multilinear formとは 意味・読み方・使い方

名詞

multilinear form (複数形 multilinear forms)

1. (linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping V ^k → K that is linear in each of its arguments;
   (more generally) a similarly multiply linear mapping M ^r → R defined for a given module M over some commutative ring R.
   • 1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,

     Finally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (かつ Hankel operators too) really should be viewed as forms, not operators.

   • 1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,

     A multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate ${\displaystyle m}$-multilinear form can more concisely be written as ${\displaystyle M\!f}$.

   • 2003, Vladislav V. Goldberg (translator), Maks A. Akivis, Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55,

     Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).