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# A Hypermatrix Analog of the General Linear Group

Communicated by *Notices* Associate Editor Emilie Purvine

Matrices are so ubiquitous and so deeply ingrained into our mathematical lexicon that one naturally asks: “Are there higher-dimensional analogs of matrices; more importantly why bother with them at all?” In short, hypermatrices are higher-dimensional matrices. Hypermatrices are important because they broaden the scope of matrix concepts such as spectra and group actions. Hypermatrix algebras also illuminate subtle aspects of matrix algebra. In this article we describe an instance where the transition from matrices to hypermatrices shines a light on subtle details of invariant theory and leads to new insights.

## 1. What are Hypermatrices?

*Hypermatrices* are multidimensional matrices. More formally a hypermatrix is a function which maps a Cartesian product of finite subsets of consecutive integers to members of a field or a skew field :

Such a hypermatrix is of order and of size Investigations on hypermatrices were initiated during the 19th century in Cayley’s work on hyperdeterminants .Cay45. To some extent, the history of hypermatrix investigations resembles the history of matrix investigations. The study of matrix and hypermatrix invariants preceded independent investigations of their algebras. The present note focuses on an invariant theory perspective on the transition from matrices to hypermatrices. There are numerous other approaches to generalize matrix results to hypermatrices. For instance, other approaches focus instead on discriminants, determinants, resultants, and spectra. GKZ94GF17Qi05DLDMV00Tuc66Lim05Lim21

## 2. Are Tensors Hypermatrices?

The distinction between tensors and hypermatrices closely mirrors the distinction between abstract linear transformations and their matrix representations. Recall that an abstract linear transformation specified over finite-dimensional spaces has an orbit of possible matrix representations. For instance if -vector represents an abstract linear transformation with respect to bases and then , represents the same linear transformation with respect to new bases and for arbitrary invertible matrices and Recall that the matrix product of . with satisfies the equality

meaning that the matrix product is equivariant under the action of the general linear group. In other words, the matrix product is a well-defined operation on tensor products of pairs of vector spaces and such that where , and denote dual spaces to and respectively. The First Fundamental Theorem of Invariant Theory ,Pro07LY13 asserts that it is impossible to specify a binary product whose operands are taken from tensor products of triplets of vector spaces and and outputs a member of a tensor product of a triplet of vector spaces. Later we describe a ternary product whose operands are taken from tensor products of triplets of vector spaces.

The *tensorial orbit* of an abstract linear transformation represented by (relative to arbitrarily chosen bases) is the subset of all matrices of the same size which account for all possible changes of bases and described by

For instance, the tensorial orbit of whose entries are taken from the finite field with two elements denoted is

In particular, the tensorial orbit of a zero matrix is always a singleton, whose unique member is the zero matrix itself. Zero matrices are the only matrices whose tensorial orbits are singletons. Members of the tensorial orbit of the identity matrix are all the elements of the general linear group over denoted GL It is comprised of all . invertible matrices whose entries are taken from the field or skew field When investigating abstract linear transformations via their matrix representations, it is crucial to distinguish matrix properties which depend on the choice of coordinate system from properties which are independent of this choice. .

A property common to all members of a tensorial orbit is called a *tensorial invariant*. Properties common to some specified suborbit of the tensorial orbit are also called tensorial invariants relative to the specified suborbit. For instance, the size of the tensorial orbit (i.e., the cardinality) of an abstract linear transformation prescribed over a finite field is a tensorial invariant. It is well known that there are distinct tensorial orbits over Each orbit collects . matrices having the same rank. In particular over (where denotes the finite field with elements), the largest orbit is GL and its size is Two other well-known matrix tensorial invariants include the rank and the nullity. .

## 3. Hypermatrix Tensorial Invariant

Classical invariant theory is the study of properties of polynomials which are unaffected by a linear change of variables. Such properties are not tied to any specific basis for the dual space generating their coordinate ring Olv99. Classically, third-order hypermatrices admit tensorial orbits very similar to their matrix counterparts. Tensorial orbits of third-order hypermatrices are also devised via a change of basis. In other words, a member of the tensorial orbit of is obtained by applying three separate invertible linear transformations to columns, rows, and depth vectors of respectively. Each column, row, and depth vector of , lies in the three spaces -vector , and , respectively. For instance, there are eight distinct tensorial orbits over , of sizes listed in increasing order:

Incidentally, hypermatrix tensorial invariants like the tensor rank are defined by analogy to their matrix counterparts. Technically, a tensor does not refer to any particular hypermatrix along its tensorial orbit but instead refers to the orbit as a whole. Unfortunately, in the majority of references which discuss applications of tensor decompositions, the word “tensor” is widely used to mean hypermatrix.

