dc.contributor.author |
Dbouk, Malak Adel. |
dc.date.accessioned |
2012-12-03T13:33:59Z |
dc.date.available |
2012-12-03T13:33:59Z |
dc.date.issued |
2012 |
dc.identifier.uri |
http://hdl.handle.net/10938/9339 |
dc.description |
Thesis (M.S.)--American University of Beirut, Department of Mathematics, 2012.;"Advisor : Dr. Khuri Makdissi Kamal, Professor, Mathematics--Members of Committee : Dr. Abu Khuzam Hazar, Professor, Mathematics Dr. Egeileh Michel, Professor, Mathematics." |
dc.description |
Includes bibliographical references (leaf 100) |
dc.description.abstract |
Throughout my thesis I will discuss three main decompositions: the Bruhat, Cartan and Iwasawa decompositions. In the first chapter I will prove these decompositions in the general linear group of invertible matrices G. The main idea is that I have the subgroup of upper triangular matrices B of G, the maximal compact subgroup K of orthogonal matrices, A the subgroup of diagonal matrices, and the Weyl subgroup W. The Bruhat decomposition states that G is equal to the disjoint union of BwB where w belongs to W. Actually we can interpret the Bruhat decomposition in terms of flags. Given any two flags then there exists a basis of Rn that generates one of the flags and whose permutation generates the other flag. I use row and column operations to prove the result. The Iwasawa Decomposition states that G= KB. This says that every flag can be generated by an orthonormal basis. The Cartan decomposition states that G=KAK. I use the spectral theorem in addidtion to the fact that any symmetric positive definite matrix has a unique square root to prove this decomposition. In chapter 2, I prove these decompositions in the orthogonal group O(p,q). In this chapter the subgroup B is no more the subgroup of upper triangular matrices and A is no more the subgroup of diagonal matrices. In the proofs of the decompositions I use an isotropic basis of Rn under which the B and the A are easier to deal with. Another obstacle I face is making sure that the row and column operations I perform belong to O(p,q). This changes some aspects of the proof and makes it more tedious. Finally in chapter 3, I prove the decompositions in the general linear group os invertible matrices whose entries are in the field of p-adic numbers. In this chapter the maximal compact subgroup is the general linear group of invertible matrices whose entries belong to the ring of p-adic integers. A and B are still the subgroup of diagonal matrices and upper triangular matrices respectively. To prove the results we use row and column operations in addition to t |
dc.format.extent |
viii, 100 leaves 30 cm. |
dc.relation.ispartof |
Theses, Dissertations, and Projects |
dc.subject.classification |
T:005652 AUBNO |
dc.subject.lcsh |
Decomposition (Mathematics) |
dc.title |
Bruhat, Cartan and Iwasawa decompositions in GLn(R), O(p, q) and GLn(Qp) / by Malak Adel Dbouk. |
dc.type |
Thesis |
dc.contributor.department |
American University of Beirut. Faculty of Arts and Sciences. Department of Mathematics. |