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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.
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.
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 ...
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 mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or | A |. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix.
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.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide Multilinear may refer to: Multilinear form, a type of ...
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.