Difference between revisions of "Affinor"
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| + | 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 \otimes V \simeq \mathop{\rm End} ( V ) $ |
| + | of linear algebra. | ||
Revision as of 16:09, 1 April 2020
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 \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=45052
Affinor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affinor&oldid=45052
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article