Difference between revisions of "Affinor"
From Encyclopedia of Mathematics
(Importing text file) |
|||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | a0111801.png | ||
| + | $#A+1 = 5 n = 0 | ||
| + | $#C+1 = 5 : ~/encyclopedia/old_files/data/A011/A.0101180 Affinor | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| + | {{TEX|auto}} | ||
| + | {{TEX|done}} | ||
| + | An [[Affine tensor|affine tensor]] of type $ (1, 1) $. | ||
| + | Specifying an affinor with components $ f _ {j} ^ { i } $ | ||
| + | is equivalent to specifying an endomorphism of the vector space according to the rule $ v ^ {i} = f _ {s} ^ { i } v ^ {s} $. | ||
| + | To the identity endomorphism there corresponds a unique affinor. The correspondence by which the matrix $ | f _ {j} ^ { i } | $ | ||
| + | is assigned to each affinor realizes an isomorphism between the algebra of affinors and the algebra of matrices. An affinor is sometimes defined in the literature as a general (affine) tensor. | ||
====Comments==== | ====Comments==== | ||
| − | I.e. one is concerned here with the isomorphism | + | I.e. one is concerned here with the isomorphism $ V^\star \otimes V \simeq \mathop{\rm End} ( V ) $ |
| + | of linear algebra. | ||
Latest revision as of 14:33, 7 April 2023
An affine tensor of type $ (1, 1) $.
Specifying an affinor with components $ f _ {j} ^ { i } $
is equivalent to specifying an endomorphism of the vector space according to the rule $ v ^ {i} = f _ {s} ^ { i } v ^ {s} $.
To the identity endomorphism there corresponds a unique affinor. The correspondence by which the matrix $ | f _ {j} ^ { i } | $
is assigned to each affinor realizes an isomorphism between the algebra of affinors and the algebra of matrices. An affinor is sometimes defined in the literature as a general (affine) tensor.
Comments
I.e. one is concerned here with the isomorphism $ V^\star \otimes V \simeq \mathop{\rm End} ( V ) $ of linear algebra.
How to Cite This Entry:
Affinor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affinor&oldid=17931
Affinor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affinor&oldid=17931
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article