Hyperdeterminant

The above determinant can be written in terms of a generalisation of the Levi-Civita symbol:

${\displaystyle \mathrm {Det} (A)=f^{ijkl}f^{nmop}f^{qrst}a_{inq}a_{jmr}a_{kos}a_{lpt}}$

where f is a generalisation of the Levi-Civita symbol which allows two indices to be the same:

${\displaystyle f^{0011}=f^{1100}=f^{0110}=f^{1001}=-1/2}$

${\displaystyle f^{0101}=f^{1010}=1}$

where the f satisfy:

${\displaystyle f^{...abc...}+f^{...bca...}+f^{...cab...}+f^{...cba...}+f^{...acb...}+f^{...bac...}=0.}$

For symmetric 2 × 2 × 2 × ⋯ hypermatrices, the hyperdeterminant is the discriminant of a polynomial. For example,

${\displaystyle a_{000}=a}$

${\displaystyle a_{001}=a_{010}=a_{100}=b}$

${\displaystyle a_{110}=a_{101}=a_{011}=c}$

${\displaystyle a_{111}=d}$

Then Det(A) is the discriminant of ${\displaystyle ax^{3}+3bx^{2}+3cx+d.}$

In the general case a hyperdeterminant is defined as a discriminant for a multilinear map f from finite-dimensional vector spaces V_i to their underlying field K which may be ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$.

${\displaystyle f:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{r}\to K}$

f can be identified with a tensor in the tensor product of each dual space V^*_i

${\displaystyle f\in V_{1}^{*}\otimes V_{2}^{*}\otimes \cdots \otimes V_{r}^{*}}$

By definition a hyperdeterminant Det(f) is a polynomial in components of the tensor f which is zero if and only if the map f has a non-trivial point where all partial derivatives with respect to the components of its vector arguments vanish (a non-trivial point means that none of the vector arguments are zero.)

The vector spaces V_i need not have the same dimensions and the hyperdeterminant is said to be of format (k₁, ..., k_r) k_i > 0, if the dimension of each space V_i is k_i + 1. It can be shown that the hyperdeterminant exists for a given format and is unique up to a scalar factor, if and only if the largest number in the format is less than or equal to the sum of the other numbers in the format.[5]

This definition does not provide a means to construct the hyperdeteriminant and in general this is a difficult task. For hyperdeterminants with formats where r ≥ 4 the number of terms is usually too large to write out the hyperdeterminant in full. For larger r even the degree of the polynomial increases rapidly and does not have a convenient general formula.

The case of formats with r = 1 deals with vectors of length k₁ + 1. In this case the sum of the other format numbers is zero and k₁ is always greater than zero so no hyperdeterminants exist.

The case of r = 2 deals with (k₁ + 1) × (k₂ + 1) matrices. Each format number must be greater than or equal to the other, therefore only square matrices S have hyperdeterminants and they can be identified with the determinant det(S). Applying the definition of the hyperdeterminant as a discriminant to this case requires that det(S) is zero when there are vectors X and Y such that the matrix equations SX = 0 and YS = 0 have solutions for non-zero X and Y.

For r > 2 there are hyperdeterminants with different formats satisfying the format inequality. For example, Cayley's 2 × 2 × 2 hyperdeterminant has format (1, 1, 1) and a 2 × 2 × 3 hyperdeterminant of format (1, 1, 2) also exists. However a 2 × 2 × 4 hyperdeterminant would have format (1, 1, 3) but 3 > 1 + 1 so it does not exist.

Since the hyperdeterminant is homogeneous in its variables it has a well-defined degree that is a function of the format and is written N(k₁, ..., k_r). In special cases we can write down an expression for the degree. For example, a hyperdeterminant is said to be of boundary format when the largest format number is the sum of the others and in this case we have[6]

${\displaystyle N(k_{2}+\cdots +k_{r},k_{2},\ldots ,k_{r})={\frac {(k_{2}+\cdots +k_{r}+1)!}{k_{2}!\cdots k_{r}!}}.}$

For hyperdeterminants of dimensions 2^r, a convenient generating formula for the degrees N_r is[7]

${\displaystyle \sum _{r=0}^{\infty }N_{r}{\frac {z^{r}}{r!}}={\frac {e^{-2z}}{(1-z)^{2}}}.}$

In particular for r = 2,3,4,5,6 the degree is respectively 2, 4, 24, 128, 880 and then grows very rapidly.

