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xyz91234@yahoo.com science forum beginner
Joined: 13 Jun 2006
Posts: 6

Posted: Mon Jul 10, 2006 3:49 am Post subject:
Tensor Decomposition



How can one generalize matrix decompositions to tensors? How can one
find the LU decomposition, singular value decomposition, jordan
elimination, of a tensor? How can one find the determinant, rank,
characteristic polynomial, etc. on a tensor.
Thank You 

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Ulysse from CH science forum beginner
Joined: 30 May 2006
Posts: 16

Posted: Mon Jul 10, 2006 12:01 pm Post subject:
Re: Tensor Decomposition



On 9 Jul 2006 20:49:58 0700, xyz91234@yahoo.com wrote:
Quote:  How can one generalize matrix decompositions to tensors? How can one
Generalize ? Matrices are not really tensors ... it can only be said 
that certain types of tensors have *in a given base* components
which may be naturally considered as forming a matrix. Mainly
this happens with 2tensors once covariant and once contravariant.
But this will not give you a true generalisation, just another
interpretation of the same things. In fact, said tensors are in
natural correspondance with endomorphisms of the given space E *)
and in a given base an endomorphism is given by a matrix which
has as elements the components of the corresponding tensor.
A change of base has the same effect on the matrix and the
tensor components. So you can interpret all you know about
matrices / endomorphisms in terms of such mixed tensors.
If instead you look at twice covariant or twice contravariant
2tensors you may of course still consider their components in
a base as forming matrices, but the behaviour of these in a change
of base will be different. For instance, twice covariant 2tensors
correspond naturally to bilinear forms (on ExE) for which a change
of base produces a transformation formula for their matrices with
the transpose of the matrix of said change instead of its inverse.
Therefore most of the concepts about matrices (that in fact concern
endomorphisms) will not make much sense for such tensors ...
And for higher order tensors I don't think there is much to do
that might be considered as such a generalisation.
Quote: 
find the LU decomposition, singular value decomposition, jordan
elimination, of a tensor? How can one find the determinant, rank,
characteristic polynomial, etc. on a tensor.
I don't know precisely what the first 3 of these things are (although 
they might be intimately connected to things I know) but probably
for all of them what I say above applies. At least the determinant
(and therefore also the char. pol.) is not an invariant for matrices
of bilinear forms unless one restricts to change of bases with
determinant +/1. For the rank we have here invariance, but
I doubt it makes sense for higher order tensors.
*) by the natural isomorphism E^* (x) F > Hom(E,F) applied to F=E:
here E^* means the dual of the vector space E and (x) the tensor
product, all spaces are over the same field, E must be finite
dimensional 

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mariano.suarezalvarez@gma science forum addict
Joined: 28 Apr 2006
Posts: 58

Posted: Mon Jul 10, 2006 6:52 pm Post subject:
Re: Tensor Decomposition



xyz91234@yahoo.com wrote:
Quote:  How can one generalize matrix decompositions to tensors? How can one
find the LU decomposition, singular value decomposition, jordan
elimination, of a tensor? How can one find the determinant, rank,
characteristic polynomial, etc. on a tensor.

I have not heard of any extension of LU decomposition,
singular value decomposition or Jordan decomposition of
"higher" tensors. I do not even have an idea of what those
terms would mean in general.
There *are*, though, extensions of the notions of determinant,
rank, and characteristic polynomial to higher tensors. The
subject, in fact, goes all the way back to Cayley.
You will find a discussion of higher determinants in the extraodinary
book "Discriminants, Resultants and Multidimensional Determinants"
(Birkhäuser, 1993), by IM Gelfand, MM Kapranov, AV Zelevinski.
Rank functions are also discussed there, IIRC.
There have been recently quite some work about a notion of
rank for tensors in the area of algebraic statistics, algebraic
philogenetic invartiants et al. The idea is very straightforward:
if V is a vector space, and t is an element of some tensor
power V x V x ... x V (here "x" means tensor power), then one
can define the rank of t to be the least number of summands in
any expression for t as a sum of elementary tensors. This
coincides with the usual notion for matrices (seen as tensors
appropriately). I've asked some people who work on this, and
it appears the connection with Gelfand et al.'s hyperdeterminants
is not worked out at all yet.
HTH
 m 

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