Determinants of Bohemian matrix families

Density plots of eigenvalues of matrices from various Bohemian families. Collection of images by Steven Thornton. To find out more about these figures, or to create your own, check out the gallery at the Bohemian Matrices website.

The adjective “Bohemian” was used for the first time in a linear algebra context by Robert Corless and Steven Thornton to describe the eigenvalues of matrices whose entries are taken from a finite discrete set, usually of integers. The term is a partial acronym for “BOunded Height Matrix of Integers”, but the origin of the term was soon forgotten, and the expression “Bohemian matrix” is now widely accepted.

As Olga Taussky observed already in 1960, the study of matrices with integer elements is “very vast and very old”, with early work of Sylvester and Hadamard that dates back to the second half of the nineteenth century. These names are the first two in a long list of mathematicians that worked on what is now known as the “Hadamard conjecture”: for any positive integer n multiple of 4, there exists an n by n matrix H, with entries -1 and +1, such that HH^T = nI.

If this is the best-known open problem surrounding Bohemian matrices, it is far from being the only one. During the 3-day workshop “Bohemian Matrices and Applications” that our group hosted in June last year, Steven Thornton released the Characteristic Polynomial Database, which collects the determinants and characteristic polynomials of billions of samples from certain families of structured as well as unstructured Bohemian matrices. All the available data led Steven to formulate a number of conjectures regarding the determinants of several families of Bohemian upper Hessenberg matrices.

Gian Maria Negri Porzio and I attended the workshop, and set ourselves the task of solving at least one of these open problems. In our recent preprint, we enumerate all the possible determinants of Bohemian upper Hessenberg matrices with ones on the subdiagonal. We consider also the special case of families with main diagonal fixed to zero, whose determinants turn out to be related to some generalizations of Fibonacci numbers. Many of the conjectures stated in the Characteristic Polynomial Database follow from our results.

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