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Title: | Properties of Block Matrices |
Authors: | Saadetoğlu, Müge (Supervisor) Dinsev, Şakir Mehmet Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics |
Keywords: | Thesis Tez Mathematics Department Algebras, Linear--Matrices Block matrix inverses determinants tensor products |
Issue Date: | Sep-2023 |
Publisher: | Eastern Mediterranean University (EMU) - Doğu Akdeniz Üniversitesi (DAÜ) |
Citation: | Dinsev, Sakir Mehmet. (2023). Properties of Block Matrices. Thesis (M.S.), Eastern Mediterranean University, Institute of Graduate Studies and Research, Dept. of Mathematics, Famagusta: North Cyprus. |
Abstract: | In this master thesis, we study the block matrices and their properties. After giving a general overview on matrices, block matrices, different types of block matrices, and multiplication of two block matrices are discussed. In the inverse section, we first examine inverses of 2×2 block diagonal and block triangular matrices, ideas of proofs here can be extended to a general n × n block diagonal or a block triangular matrix. Then we give the inverse formula for 2 × 2 block matrix, in the case that one of the blocks is invertible. We then generalise this to any n×n block matrix by splitting it into 4 blocks (by producing a 2×2 block matrix). Determinant chapter is covered by two different methods, existing in the literature. First we revise a formulae for determinant of a block matrix where the blocks (matrices) belong to a commutative subring of Mn×n(F), where F is a field or a commutative ring. Then we give the general formula which would work for any block matrix, without any commutativity condition between the blocks. We also present formulas for the determinant of tensor product of two given matrices. Keywords: block matrix, inverses, determinants, tensor products ÖZ:
Bu yüksek lisans tezinde, blok matrisler ve özellikleri incelenmi¸stir. Matrislere genel
bir bakı¸s verildikten sonra, blok matrisler, farklı blok matris türleri ve iki blok
matrisin çarpımı ele alınmı¸stır. Blok matrislerin tersleri bölümünde, önce 2×2 blok
kö¸ segen ve blok üçgensel matrislerin tersi incelenmi¸stir. Buradaki ispat yöntemleri
genel bir n×n blok kö¸segen veya blok üçgensel matrisine geni¸sletilebilir. Daha sonra
bloklarin herhangi birinin tersinin olması ko¸suluna dayanarak 2 × 2 block
matrislerinin terslerinin formülü verilmi¸stir. Ayrıca bu formül n×n blok matrisini 4
tane blo˘ ga bölerek genelle¸stirilebilir (2 × 2 blok matris üreterek). Determinant
bölümü, literatürde var olan iki farklı yöntemle ele alınmı¸stır.
˙ Ilk olarak
blokların(matrislerin), Mn×n(F)’ nin de˘ gi¸sme özelli˘ gi olan alt-halkasına ait olması
durumunda (buradaki F bir cisim veya de˘ gi¸sme özelli˘ gi olan bir halkadır) blok
matrisin determinant formülü revize edilmi¸stir. Bunun yanında bloklar arasında
herhangi bir de˘ gi¸sme ko¸sulu olmaksızın determinant formülü incelenmi¸stir. Ayrıca
verilen iki matrisin tensör çarpımının determinantı formülleri sunulmu¸stur. |
Description: | Master of Science in Mathematics. Institute of Graduate Studies and Research. Thesis (M.S.) - Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2023. Supervisor: Assist. Prof. Dr. Müge Saadetoğlu. |
URI: | http://hdl.handle.net/11129/6444 |
Appears in Collections: | Theses (Master's and Ph.D) – Mathematics
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