Hyperdeterminant - Cayley's Second Hyperdeterminant Det

Cayley's Second Hyperdeterminant Det

In the special case of a 2×2×2 hypermatrix the hyperdeterminant is known as Cayley's Hyperdeterminant after the British mathematician Arthur Cayley who discovered it. The quartic expression for the Cayley's hyperdeterminant of hypermatrix A with components a_ijk, i,j,k = 0 or 1 is given by

Det(A) = a₀₀₀2a₁₁₁2 + a₀₀₁2a₁₁₀2 + a₀₁₀2a₁₀₁2 + a₁₀₀2a₀₁₁2

− 2a₀₀₀a₀₀₁a₁₁₀a₁₁₁ − 2a₀₀₀a₀₁₀a₁₀₁a₁₁₁ − 2a₀₀₀a₀₁₁a₁₀₀a₁₁₁ − 2a₀₀₁a₀₁₀a₁₀₁a₁₁₀ − 2a₀₀₁a₀₁₁a₁₁₀a₁₀₀ − 2a₀₁₀a₀₁₁a₁₀₁a₁₀₀

+ 4a₀₀₀a₀₁₁a₁₀₁a₁₁₀ + 4a₀₀₁a₀₁₀a₁₀₀a₁₁₁

This expression acts as a discriminant in the sense that it is zero if and only if there is a non-zero solution in six unknowns xi, yi, zi, (with superscript i = 0 or 1) of the following system of equations

a₀₀₀x0y0 + a₀₁₀x0y1 + a₁₀₀x1y0 + a₁₁₀x1y1 = 0

a₀₀₁x0y0 + a₀₁₁x0y1 + a₁₀₁x1y0 + a₁₁₁x1y1 = 0

a₀₀₀x0z0 + a₀₀₁x0z1 + a₁₀₀x1z0 + a₁₀₁x1z1 = 0

a₀₁₀x0z0 + a₀₁₁x0z1 + a₁₁₀x1z0 + a₁₁₁x1z1 = 0

a₀₀₀y0z0 + a₀₀₁y0z1 + a₀₁₀y1z0 + a₀₁₁y1z1 = 0

a₁₀₀y0z0 + a₁₀₁y0z1 + a₁₁₀y1z0 + a₁₁₁y1z1 = 0

The hyperdeterminant can be written in a more compact form using the Einstein convention for summing over indices and the Levi-Civita symbol which is an alternating tensor density with components εij specified by ε00 = ε11 = 0, ε01 = −ε10 = 1:

b_kn = (1/2)εilεjma_ijka_lmn

Det(A) = (1/2)εilεjmb_ijb_lm

Using the same conventions we can define a multilinear form

f(x,y,z) = a_ijkxiyjzk

Then the hyperdeterminant is zero if and only if there is a non-trivial point where all partial derivatives of f vanish.