14419

13030

كارشناسي ارشد

رياضي محض

اصفهان : دانشگاه صنعتي اصفهان

1397

[هفت]، 73ص.: مصور

محمود منجگاني

انگليسي به فارسي; فارسي به انگليسي

رسول نصر اصفهاني، فريد بهرامي

1397/12/08

كتابنامه

رياضي

رياضي

1397/12/14

ID13030

Heinz inequalities for matrices Maryam Momeni m momeni 459@gmail com January 14 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 ir2000 MSC 47A63 47A30 15A42 15A60 ssKeywords Heinz mean inequality Young inequality unitarily invariant norm Abstract In this thesis we persent matrix form of Heinz inequality and its refinements for matrix invariant norms Also westudy the reverse of matrix form of Heinz inequality with using scalar Young inequality Let a and b be nonnegative real numbers The Heinz means are defined as a b1 b a1 0 1 H a b 2Let Mn be the space of n n complex matrices and stand for any unitarily invariant norm on Mn i e U AV A for all A Mn and for all unitary matrices U V Mn For A aij Mn theHilbert Schmidt norm of A is defined by n A 2 aij 2 i j 1It is known that the Hilbert Schmidt norm is unitarily invariant In this thesis we always suppose that A B X Mnwith A B positive semidefinite The well known Heinz mean inequality says that for every positive semidefinitematrices A and B and for every unitarily invariant norm Ap B q Aq Ap Ap q B p qwhere p and q are positive real numbers A related inequality to the Heinz mean inequality is that Ap B q B p Aq Ap q B p q 1 1Replacing A and B by A p q B p q recepectively the we get the following equivalent inequality A B At B 1 t B t A1 twhere t 0 1 If X Mn is a Hermitian matrix then the eigenvalue vector of X is denoted by X 1 X n X with 1 X 2 X n X The singular values of X are the eigenvalues 1of positive semidefinite matrix X X X 2 they are numerated as s1 X s1 X sn X Thevector of the singular values is denoted S X s1 X sn X Let x x1 xn and y y1 yn be two vectors in Rn The vector x is said to be weakly majorizedby y and denoted by x w y if and only if n n x k 1 2 n yj j j 1 j 1

محمود منجگاني

رسول نصر اصفهاني، فريد بهرامي