توصيفگر ها :
مقدار ويژه , پوش , منحني مكعبي , برد عددي , ماتريس توپليتز سه قطري , ماتريس انتقال دهنده ي بلوكي , بلوك جردن
چكيده انگليسي :
The envelope of tridiagonal Toeplitz matrices and block shift matrices ZAHRA KORKEI z korkei@math iut ac ir March 3 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Seyed Mahmoud Manjegani manjgani@cc iut ac irAdvisor Dr Farid Bahrami fbahrami@cc iut ac ir2000 MSC 15A18 15A60Keywords Envelope Cubic curve Numerical range Tridiagonal Toeplitz matrix block shift matrix Jordan blockAbstract This M Sc thesis is based on the following papers Aretaki Aik Psarrakos P and Tsatsomeros M The envelope of tridiagonal Toeplitz matrices and block shift matrices Linear Algebra and its Applications 532 2017 60 85 The concept and definition of the envelope are based on an inequality proven in 1 that the eigenvalues of A mustsatisfy This inequality allows one to replace the half plane to the left of the largest eigenvalue of the hermitian part ofA by a smaller region that contains the spectrum of A Thus upon rotating a matrix A through all angles in 0 2 the envelope arises as a region that contains the eigenvalues and is contained in the numerical range F A Let Mn C denote the algebra of n n complex matrices The classical numerical range of a matrix A is defined as the compact and convex set F A x Ax x S 1 We will show that F A can be written as an infinite intersection of closed half planes namely e i H F A 0 2 In this thesis the basic properties of numerical range introduced and studied Among them is the well known spectralcontainment property A F A Consider the real quantities v A S A y1 2 u A Im y1 S A y1 2 where 2 denotes 2 norm and u A y1 S A y1 v A Adam and Tsatsomeros introduced and studiedthe cubic curve which is defined as below 2 2 A z C 1 A Re z u A Im z 2 A Re z 1 A Re z v A u A 0 2Showing that all the eigenvalues of A lie to its left namely A lies in the unbounded closed region 2 2 in A z C 1 A Re z u A Im z 2 A Re z 1 A Re z v A u A 0 2
