Doubly Stochastic Interval Matrices

Çankaya University Journal of Science and Engineering Volume 12, No. 2 (2015) 012–019 Doubly Stochastic Interval Matrices Azim Rivaz, M. Mohseni Moghadam, S. Zangoei Zadeh Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran e-mail: , , Abstract: Interval matrices have many applications in intelligent engineering problems such as robotics in computer science. In this paper, we will first describe the concepts of interval matrices. Next, we will introduce a new class of interval matrices, namely, doubly stochastic interval matrices. Finally, we will present some properties of this new class of matrices. Keywords: Stochastic matrix, interval matrix, doubly stochastic interval matrix. 1. Introduction Many real-life problems originate from diverse uncertainties, for example due to data measurement errors. The elements of a matrix, occurring in practice, are usually obtained from experiments, hence they may appear with uncertainties. We represent the uncertain elements in interval forms instead of fixed real numbers. The first systematic treatment of interval vectors and matrices was given by Apostolatos and Kulisch (1968). An interval matrix AI is a matrix whose elements are intervals and will be written as AI = [A, A] = (aIi j )(m×n) , where aIi j = [ai j , ai j ]. For this matrix, | AI | is a real matrix defined as | AI |= (| aIi j |)(m×n) , where | aIi j |= max{| ai j |, | ai j |}. For the interval matrix AI = [A, A], the center matrix denoted by Ac and the radius matrix denoted by ∆ are respectively defined as 1 1 Ac = (A + A) , ∆ = (A − A). 2 2 We assume that the reader is familiar with basic interval arithmetic, otherwise see [8]. An interval matrix AI is said to be nonnegative if A ≥ 0. We will say that an interval matrix AI is cogredient to an interval matrix BI if there exists a permutation matrix P such that AI = PT BI P. ISSN 1309 - 6788 c 2015 Çankaya University CUJSE 12, No. 2 (2015) Doubly Stochastic Interval Matrices 13 The real eigenvalue set of a square interval matrix is defined as Λ = {λ ∈ R; Ax = λ x, x 6= 0, A ∈ AI }. (1) A real vector x is called a real eigenvector of AI if it is a real eigenvector of some matrix A ∈ AI , [10]. A real nonnegative matrix A is said to be an r-doubly stochastic matrix if each of its row and column sums is r. The set of n × n r-doubly stochastic matrices is denoted by Γrn , [7]. Doubly stochastic matrices are very important in engineering and robotic problems: motion planning, localization and navigation [3] to name a few. In robotic networks, the discrete time consensus algorithm requires the adjacency matrix to be doubly stochastic [2]. On the other hand, dealing with uncertainties is unavoidable in these problems [5]. In fact, interval analysis is used to solve many robotic problems such as the clearance effect, robot reliability, motion planning, localization and navigation [6]. Therefore, we motivate the concept of a doubly stochastic interval matrix by the above observation. In the next section, we will introduce doubly stochastic interval matrices and study some of the properties of these matrices. Finally we show the application of these matrices in the field of robotics. 2. Basic Definitions and Main Results As mentioned previously, real r-doubly stochastic matrices have useful and interesting properties. In this section, we apply the existence of uncertainties in the elements of these matrices. At first, we introduce the doubly stochastic interval matrices and then we present some results for these matrices. Definition 1. A nonnegative n × n interval matrix AI = [A, A] is said to be [α , β ]-doubly stochastic interval matrix and denoted by AI[α ,β ] if A and A are α -doubly stochastic and β -doubly stochastic matrices, respectively. α +β Clearly, the center matrix of a doubly stochastic interval matrix AI[α ,β ] belongs to Γn 2 and its β −α radius matrix belongs to Γn 2 . Remark 2.1. Each [α , β ]-doubly stochastic interval matrix contains numerous r-doubly stochastic matrices, where α ≤ r ≤ β . The following lemma shows the existence of at least one r-doubly stochastic matrix in AI , for each α ≤ r ≤ β. 14 A. Rivaz et al. Lemma 1. Let AI[α ,β ] be an n × n doubly stochastic interval matrix. For each α ≤ r ≤ β , there exists at least one r-doubly stochastic matrix A ∈ AI . Proof. Define the map φ : [0, 1] −→ AI as follows: φ (t) = A + t(A − A). If we choose t = βr−−αα for each α ≤ r ≤ β , it is clear that φ (t) ∈ AI . Moreover, we have n ∑ (φ (t))i j = r, i = 1, · · · , n, j=1 and n ∑ (φ (t))i j = r, j = 1, · · · , n. i=1 In the following example, the existence of several 5.5-doubly stochastic interval matrices in AI has been shown. Example 2.1. Suppose  [2, 3] [0, 1] [1, 4]  AI =   [1, 5] [2, 3] 0 0 [1, 4] [2, 4]   .  2.5 Then AI is a [3, 8]-doubly stochastic interval matrix. Ac ∈ Γ5.5 3 and ∆ ∈ Γ3 . some other 5.5-doubly stochastic matrices in AI are         2 1 2.5 2 0.5 3 2.5 0 3 3 0.5 2             A= 0  0  0   ,  3.5 2  ,  3 2.5 0  ,  2.5 3  3.5 2 . 0 2.5 3 0 3 2.5 0 3 2.5 0 2 3.5 Arndt in [1] says that a nonnegative interval matrix AI is called (ir)reducible if | AI | is (ir)reducible. We extend the definition of reducibility of interval matrices, as the following definition. In fact, this definition is an extension of the reducibility of real matrices [7]. Definition 2. A nonnegative n-square interval matrix AI , for n ≥ 2, is strongly reducible if there exists a permutation matrix P such that PT AI P = AI1 0 AI3 AI2 ! , where AI1 and AI2 are (n-k)-square and k-square matrices, 1 ≤ k < n, respectively. Otherwise, AI is weakly irreducible. In the following lemma, some equivalent statements to reducibility of interval matrices are given. CUJSE 12, No. 2 (2015) Doubly Stochastic Interval Matrices 15 Lemma 2. For the square nonnegative interval matrix AI , the following statements are equivalent: (i) AI is reducible, (ii) A is reducible, (iii) all A ∈ AI are reducible. Proof. Since AI is nonnegative, we have | AI |= A. This implies (i) ⇒ (ii) and (iii) ⇒ (i). (ii) ⇒ (iii): Let A be reducible. Then there exists a permutation matrix P such that ! A1 0 T , P AP = A3 A2 where A1 and A2 are square matrices. Now suppose A ∈ AI . Then from 0 ≤ A ≤ A, it follows that ! ! 0 0 A1 A4 T ≤ PT AP = ≤ P AP = A3 A2 0 0 A1 0 A3 A2 ! . This implies that A4 = 0, and hence A is reducible. By the following example, we will show that if AI is irreducible, then every A ∈ AI is not neces- sarily irreducible. Example 2.2. The matrix  [2, 3] [2, 3] [1, 1]     AI =   [3, 4] [0, 1] [2, 3]  [0, 2] [0, 3] [2, 4] is irreducible, since | AI |= A is irreducible, but A is reducible. The following lemma shows that the strong reducibility is higher than reducibility of an interval matrix. Lemma 3. Every strongly reducible interval matrix is a reducible interval matrix. Proof. If AI = [A, A] i (...truncated)