . Then T → preserves if and only if r = r , Sgn a = Sgn d = Sgn b = Sgn c . ≺rs 1 2 { } { } 6 { } { } 2 2 Theorem 1.5. A linear function T : R R strongly preserves rs if and only if → ≺ [T ] = αI for some α R 0 . ∈ \{ } 2 Main results Till the end of this section, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs on R will be specify. ≺ Theorem 2.1. Let B, B Ω (e), and let D = κ(B) and D = κ(B). Then ∈ 2 λ(B) λ(B) λ(D) λ(D) ≺rs ⇐⇒ ≺rs e e e e e 409 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 3{4 α α β β Proof. Since B, B Ω2(e), there exist α,β 0 such that B = α− α , B = β− β , and ∈ ≥ − − 0, 2α and 0, 2β are the set of eigenvalues of B and B, respectively. By the definition of { } 0{ 4α } 0 4β κ, D = ande D = . So 4α, 4α and 4β, 4β are thee set of eigenvalues 4α 0 4β 0 {− } {− } e of D and D, respectively. We see that λ(B) e λ(B) if and only if B = 0. Also, if λ(D) λ(D) if and only if ≺rs ≺rs D = 0. Hencee λ(B) λ(B) if and only if λ(D) λ(D). ≺rs ≺rs e e e e References [1] A. Y. Alfakih, On the eigenvalues of Euclidean distance matrices, Comput. Appl. Math., 27, 2008, 237-250. [2] T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenval- ues, Linear Algebra Appl., 118, 1989, 163-248. [3] A. Armandnejad, Right gw-majorization on Mn,m, Bull. Iranian math. Soc., 35, 2009, no. 2, 69-76. [4] A. Armandnejad and H. Heydari, Linear preserving gd-majorization functions from Mn,m to Mn,k, Bull. Iranian math. Soc., 37, 2011, no. 1, 215-224. [5] A. Armandnejad and A. Ilkhanizadeh Manesh, Gut-majorization on Mn,m and its linear preservers, Electron. J. Linear Algebra, 23, 2012, 646-654. [6] A. Armandnejad and A. Salemi, On linear preservers of lgw-majorization on Mn,m, Bull. Malays. Math. Soc., 35, 2, 2012, no. 3, 755-764. [7] A. Armandnejad and A. Salemi, The structure of linear preservers of gs- majorization, Bull. Iranian Math. Soc., 32, 2006, no. 2, 31-42. [8] H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers, Linear and Multilinear Algebra, 53, 2005, 1-11. [9] A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majoriza- tion, Linear Algebra Appl., 423, 2007, 255-261. [10] A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly pre- serving majorizations of matrices, Electronic Journal of Linear Algebra, 15, 2006, 260-268. [11] A. Ilkhanizadeh Manesh, Linear functions preserving sut-majorization on Rn, Iranian Journal of Mathematical Sciences and Informatics, (submission). [12] A. Ilkhanizadeh Manesh, Right gut-Majorization on Mn,m, Electron. J. Linear Alge- bra, (submission). [13] A. Ilkhanizadeh Manesh and A. Armandnejad, Ut-Majorization on Rn and its Linear Preservers, Operator Theory: Advances and Applications, 242, 2014, 253-259. 410 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 4{4 Email: [email protected]; [email protected] Email: [email protected] 411 www.SID.ir .
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