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 hypermatrix (for ) is given by
(1)
The above hyperdeterminant vanishes iff the following system
of equations in six unknowns has a nontrivial solution,
(2)
(3)
(4)
(5)
(6)
(7)
Glynn (1998) has found the only known multiplicative hyperdeterminant in dimension larger than two.
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.Glynn, D. G. "The Modular
Counterparts of Cayley's Hyperdeterminant." Bull. Austral. Math. Soc.57,
479-497, 1998.Schläfli, L. "Über die Resultante eine
Systemes mehrerer algebraischer Gleichungen." Denkschr. Kaiserl. Akad. Wiss.,
Math.-Naturwiss. Klasse4, 1852.
hypermatrix 
(for 
) is given by
