Schurov komplement : magistrsko delo

Mihić, Željka (author), Oblak, Polona (mentor)

Na realnih bločnih matrikah definiramo Schurov komplement. Motivacija za definicijo izvira iz sistema linearnih enačb, ki se ga s pomočjo Schurovega komplementa lahko zreducira na reševanje manjšega sistema linearnih enačb. Na več načinov zapišemo inverz matrike, izražen s Schurovim komplementom, ki ga je že leta 1939 zapisal matematik Aitken. Dokažemo kvocientno formulo in izračunamo Schurov komplement vgnezdene matrike. Ogledamo si tudi nekaj lastnosti determinante matrik s pomočjo Schurovega komplementa. Na simetričnih bločnih matrikah dokažemo izrek o pozitivni definitnosti bločne podmatrike in njegovih Schurovih komplementov. S pomočjo Schurovega komplementa izpeljemo tudi razcep Choleskega simetričnih pozitivno definitnih matrik.

• Language: Slovenian
• Keywords: Schurov komplement, inverz, determinanta, simetrične matrike
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-104457
• UDC: 512
• COBISS.SI-ID: 18458457
• Publication date in RUL: 07.10.2018

Secondary language

• Language: English
• Title: Schur complement

Abstract

We define the Schur complement on a block matrix over real numbers. Motivation for the definition comes from the system of linear equations which can be reduced to a smaller system of linear equations by using the Schur complement. We write the inverse matrix which blocks are expressed by Schur complement. Those equalities were established already in 1939 by mathematician Aitken. We prove the quotient property of nested Schur complement. We also investigate the properties of determinants using Schur complement. We develop some properties of positive semidefiniteness of Schur complement of a symmetric matrix. We use Schur complement to construct the Cholesky decomposition of a symmetric positive definite matrix.

• Keywords: Schur complement, inverse, determinant, symmetric matrices