Bohemian Matrices

The term was first used to refer to matrices whose entries are integers of bounded height, hence the name, derived from the acronym BOunded Height Matrix of Integers (BOHEMI). Since then, other populations have been studied. There is no single family of Bohemian matrices. Instead, a matrix can be said to be Bohemian with respect to a set from which its entries are drawn. Bohemian matrices may possess additional structure. For example, they may be Toeplitz matrices or upper Hessenberg matrices.

Software testing

One of the applications of Bohemian matrices is in software testing using linear algebra. Bohemian matrices are typically distinctly represented on computers, and are identifiable for extreme behaviour either by exhaustive search (for small dimensions), random sampling, or optimization techniques.

For example, Steven E. Thornton used this concept to develop a tool that solved over two trillion eigenvalue problems. In doing so, he uncovered instances of convergence failure in some popular software systems.

An anymatrix is an extensible MATLAB matrix collection capturing a set of optimal classes of Bohemian matrices.

Improved bounds

In a talk given at the 2018 Bohemian Matrices and Applications Workshop, Nick Higham explained how he utilised genetic algorithms on Bohemian matrices with population P={-1, 0, 1} to sharpen lower bounds on the maximal growth factor for rook pivoting.

Random matrices

Bohemian matrices can be studied through random sampling. Such studies might be considered part of the field of random matrices. However, the study of random matrices to date has mostly focused on real symmetric or Hermitian matrices, or on matrices whose entries are drawn from a continuous distribution (such as Gaussian ensembles). Notable exceptions include the work of Terence Tao and Van Vu.

Bernoulli and Hadamard matrices

Matrices with entries only from ±1 are sometimes called Bernoulli matrices and are therefore examples of Bohemian matrices. A Hadamard matrix is a Bernoulli matrix that satisfies a further property, namely that its determinant is maximal. Therefore, Hadamard matrices are also Bohemian. Hadamard matrices (and indeed Bernoulli matrices) have been studied for much longer than the name "Bohemian matrix" has existed, but the kinds of questions one can ask about Hadamard matrices (the original question was which matrices have maximal determinant) can also be asked about other Bohemian matrices. One of the first generalisations of Hadamard matrices was to what are sometimes called Butson-type Hadamard matrices, whose entries are qth roots of unity for q > 2. These can also be considered prototypical Bohemian matrices.

Graph theory

Matrices with discrete entries, particularly incidence matrices, play a crucial role in understanding graph theory. The outcomes of graph theory aid in explaining certain phenomena encountered in Bohemian matrix experiments. Conversely, experiments conducted in the manner advocated by Bohemian matrix workers can provide useful information for graphs.

Combinatorics

Numerous open problems listed in the Encyclopedia of Integer Sequences under Bohemian matrices are combinatoric in nature. For instance, A306782 lists a table of the number of distinct minimal polynomials for Bernoulli matrices (that is, Bohemian matrices with population ±1) up to and including dimension 5. The numbers for higher dimensions are not known. The number of Bernoulli matrices of dimension 6 is 2³⁶=68,719,476,736; while this set could likely be searched exhaustively (it is delightfully parallel), the greater-than-exponential growth of the number of matrices soon defeats brute force. There are symmetries that might be taken advantage of, as is done for zero-one matrices, but these require sophisticated combinatoric knowledge.

Number theory

Number theorists have worked on polynomials with restricted coefficients over the years. For instance, Littlewood polynomials have coefficients ±1 when expressed in the monomial basis. Kurt Mahler was concerned with the problem, as were Andrew Odlyzko, Bjorn Poonen and Peter Borwein.

By using companion matrices, each of these restricted-coefficient polynomial problems can be considered to be a Bohemian matrix problem. However, the characteristic polynomial of a Bohemian matrix might have coefficients that are exponentially large in the dimension, so the converse is not true and these areas of study are not equivalent.

Connections to Magic Squares are explored in Kathleen Ollerenshaw's book with D. Brée. The following paper explicitly connects Bohemian matrices to quadratic forms.

Solution of polynomial equations

A commonly-used technique to find polynomial roots is to construct a companion matrix for it and use numerical eigenvalue solvers to find the eigenvalues, which are then the roots of the original polynomial. While seemingly counter-intuitive and wasteful, this technique can be practical. This is the default method of NumPy's Polynomial package. The method is frequently numerically stable, but occasionally does not work well if the polynomial coefficients are large or the polynomial is otherwise ill-conditioned.

It is possible to improve the situation by finding a minimal height companion matrix for the polynomial by looking for such in a Bohemian matrix family. However, no efficient general-purpose techniques are known.

The name "Bohemian matrices", together with the idea that it might be useful to categorise problems in this way, appeared in print by ISSAC 2016. The name arises from the mnemonic BOunded HEight Matrix of Integers (BOHEMI), although the classification has since been extended to other discrete populations including non-integer populations (including for example Gaussian integers). The breadth and utility of making this categorisation is becoming increasingly apparent: the first significant journal publication was although smaller publications appeared before that. As of March 2022, publications explicitly using the name, aside from those cited elsewhere in this article, include

The first workshop on the subject was held in 2018 at the University of Manchester and was entitled Bohemian Matrices and Applications. The idea can be considered to be a specialization in the sense of George Pólya, in that Bohemian matrix families are restricted to having certain entries and are thus special matrices. The idea can also be considered a generalisation, as instead of merely having entries either 0 or 1 as in an incidence matrix of a graph or -1 and 1 as in the companion matrix of a Littlewood polynomial, the Bohemian family can have entries that are otherwise bounded, discrete, or (usually) both.

The idea is somewhat similar to that of sign pattern matrices, in which two matrices with real entries are considered equivalent if the corresponding entries have the same sign, and that theory looks for common properties. A Bohemian matrix with population P={-1, 0, 1} is an example of a sign pattern matrix, and so it has all the properties of such matrices, but it may also have special properties owing to its Bohemian nature.