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On commutativity and lie nilpoten y in matrix algebras
Thesis (MSc)--Stellenbosch University, 2015
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2015
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| _version_ | 1867613951930400768 |
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| access_status_str | Open Access |
| author | Sehoana, Mahlare Gerald |
| author2 | Van Wyk, L. |
| author_browse | Sehoana, Mahlare Gerald Van Wyk, L. |
| author_facet | Van Wyk, L. Sehoana, Mahlare Gerald |
| author_sort | Sehoana, Mahlare Gerald |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2015 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/98118 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:44:18.862Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2015 |
| publishDateRange | 2015 |
| publishDateSort | 2015 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/98118 On commutativity and lie nilpoten y in matrix algebras Sehoana, Mahlare Gerald Van Wyk, L. Stellenbosch University. Faculty of Science. Department Mathematical Sciences (Mathematics) Matrix algebras Matrices Lie algebras Schur's theorem Cayley–Hamilton theorem Commutativity (mathematics) Thesis (MSc)--Stellenbosch University, 2015 ENGLISH ABSTRACT : In this thesis we first discuss the proof by Mirzakhani [9] of Schur's Theorem which gives the maximum number of linearly independent matrices in a commutative algebra of n×n matrices over a field F. An example illustrating the application of Schur's Theorem is given. Secondly, we discuss the Cayley-Hamilton Theorem which asserts that any n×n matrix A satisfies its characteristic polynomial. A deduction of a Cayley-Hamilton trace identity for a 2 × 2 matrix A over a commutative ring from the Cayley-Hamilton Theorem is shown. We then discuss the Cayley-Hamilton trace identity for any matrix A ∈ M2(R) when (i) R is commutative, (ii) R is not necessarily commutative, (iii) R is not necessarily commutative and sp(A) = 0, (iv) R is not necessarily commutative and satisfies the identity [[x, y], [x, z]] = 0. Lastly, we discuss the matrix algebras U∗n(R), in particular the matrix algebras U∗3 (R) and U∗4 (R), in relation to polynomial identities [[. . . [[x1, x2], x3], . . .], xn] = 0, [x, y][w, z] = 0 and [[x, y], [w, z]] = 0. AFRIKAANSE OPSOMMING : In hierdie tesis beskryf ons eerstens die bewys deur Mirzakhani [9] van Schur se Stelling wat die maksimum aantal lineêr onafhanklike matrikse in 'n kommutatiewe algebra van n × n matrikse oor 'n liggaam F gee. 'n Voorbeeld word gegee wat die toepassing van Schur se Stelling illustreer. Tweedens bespreek ons die Cayley-Hamilton Stelling wat beweer dat elke n×n matriks A sy karakteristieke polinoom bevredig. 'n Afleiding van 'n Cayley-Hamilton spoor identiteit vir 'n 2 × 2 matriks A oor 'n kommutatiewe ring vanuit die Cayley-Hamilton Stelling word gegee. Ons bespreek dan die Cayley-Hamilton spoor identiteit vir enige matriks A ∈ M2(R) wanneer (i) R kommutatief is, (ii) R nie noodwendig kommutatief is nie, (iii) R nie noodwendig kommutatief is nie en sp(A) = 0, (iv) R nie noodwendig kommutatief is nie en die identiteit [[x, y], [x, z]] = 0 bevredig. Laastens bespreek ons die matriksalgebras U∗n(R), in besonder die matriksalgebras U∗3 (R)en U∗ 4 (R), met betrekking tot die polinoom identiteite [[. . . [[x1, x2], x3], . . .], xn] = 0, [x, y][w, z] = 0 en [[x, y], [w, z]] = 0. 2015-12-14T07:44:19Z 2015-12-14T07:44:19Z 2015-12 Thesis http://hdl.handle.net/10019.1/98118 en_ZA Stellenbosch University vi, 88 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Matrix algebras Matrices Lie algebras Schur's theorem Cayley–Hamilton theorem Commutativity (mathematics) Sehoana, Mahlare Gerald On commutativity and lie nilpoten y in matrix algebras |
| title | On commutativity and lie nilpoten y in matrix algebras |
| title_full | On commutativity and lie nilpoten y in matrix algebras |
| title_fullStr | On commutativity and lie nilpoten y in matrix algebras |
| title_full_unstemmed | On commutativity and lie nilpoten y in matrix algebras |
| title_short | On commutativity and lie nilpoten y in matrix algebras |
| title_sort | on commutativity and lie nilpoten y in matrix algebras |
| topic | Matrix algebras Matrices Lie algebras Schur's theorem Cayley–Hamilton theorem Commutativity (mathematics) |
| url | http://hdl.handle.net/10019.1/98118 |
| work_keys_str_mv | AT sehoanamahlaregerald oncommutativityandlienilpotenyinmatrixalgebras |