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Download as PDF; Printable version; ... This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the ...
Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .; The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns). The multilinear polynomials in n {\displaystyle n} variables form a 2 n {\displaystyle 2^{n}} -dimensional vector space , which is also the basis used in the ...
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map: that is separately -linear in each of its arguments. [1] More generally, one can define multilinear forms on a module over a commutative ring.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide Multilinear may refer to: Multilinear form, a type of ...
The Theory of the Determinant in the Historical Order of Development. 4 vols. New York: Dover Publications 1960; A Treatise on the Theory of Determinants. Revised and Enlarged by William H. Metzler. New York: Dover Publications 1960 "A Second Budget of Exercises on Determinants", American Mathematical Monthly, Vol. 31, No. 6. (June, 1924), pp ...
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal.