Maximum Absolute Determinants of Upper Hessenberg Bohemian Matrices

Upper Hessenberg and Toeplitz Bohemians

We look at Bohemians, specifically those with population $\{-1, 0, {+1}\}$ and sometimes $\{0,1,i,-1,-i\}$. More, we specialize the matrices to be upper Hessenberg Bohemian. From there, focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the numbers o…

波埃米安上赫森伯格矩阵

Bohemian Upper Hessenberg Matrices

本文研究了波埃米安矩阵，特别是其元素取{-1,0,+1}的上赫森伯格矩阵，并探讨了其特性，推导出特征多项式递推公式，确定了最大高度的下界，并统计了稳定、正交和中性矩阵的数量。

博埃米安上赫森伯格托普利茨矩阵

Bohemian Upper Hessenberg Toeplitz Matrices

研究聚焦于具有{-1, 0, +1}元素的博埃米安上赫森伯格托普利茨矩阵，通过分析其特征多项式最大高度，确定了多项式并给出了高度的指数下界。

Bohemian Matrix Geometry

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers -- hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI) -- but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questio…