Three other special formulae for computing the degree of hyperdeterminants are given in [7]

for 2 × m × m use N(1, m − 1, m − 1) = 2m(m − 1)

for 3 × m × m use N(2, m − 1, m − 1) = 3m(m − 1)²

for 4 × m × m use N(3, m − 1, m − 1) = (2/3)m(m − 1)(m − 2)(5m − 3)

A general result that follows from the hyperdeterminants product rule and invariance properties listed below is that the least common multiple of the dimensions of the vector spaces on which the linear map acts divides the degree of the hyperdeterminant, that is,

lcm(k₁ + 1, ..., k_r + 1) | N(k₁, ..., k_r).

One of the most familiar properties of determinants is the multiplication rule which is sometimes known as the Binet-Cauchy formula. For square n × n matrices A and B the rule says that

det(AB) = det(A)det(B)

This is one of the harder rules to generalize from determinants to hyperdeterminants because generalizations of products of hypermatrices can give hypermatrices of different sizes. The full domain of cases in which the product rule can be generalized is still a subject of research. However, there are some basic instances that can be stated.

Given a multilinear form f(x₁, ..., x_r) we can apply a linear transformation on the last argument using an n × n matrix B, y_r = B x_r. This generates a new multilinear form of the same format,

g(x₁, ..., x_r) = f(x₁, ..., y_r)

In terms of hypermatrices this defines a product which can be written g = f.B

It is then possible to use the definition of the hyperdeterminant to show that

det(f.B) = det(f)det(B)^N/n

where n is the degree of the hyperdeterminant. This generalises the product rule for matrices.

Further generalizations of the product rule have been demonstrated for appropriate products of hypermatrices of boundary format.[8]

Cayley's first hyperdeterminant det₀ is multiplicative in the following sense. Let A be a r-dimensional n × ... × n hypermatrix with elements a_{i, ..., k}, B be a s-dimensional n × ... × n hypermatrix with elements b_..., and C be a (r + s − 2)-dimensional n × ... × n hypermatrix with elements c_... such that (using Einstein notation)

c_{i, ..., j, l, ..., m} = a_{i, ..., j}^kb_{k, l, ..., m},

then

det₀(C) = det₀(A) det₀(B).

A determinant is not usually considered in terms of its properties as an algebraic invariant but when determinants are generalized to hyperdeterminants the invariance is more notable. Using the multiplication rule above on the hyperdeterminant of a hypermatrix H times a matrix S with determinant equal to one gives

det(H.S) = det(H)

In other words, the hyperdeterminant is an algebraic invariant under the action of the special linear group SL(n) on the hypermatrix. The transformation can be equally well applied to any of the vector spaces on which the multilinear map acts to give another distinct invariance. This leads to the general result,

The hyperdeterminant of format ${\displaystyle (k_{1},\ldots ,k_{r})}$ is an invariant under an action of the group ${\displaystyle \mathrm {SL} (k_{1}+1)\otimes \cdots \otimes \mathrm {SL} (k_{r}+1)}$

For example, the determinant of an n × n matrix is an SL(n)² invariant and Cayley's hyperdeterminant for a 2 × 2 × 2 hypermatrix is an SL(2)³ invariant.

A more familiar property of a determinant is that if you add a multiple of a row (or column) to a different row (or column) of a square matrix then its determinant is unchanged. This is a special case of its invariance in the case where the special linear transformation matrix is an identity matrix plus a matrix with only one non-zero off-diagonal element. This property generalizes immediately to hyperdeterminants implying invariance when you add a multiple of one slice of a hypermatrix to another parallel slice.

A hyperdeterminant is not the only polynomial algebraic invariant for the group acting on the hypermatrix. For example, other algebraic invariants can be formed by adding and multiplying hyperdeterminants. In general the invariants form a ring algebra and it follows from Hilbert's basis theorem that the ring is finitely generated. In other words, for a given hypermatrix format, all the polynomial algebraic invariants with integer coefficients can be formed using addition, subtraction and multiplication starting from a finite number of them. In the case of a 2 × 2 × 2 hypermatrix, all such invariants can be generated in this way from Cayley's second hyperdeterminant alone, but this is not a typical result for other formats. For example, the second hyperdeterminant for a hypermatrix of format 2 × 2 × 2 × 2 is an algebraic invariant of degree 24 yet all the invariants can be generated from a set of four simpler invariants of degree 6 and less.[9]