Difference between revisions of "Multilinear algebra"
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The branch of algebra dealing with multilinear mappings (cf. [[Multilinear mapping|Multilinear mapping]]) between modules (in particular, vector spaces). The first sections of multilinear algebra were the theory of bilinear and quadratic forms, the theory of determinants, and the Grassmann calculus that extends this (see [[Exterior algebra|Exterior algebra]]; [[Bilinear form|Bilinear form]]; [[Quadratic form|Quadratic form]]; [[Determinant|Determinant]]). A basic role in multilinear algebra is played by the concepts of a [[Tensor product|tensor product]], a [[Tensor on a vector space|tensor on a vector space]] and a [[Multilinear form|multilinear form]]. The applications of multilinear algebra to geometry and analysis are related mainly to [[Tensor calculus|tensor calculus]] and differential forms (cf. [[Differential form|Differential form]]). | The branch of algebra dealing with multilinear mappings (cf. [[Multilinear mapping|Multilinear mapping]]) between modules (in particular, vector spaces). The first sections of multilinear algebra were the theory of bilinear and quadratic forms, the theory of determinants, and the Grassmann calculus that extends this (see [[Exterior algebra|Exterior algebra]]; [[Bilinear form|Bilinear form]]; [[Quadratic form|Quadratic form]]; [[Determinant|Determinant]]). A basic role in multilinear algebra is played by the concepts of a [[Tensor product|tensor product]], a [[Tensor on a vector space|tensor on a vector space]] and a [[Multilinear form|multilinear form]]. The applications of multilinear algebra to geometry and analysis are related mainly to [[Tensor calculus|tensor calculus]] and differential forms (cf. [[Differential form|Differential form]]). | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Marcus, "Finite dimensional multilinear algebra" , '''I-II''' , M. Dekker (1973) {{MR|0352112}} {{ZBL|0284.15024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Marcus, "Finite dimensional multilinear algebra" , '''I-II''' , M. Dekker (1973) {{MR|0352112}} {{ZBL|0284.15024}} </TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR> | ||
| + | </table> | ||
Latest revision as of 14:31, 7 April 2023
The branch of algebra dealing with multilinear mappings (cf. Multilinear mapping) between modules (in particular, vector spaces). The first sections of multilinear algebra were the theory of bilinear and quadratic forms, the theory of determinants, and the Grassmann calculus that extends this (see Exterior algebra; Bilinear form; Quadratic form; Determinant). A basic role in multilinear algebra is played by the concepts of a tensor product, a tensor on a vector space and a multilinear form. The applications of multilinear algebra to geometry and analysis are related mainly to tensor calculus and differential forms (cf. Differential form).
References
| [a1] | M. Marcus, "Finite dimensional multilinear algebra" , I-II , M. Dekker (1973) MR0352112 Zbl 0284.15024 |
| [a2] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207 |
How to Cite This Entry:
Multilinear algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multilinear_algebra&oldid=53619
Multilinear algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multilinear_algebra&oldid=53619
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article