Skew-symmetric tridiagonal Bohemian matrices

Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous "numbers" dress; photo courtesy Caltech Archives. Abstract: Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent. This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment. I also give a terrible pun. Don't say you weren't warned