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2 Talk Eigenvalues of Euclidean matrices and rs-majorization on R pp.: 1–4

Eigenvalues of 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 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 D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean (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 = − PDP with P = I ee , where I is the n n , 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) = − PDP 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 . 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

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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 R ≺ is called row 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 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

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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

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Email: [email protected]; [email protected] Email: [email protected]

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