Determinanty složitějších matic

Bc. Ján Húska

Bachelor's thesis

Determinants of complicated matrices

Abstract:

In this bachelor's thesis, we discuss the determinants of complicated matrices. In particular, it will be a description of various n-dimensional determinants whose evaluation will be in some closed form. We will give various proofs for the Vandermond determinant and the Cauchy double alternant. We will encounter a summary of various methods such as polynomial arguments, the direct method, Krattenthaler's lemma, and Dodgson condensation. We apply the above methods and mention also the methods of differential calculus, LU-factorization and others. Next, we describe the determinants of the Hankel matrix formed by the Catalan numbers and the circulant Toeplitz matrix. As applications in other areas of mathematics, we mention polynomial interpolation and resultant theory.

Abstract:

V tejto bakalárskej práci sa venujeme determinantom zložitejších matíc. Pôjde najmä o popis rôznych n-rozmerných determinantov, ktorých vyhodnotenie bude v nejakom uzavretom tvare. Poskytneme rôzne dôkazy pre Vandermondov determinant a Cauchy dvojitý alternant. Stretneme sa so súhrnom rôznych metód, ako sú polynomiálne úvahy, priama metóda, Krattenthalerovo lema a Dodgsonová kondenzácia. Vyššie uvedené metódy aplikujeme a spomenieme aj metódy diferenciálneho počtu, LU-faktorizácie a iné. Ďalej popíšeme determinanty Hankelovej matice tvorenej Catalanovými číslami a cirkulantnú Toeplitzovou maticu. Ako využitie v iných oblastiach matematiky uvádzame polynomiálnu interpoláciu a teóriu rezultánt.

Keywords

Determinant, Vandermondov determinant, Cauchyho dvojitý alternant, Krattenthaler, Laplaceov rozvoj, Dodgsonova kondenzácia, Hankelove matice, Toeplitzove matice, polynomiálna interpolácia, rezultanta, Vandermonde determinant, Cauchy double alternant, Laplace expansion, Dodgson condensation, Hankel matrix, Toeplitz matrix, polynomial interpolation, resultant

Language used: Slovak

Date on which the thesis was submitted / produced: 6. 5. 2024

Thesis defence

• Date of defence: 25. 6. 2024
• Supervisor: doc. Mgr. Josef Šilhan, Ph.D.
• Reader: Mgr. Milan Lekár, PhD.

Masaryk University

Faculty of Science

Bachelor programme / field:
Mathematics / Mathematics