|
The hyperdeterminant is a generalization of the determinant in algebra. Whereas a determinant is a scalar valued function defined on an ''n'' × ''n'' square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes. There are at least three definitions of hyperdeterminant. The first was discovered by Cayley in 1843 (published in 1849). It is usually denoted by det0. The second Cayley hyperdeterminant originated in 1845 and is often called "Det." This definition is a discriminant for a singular point on a scalar valued multilinear map. Cayley's first hyperdeterminant is defined only for hypercubes having an even number of dimensions (although variations exist in odd dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions). The third hyperdeterminant, most recently defined by Glynn (1998), occurs only for fields of prime characteristic ''p''. It is denoted by det''p'' and acts on all hypercubes over such a field. Only the first and third hyperdeterminants are "multiplicative", except for the second hyperdeterminant in the case of "boundary" formats. The first and third hyperdeterminants also have closed formulae as polynomials and therefore their degrees are known, whereas the second one does not appear to have a closed formula or degree in all cases that is known. The notation for determinants can be extended to hyperdeterminants without change or ambiguity. Hence the hyperdeterminant of a hypermatrix ''A'' may be written using the vertical bar notation as |''A''| or as ''det''(''A''). A standard modern textbook on Cayley's second hyperdeterminant Det (as well as many other results) is "Discriminants, Resultants and Multidimensional Determinants" by Gel'fand, Kapranov, Zelevinsky() referred to below as GKZ. Their notation and terminology is followed in the next section. == 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'') = ''a''0002''a''1112 + ''a''0012''a''1102 + ''a''0102''a''1012 + ''a''1002''a''0112 :: − 2''a''000''a''001''a''110''a''111 − 2''a''000''a''010''a''101''a''111 − 2''a''000''a''011''a''100''a''111 − 2''a''001''a''010''a''101''a''110 − 2''a''001''a''011''a''110''a''100 − 2''a''010''a''011''a''101''a''100 + 4''a''000''a''011''a''101''a''110 + 4''a''001''a''010''a''100''a''111 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 ''x''i, ''y''i, ''z''i, (with superscript i = 0 or 1) of the following system of equations :''a''000''x''0''y''0 + ''a''010''x''0''y''1 + ''a''100''x''1''y''0 + ''a''110''x''1''y''1 = 0 :''a''001''x''0''y''0 + ''a''011''x''0''y''1 + ''a''101''x''1''y''0 + ''a''111''x''1''y''1 = 0 :''a''000''x''0''z''0 + ''a''001''x''0''z''1 + ''a''100''x''1''z''0 + ''a''101''x''1''z''1 = 0 :''a''010''x''0''z''0 + ''a''011''x''0''z''1 + ''a''110''x''1''z''0 + ''a''111''x''1''z''1 = 0 :''a''000''y''0''z''0 + ''a''001''y''0''z''1 + ''a''010''y''1''z''0 + ''a''011''y''1''z''1 = 0 :''a''100''y''0''z''0 + ''a''101''y''0''z''1 + ''a''110''y''1''z''0 + ''a''111''y''1''z''1 = 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)ε''il''ε''jm''''a''''ijk''''a''''lmn'' :''Det''(''A'') = (1/2)ε''il''ε''jm''''b''''ij''''b''''lm'' Using the same conventions we can define a multilinear form :''f''(x,y,z) = ''a''''ijk''''x''''i''''y''''j''''z''''k'' Then the hyperdeterminant is zero if and only if there is a non-trivial point where all partial derivatives of ''f'' vanish. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hyperdeterminant」の詳細全文を読む スポンサード リンク
|