Hyperdeterminant

A technically defined extension of the ordinary Determinant to ``higher dimensional'' Hypermatrices. Cayley (1845) originally coined the term, but subsequently used it to refer to an Algebraic Invariant of a multilinear form. The hyperdeterminant of the $2\times 2\times 2$ Hypermatrix $A=a_{ijk}$ (for $i,j,k=0$, 1) is given by

$\mathop{\rm det}(A)=({a_{000}}^2{a_{111}}^2+{a_{001}}^2{a_{110}}^2+{a_{010}}^2{a_{101}}^2+{a_{011}}^2{a_{100}}^2)$
$\quad -2(a_{000}a_{001}a_{110}a_{111}+a_{000}a_{010}a_{101}a_{111}+a_{000}a_{011}a_{100}a_{111}$
$\quad +a_{001}a_{010}a_{101}a_{110}+a_{001}a_{011}a_{110}a_{100}+a_{010}a_{011}a_{101}a_{100})$
$\quad +4(a_{000}a_{011}a_{101}a_{110}+a_{001}a_{010}a_{100}a_{111}).$

The above hyperdeterminant vanishes Iff the following system of equations in six unknowns has a nontrivial solution,

$$a_{000}x_0y_0+a_{010}x_0y_1+a_{100}x_1y_0+a_{110}x_1y_1 = 0$$

$$a_{001}x_0y_0+a_{011}x_0y_1+a_{101}x_1y_0+a_{111}x_1y_1 = 0$$

$$a_{000}x_0z_0+a_{001}x_0z_1+a_{100}x_1z_0+a_{101}x_1z_1 = 0$$

$$a_{010}x_0z_0+a_{011}x_0z_1+a_{110}x_1z_0+a_{111}x_1z_1 = 0$$

$$a_{000}y_0z_0+a_{001}y_0z_1+a_{010}y_1z_0+a_{011}y_1z_1 = 0$$

$$a_{100}y_0z_0+a_{101}y_0z_1+a_{110}y_1z_0+a_{111}y_1z_1 = 0.$$

See also Determinant, Hypermatrix

References

Cayley, A. ``On the Theory of Linear Transformations.'' Cambridge Math. J. 4, 193-209, 1845.

Gel'fand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V. ``Hyperdeterminants.'' Adv. Math. 96, 226-263, 1992.

Schläfli, L. ``Über die Resultante eine Systemes mehrerer algebraischer Gleichungen.'' Denkschr. Kaiserl. Akad. Wiss., Math.-Naturwiss. Klasse 4, 1852.