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Hauke Reddmann
science forum Guru Wannabe
Joined: 03 May 2005
Posts: 112
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 Posted: Wed Jun 28, 2006 8:16 am Post subject:
Tensor Inverse?
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Is there something that could be termed "inverse" to
a tensor like, say, T^i_jk? If the "determinant" is
nonzero? And even more, if yes, does T*U=T*V imply
U=V?
--
Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn |
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Robert Low
science forum Guru
Joined: 01 May 2005
Posts: 1063
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| Posted: Wed Jun 28, 2006 8:41 am Post subject:
Re: Tensor Inverse?
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Hauke Reddmann wrote:
| Quote: | Is there something that could be termed "inverse" to
a tensor like, say, T^i_jk? If the "determinant" is
nonzero? And even more, if yes, does T*U=T*V imply
U=V?
|
You can think of a tensor as a linear operator
(generally in many different ways). No reason
why some of them shouldn't have inverses.
For example, you might have some vector space
V, and T is in V \otimes V \otimes V* \otimes V*
(where \otimes means tensor product). Then you
can think of T as a linear map from V* \otimes V*
to V* \otimes V*, or as a linear map from
V \otimes V* to V \otimes V*, or from V to
V \otimes V \otimes V*, or ....
Then you can pick bases for the domain and codomain
of the tensor thought of as a linear mapping with that
domain and codomain, express your tensor with respect
to those bases, and do what you'd normally do with
the resulting matrix. |
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Timothy Murphy
science forum Guru Wannabe
Joined: 29 Apr 2005
Posts: 275
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| Posted: Thu Jun 29, 2006 2:29 pm Post subject:
Re: Tensor Inverse?
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Hauke Reddmann wrote:
| Quote: | Is there something that could be termed "inverse" to
a tensor like, say, T^i_jk? If the "determinant" is
nonzero? And even more, if yes, does T*U=T*V imply
U=V?
|
It isn't clear what you mean by T*U .
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland |
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Hauke Reddmann
science forum Guru Wannabe
Joined: 03 May 2005
Posts: 112
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| Posted: Fri Jun 30, 2006 9:32 am Post subject:
Re: Tensor Inverse?
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Timothy Murphy <tim@birdsnest.maths.tcd.ie> wrote:
| Quote: | Hauke Reddmann wrote:
Is there something that could be termed "inverse" to
a tensor like, say, T^i_jk? If the "determinant" is
nonzero? And even more, if yes, does T*U=T*V imply
U=V?
It isn't clear what you mean by T*U .
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U and V are tensors, T*U is standard multiplication in
index form (and that may even contain some implicit
Einstein summation - could be relevant)
--
Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn |
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Timothy Murphy
science forum Guru Wannabe
Joined: 29 Apr 2005
Posts: 275
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| Posted: Fri Jun 30, 2006 11:37 am Post subject:
Re: Tensor Inverse?
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Hauke Reddmann wrote:
| Quote: | Timothy Murphy <tim@birdsnest.maths.tcd.ie> wrote:
Hauke Reddmann wrote:
Is there something that could be termed "inverse" to
a tensor like, say, T^i_jk? If the "determinant" is
nonzero? And even more, if yes, does T*U=T*V imply
U=V?
It isn't clear what you mean by T*U .
U and V are tensors, T*U is standard multiplication in
index form (and that may even contain some implicit
Einstein summation - could be relevant)
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That still doesn't clarify it, at least for me.
What exactly are U,V?
Incidentally, the tensor T^i_jk has a unique
Penrose/Moore generalized inverse X_i^jk ,
at least of you are working over the complex numbers.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland |
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