TOPICS
Search

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 2×2×2 hypermatrix A=a_(ijk) (for i,j,k=0,1) is given by

 det(A)=(a_(000)^2a_(111)^2+a_(001)^2a_(110)^2+a_(010)^2a_(101)^2+a_(011)^2a_(100)^2)-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)+a_(001)a_(010)a_(101)a_(110)+a_(001)a_(011)a_(110)a_(100)+a_(010)a_(011)a_(101)a_(100))+4(a_(000)a_(011)a_(101)a_(110)+a_(001)a_(010)a_(100)a_(111)).
(1)

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
(2)
a_(001)x_0y_0+a_(011)x_0y_1+a_(101)x_1y_0+a_(111)x_1y_1=0
(3)
a_(000)x_0z_0+a_(001)x_0z_1+a_(100)x_1z_0+a_(101)x_1z_1=0
(4)
a_(010)x_0z_0+a_(011)x_0z_1+a_(110)x_1z_0+a_(111)x_1z_1=0
(5)
a_(000)y_0z_0+a_(001)y_0z_1+a_(010)y_1z_0+a_(011)y_1z_1=0
(6)
a_(100)y_0z_0+a_(101)y_0z_1+a_(110)y_1z_0+a_(111)y_1z_1=0.
(7)

Glynn (1998) has found the only known multiplicative hyperdeterminant in dimension larger than two.


See also

Determinant, Hypermatrix

Explore with Wolfram|Alpha

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.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. Klasse 4, 1852.

Referenced on Wolfram|Alpha

Hyperdeterminant

Cite this as:

Weisstein, Eric W. "Hyperdeterminant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hyperdeterminant.html

Subject classifications