Around the early 1990s, D. M. Mesner and P. Bhattacharya MB90MB94 generalized association schemes to higher-order combinatorial structures. In the process, they pioneered a ternary hypermatrix product operation which generalizes the matrix product. The Bhattacharya–Mesner product opened up a path to investigations of new kinds of tensorial orbits defined over skew fields, which we call *non-classical tensorial orbits*. The main motivation for investigating non-classical tensorial orbits was to characterize invariants of linear actions prescribed over skew fields GER11GF17GF20. By contrast, the main motivation for investigating classical tensorial orbits is to determine invariants of linear actions prescribed over fields. Recall that an ordered matrix pair is conformable if the number of columns of the first matrix in the pair matches the number of rows in the second matrix of the pair. In other words, conformable pairs are matrix pairs for which the matrix product is meaningful in the specified order. The Bhattacharya–Mesner product naturally generalizes the matrix product to a ternary product operation defined for a conformable triplet of third-order hypermatrices

denoted with entries given by

Accordingly, the transpose of is denoted and permutes the entries of , by affecting a cyclic permutation of indices,

For notational convenience we adopt the notation convention

When is a field, the following transpose identity reminiscent of the matrix counterpart holds:

When ,Prod corresponds to a third-order analog of an inner product. Its operands form a conformable triplet made of a row vector, a depth vector, and a column vector. In the case where ,Prod expresses the natural action of on a pair of matrix depth slices (also called frontal slices) More precisely, in this case .Prod yields a matrix whose entry is the 3-way inner product of the row of th (as a matrix), column of th (as a matrix), and the line of (a vector). Finally, when ,Prod corresponds to a third-order analog of an outer product. Its operands form a conformable triplet made of a column slice (i.e., an element of ), a depth slice (i.e., an element of and a row slice (i.e., an element of ), ). The Bhattacharya–Mesner algebra is therefore a non-associative algebra with a ternary composition rule. Note that the Bhattacharya–Mesner outer product corresponds to the trilinear form associated with the well-known matrix multiplication tensor whose fundamental importance was discovered by Strassen Str69. More precisely, recall from Strassen’s investigations of the arithmetic complexity of matrix multiplication over Str69 the trilinear form

We see that the hypermatrix which underlies the said trilinear form expresses an instance of the Bhattacharya–Mesner outer product. Naturally, the Bhattacharya–Mesner rank of is the smallest number of Bhattacharya–Mesner outer-product summands which add up to Recall that every outer product is a Bhattacharya–Mesner product instance where . More importantly, the Bhattacharya–Mesner rank of a third-order hypermatrix gives a lower bound on the number of ordinary matrix products required to simultaneously express its entries. In short, the Bhattacharya–Mesner rank gives a lower bound on the matrix multiplication complexity, a third-order hypermatrix. Hereafter unless otherwise specified, hypermatrix outer products will refer to the Bhattacharya–Mesner outer product (i.e., a product instance where . ).

In hindsight, the Bhattacharya–Mesner product compactly describes general systems of linear equations over skew fields such as the field of quaternions. For instance, if we wanted to express a general linear system of equations with unknowns, for and we would have

Hence we could do so with hypermatrices

via a single constraint in the entries of the

When

where the matrix

We explain here how non-classical invariants depart from their classical counterparts. For this purpose, we draw attention to the simplest non-trivial illustration. Namely the determination of fundamental invariants of quadratic binary forms. Classically, a quadratic binary form is a homogeneous polynomial of degree

where

on the polynomial

This action maps

where

It is well known and easy to see that

expresses an invariant of

In contrast with the classical setting, in non-classical settings the simplest quadratic binary forms are polynomials in non-commuting variables

where

One recognizes at once from such a map the Bhattacharya–Mesner product of a conformable triplet made up of hypermatrices of sizes

Unfortunately, very little is known about invariants of polynomials in non-commutative variables. As an initial step in the study of non-classical invariants, we investigate non-commutative invariants of linear forms. Such investigations suggest distinctive third-order analogues of general linear groups. But first we briefly mention an alternative formulation of the eigenvalue/eigenvector problem over skew fields suggested by 3. In this context, the eigenvalue/eigenvector problem is specified via a pair of matrix slices

by an equation of the form

The vector

## 4. Third-Order Analog of the General Linear Group

We now describe a third-order analog of invertible matrices. Recall that the general linear group over an arbitrary field

We preface our description of a third-order analog of

by a map of the form

Since entries of

In short, third-order analogues of GL*hypermatrix inverse pairs *

A computer search reveals that there are

We define a group composition law as follows:

The pair *scaling hypermatrices*. A pair

If

The entry constraints for the pair