Antitrikotni (Batmanov) razcep za simetrične matrike : delo diplomskega seminarja

Metličar, Samo (author), Plestenjak, Bor (mentor)

V delu obravnavamo antitrikotni razcep za simetrične nedefinitne matrike, znan tudi kot Batmanov razcep, katerega obstoj tudi dokažemo. Obravnavamo algoritem, ki z množenjem z ortogonalnimi matrikami pretvori vhodno matriko v bločno antitrikotno matriko, iz katere lažje razberemo inercijo in ocenimo lastne vrednosti. Omenjena dejstva so podprta tudi s primeri. V delo je vključena tudi analiza časovne zahtevnosti algoritma, ki lahko pri različnih vhodnih podatkih močno varira.

• Language: Slovenian
• Keywords: Batmanov razcep, simetrične matrike, bločne antitrikotne matrike, lastne vrednosti, inercija, algoritem, časovna zahtevnost
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-110401
• UDC: 519.6
• COBISS.SI-ID: 18817881
• Publication date in RUL: 14.09.2019

Secondary language

• Language: English
• Title: Antitriagonal (Batman) decomposition for symmetric matrices

Abstract

In this work the antitriangular decomposition for symmetric matrices, also known as the Batman decomposition, is examined and it's existence is proved. An algorithm which uses the multiplication by orthogonal matrices to transform the input matrix to a block antitriangular matrix is presented. This algorithm allows us to determine the inertia and estimate the eigenvalues more efficiently. This claim is supported by examples. Also included in the work is the analysis of time complexity of the algorithm, which can variate strongly depending on the input data.

• Keywords: Batman decomposition, symmetric matrices, block antitriangular matrices, eigenvalues, inertia, algorithm, time complexity