Peter Zizler | Searcharxiv

arXiv2025

Convolution and Combination Matrices in Non-stationary Filtering

Time independent convolution yields circulant matrices whose eigenvectors are the Fourier exponentials with the eigenvalues being the Fourier transform of the mask. The case of time dependent convolution, the non-stationary case, no longer has this property and two matrices are then introduced, the cyclic convolution matrix and the cyclic combination matrix. In our paper, we prove results on the properties of these matrices. We give results in the context of the non-stationary frequency response in the Fourier domain, where the Fourier matrix is a full matrix in general. The techniques used here are attainable at the advanced undergraduate linear algebra settings and can be introduced into a relevant linear algebra undergraduate course.

math.GM↗

arXiv2023

Singular Value Decomposition at FIFA 2022

The singular value decomposition is arguably one of the most fundamental results in linear algebra. While rigorous proof of this result is of importance, equally important is the motivation in the applied settings. We provide a lively and quite intuitive presentation on the appearance of the singular value decomposition of a matrix in the round robin tournaments, as well as the polar decomposition, all in the context of FIFA 2022 World Cup. This exposition is intended to be implemented in a class setting or given as a take home project in the second linear algebra course. Given the popularity of this world event we expect students to be interested and engaged throughout the analysis and ensuing conversations. Discussions among students should be enriching as they attempt to analyse the various FIFA groups and give comparisons. Furthermore, ideas in regards how to best interpret the singular vectors should be quite interesting. Basic notions and results from the first year linear algebra course are assumed.

math.HO↗