Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1912.09991

The $2\times2$ Matrix Mortality Problem and Invertible Matrices

Christopher Carl Heckman
View original source ↗Read source document ↗Export metadata ↓

Abstract

By modifying the proof of a paper by O. Bournez and M. Branicky, we establish that the Matrix Mortality Problem is decidable with any finite set of $2\times2$ matrices which has at most one invertible matrix. The same modification also shows that the number of non-invertible matrices is irrelevant.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christopher Carl Heckman. 2019-12-18. The $2\times2$ Matrix Mortality Problem and Invertible Matrices. https://arxiv.org/abs/1912.09991

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Graphs of Moore-Penrose inverse of matrices possessing the treeangle property

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

math.RA↗
arXiv2026

Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

math.RA↗
arXiv2026

An introduction to the algebra of rings and fields

This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some number-theoretical applications (such as a proof of quadratic reciprocity and Jacobsthal's formulas for $p = x^2 + y^2$), and tastes of Gröbner bases and the Smith normal form. Familiarity with groups and vector spaces is assumed, though no deep results from either theory are used. Over 250 exercises are included (mostly without solutions).

math.RA↗