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 questions posed about the distributions of eigenvalue densities.

One useful standard method to compute eigenvalues of matrix polynomials P(z)∈ C n x n [z] of degree at most ℓ in z (denoted of grade ℓ, for short) is to first transform P(z) to an equivalent linear matrix polynomial L(z)=zB-A, called a companion pencil, where A and B are usually of larger dimension than P(z) but L(z) is now only of grade 1 in z. The eigenvalues and eigenvectors of L(z) can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of P(z), including those for infinite eigenvalues, can also be recovered from eigenvectors of L(z) if L(z) is what is called a "strong linearization'' of P(z). In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors E(z) and F(z) which, via the equation E(z)L(z)F(z) = diag(P(z), In, …, I_n), explicitly show the equivalence of P(z) and P(z). By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.