Bohemian Upper Hessenberg Matrices
This work provides theoretical insights and enumeration results for a specialized class of matrices, but the results are incremental and primarily of interest to researchers in matrix theory and combinatorial matrix analysis.
The paper studies Bohemian upper Hessenberg matrices with entries from {-1,0,+1} and subdiagonal entries ±1, deriving recursive formulas for characteristic polynomials and identifying polynomials with maximal height, which grows exponentially with matrix order. It also counts stable, normal, and neutral matrices within this family.
We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $\pm1$. Many properties remain after these specializations, some of which surprised us. We find two recursive formulae for the characteristic polynomials of upper Hessenberg matrices. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix. We count stable matrices, normal matrices, and neutral matrices, and tabulate the results of our experiments. We prove a theorem about the only possible kinds of normal matrices amongst a specific family of Bohemian upper Hessenberg matrices.