Bounds for the Right Spectral Radius of Quaternionic Matrices
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BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES I. Ali
UDC 517.5 In the present paper, we establish bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we establish bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces that contains all Gerˇsgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate our results.
1. Introduction The problems of quaternion division algebra have received much attention in the literature due to their applications in pure and applied sciences, such as quantum physics, control theory, altitude control, computer graphics, and signal processing (see, e.g., [1, 2, 4–6, 12, 14, 20–22] and the references therein). There are numerous published research papers on the location and estimation of the left and right eigenvalues of a quaternionic matrix [8, 16, 20, 22, 23]. The stability of linear difference-differential equations with quaternionic matrix coefficients is based on the location of right eigenvalues of their corresponding quaternionic block matrices [10, 11, 15]. The upper bounds for the left and right spectral radii of a quaternionic matrix were proposed by F. Zhang [22] in terms of the operator norm of a quaternionic matrix. The bounds for the sum of the norms of left eigenvalues were derived with the help of localization theorems for the left eigenvalues of the quaternionic matrix [8]. The first attempts to locate zeros of the quaternionic polynomials were made by G. Opfer [9] by direct calculations. In the first part of the paper, we present bounds for the sum of the absolute values of right eigenvalues of the quaternionic matrix. We further discuss bounds for the right spectral radius of the quaternionic matrix by applying the theory mentioned above. In the second part of the paper, we present the minimal ball that contains all Gerˇsgorin balls of the quaternionic matrix specified above. Further, we present the localization theorems for the right eigenvalues of quaternionic matrix with the help of the minimal ball mentioned above. The paper is organized as follows: Section 2 reviews some existing results. In Section 3, we discuss the upper bounds for the sum of the norms of right eigenvalues and the right spectral radius of the quaternionic matrix. Finally, in Section 4, we present the minimal ball and locations for the right eigenvalues of the quaternionic matrix. 2. Preliminaries Throughout the paper, R and C denote the fields of real and complex numbers, respectively. The set of real quaternions is defined by H := {q = q0 + q1 i + q2 j + q3 k : q0 , q1 , q2 , q3 2 R} with i2 = j2 = k2 = ijk = −1. This relation implies that ij = −ji = k, jk = −kj = i, and ki = −ik = j. School of Basic Sciences, Indian Institute of Technology Indore, Simrol, India; e-mail: [email protected]. Published in Ukrains’kyi Matematyc
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