TAILIEUCHUNG - Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix

The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationships between certain orthogonal polynomials and the determinant of the Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2 -periodic generalization. Then we compute its spectrum by left eigenvectors with a similarity trick. | Turk J Math (2016) 40: 80 – 89 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix 1 Emrah KILIC ¸ 1 , Talha ARIKAN2,∗ Department of Mathematics, TOBB Economics and Technology University, Ankara, C ¸ ankaya, Turkey 2 Department of Mathematics, Hacettepe University, Ankara, C ¸ ankaya, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: The Sylvester matrix was first defined by JJ Sylvester. Some authors have studied the relationships between certain orthogonal polynomials and the determinant of the Sylvester matrix. Chu studied a generalization of the Sylvester matrix. In this paper, we introduce its 2 -periodic generalization. Then we compute its spectrum by left eigenvectors with a similarity trick. Key words: Sylvester matrix, spectrum, determinant 1. Introduction There has been increasing interest in tridiagonal matrices in many different theoretical fields, especially in applicative fields such as numerical analysis, orthogonal polynomials, engineering, telecommunication system analysis, system identification, signal processing (., speech decoding, deconvolution), special functions, partial differential equations, and naturally linear algebra (see [2, 6, 7, 8, 15]). Some authors consider a general tridiagonal matrix of finite order and then describe its LU factorization and determine the determinant and inverse of a tridiagonal matrix under certain conditions (see [3, 9, 12, 13]). The Sylvester type tridiagonal matrix Mn (x) of order (n + 1) is defined as x 1 n x 0 n−1 Mn (x) = . . 0 0 0 0 0 0 0 2 0 0 x . 3 . 0 0 0 . 0 0 ··· ··· . . . ··· ··· 0 0 0 0 . . n−1 0 x n 1 x and Sylvester [14] gave its determinant as det Mn (x) = n ∏ (x + n − .

• Toán học
• Sylvester matrix
• 2 periodic tridiagonal Sylvester matrix
• Orthogonal polynomials
• Determinant of the Sylvester matrix
• 2 periodic generalization
• Vietnam Journal of Mechanics
• The dispersion of rayleigh waves
• Orthotropic layered half space using matrix method
• Identical principle axes
• The Sylvester’s theorem