هندسه‌ي ريماني و ميانگين‌هاي هندسي ماتريسي

چكيده انگليسي :

Riemannian geometry and matrix geometric means FATEMEH SHARIF MOGHADAM f sharif@math iut ac ir March 10 2019 Master of Science Thesis in Farsi Department 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 15A45 15A48 53B21 53C22Keywords Positive definite matrix Geometric mean Riemannian manifold Semi parallelogram law GradientAbstract This M Sc thesis is based on the following paper Riemannian geometry and matrix geometric means Linear Algebra Appl 413 2006 R B J H594 618 Operator theorists physicists engineers and statisticians have long been interested in various averaging operations means on positive definite matrices When just two matrices are involved the theory is very well developed Seethe foundational work of Kubo and Ando 15 The concept of geometric mean of two positive definite matrices was first introduced by Pusz and Woronowicz 26 The studies of these mathematicians can not be easily extended to three matrices and it has been along standingproblem to define a natural geometric mean of three positive definite matrices In some recent paper a new concept of geometric mean of two positive definite matrices has been achieved by identi fying the geometric mean of A and B as the midpoint of geodesic joining A and B which is stated below The basicdefinitions for this concept have been a lot of depth and ideas related to our work inspired by references 2 and 19 The thesis proceeds as follows In the first chapter we introduce the basic concepts and definitions needed In the second chapter first we analyzesome of the basic concepts of Riemannian geometry based on matrix analysis and explain how these concepts willhelp us better understand the geometric mean of two positive definite matrices To do this we write the definition ofthe geometric mean of two positive definite matrices A and B from 8 as follows 1 1 1 1 1 A#B A 2 A BA 2 A2 2 2The above definition is a valid interpretation of geometric mean A#B for two positive definite matrices A and B According to this interpretation there is a non positive curvature hyperbolic geometry on the space Pn consists ofn n positive matrices in which this A#B has a pleasing conceptual meaning A#B is the midpoint of the geodesic joining A and B But an acceptable interpretation of the geometric mean ofthree positive definite matrices A B C in Pn is not provided Ando Li and Mathias 2 proposed a list of ten properties the ALM properties that a good geometric mean of kmatrices should satisfy In the third chapter we describe these ten properties and according to them we give severaldefinitions for the geometric mean of two positive matrices We show which of these definitions of geometric meansof two positive definite matrices is generalized to three positive definite matrices Finally we present a valid interpretation of the geometric mean of three positive definite matrices which is expressedin 2 and denote by alm A B C that is obtained by a limit procedure successively replacing of the triangle by thegeometric means of its sides

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هندسه‌ي ريماني و ميانگين‌هاي هندسي ماتريسي

ميانگين هندسي , منيفلد ريماني , قانون شبه متوازي الاضلاع , گراديان , ماتريس معين مثبت