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Can someone please explain to me what is the tensor product and any good elementary tensor algerbra books?

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HallsofIvy

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Thanks alot hall, its starting to make a bit of sense now! Could you also try and explain to me what wedge is?

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Daniel.

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http://en.wikipedia.org/wiki/Wedge_product

Daniel.

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The cross product of two vectors is actually a wedge product, although a very lucky one.

[tex](\bold{u}\times\bold{v})_i = \epsilon_{ijk}u_jv_k[/tex]

So the vector product can be written with the Levi-Civita symbol, which takes care of the anti-symmetry.

But we can write this as

[tex]\epsilon_{ijk}u_jv_k = \frac{1}{2}\epsilon_{ijk}(u_jv_k - u_kv_j)[/tex]

[tex] = \epsilon_{ijk}(u\wedge v)_{jk}[/tex]

[tex] = \ast(u\wedge v)_i[/tex]

So the vector product is related to the wedge product by the Hodge star operator.

Note: The last equality arises due to the fact that the Hodge star of the identity element in 3 dimensions is:

[tex](\ast 1)_{ijk} = \epsilon_{ijk}[/tex]

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[tex]r' \circ T = r[/tex]

Can someone explain to me why this is an important definition. What do they mean when they say "has the universal property that..."?

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Hurkyl

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When aWhat do they mean when they say "has the universal property that..."?

In this case, we're interested in bilinear maps:

VxW ----------------> R

Well, the tensor product (along with its canonical map)

VxW ---> V (x) W

is the best possible bilinear map from VxW. Because when we have any other bilinear map (such as the one written above), we can uniquely factor it as:

VxW ---> V (x) W ---> R

Lots of useful things can be described in terms of universal objects (or more interesting constructions). In fact, these descriptions are often very important, because they're the way in which you'd actually

For example, we are interested in all sorts of bilinear products on VxV. For example:

Any inner product: VxV --->

The cross product:

The outer product:

(M_(m, n) is the vector space of all m by n matrices)

Each of these can be described as a bilinear map on the tensor product. In fact, because of the universal property of the tensor product, it's often easier to define new products in terms of the tensor product! For example, the

v/\w := v(x)w - w(x)v

and its values live in the appropriate subspace of V(x)V.

(There is an equivalent definition in terms of a quotient space)

Actually, the wedge product is another example of a

VxV --------------> R

can be factored uniquely as

VxV ---> /\²(V) ---> R

In fact, we've seen this universal property at work in the other examples: Oxymoron showed how to define the cross product in terms of the wedge product!

Anyways, since one of the whole points of

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mathwonk

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I can't think of a book right now but ifSeReNiTy said:

[tex]/boldfacec[/b] = /boldface a O /boldface b [/tex]

Where O represents the tensor product symbol. It really should have an X through it but I can't find the Latex rules that use to be posted around here. Once I find it I'll finish this post.

Pete

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Lisa!

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I think you mean this thread:mathwonk said:

https://www.physicsforums.com/showthread.php?t=67268

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mathwonk

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A tensor product of X and Y is a "universal" bilinear map XxY-->T, i.e. one such that any other bilinear map XxY-->Z, has a unique factorization XxY-->T-->Z where T-->Z is linear.

a wedge product is way of linearizing alternating bilinear maps.

i.e. given X,Y you probably know what an alternating bilibear map XxY-->Z is.

then you tell me what a wedge product is:....

see how useful and trivial, category theory makes all definitions?

them construction on the other ahnd are another matter.

i.e. do these objects just defined actually exist?

old fashioned people refer to describing the constructions as telling what a tensor or wedge product is. new fangled persons prefer to give the properties of the objects rather tha the detailed constructions, in order to say what they "are".

this new fangled approach began over 100 years ago but some people are still not on board.

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[tex]

T= V \otimes W

[/tex]

where V and W are two vectors in R2. Then what do the eigen values and eigen vectors of T mean?

Thanks in advance.

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mathwonk

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i.e. if T is a transformation such that Tx = cx for some number c and non zero vector x, then x is an eigenvector and c an eigenvalue.

since your question implied you know how to make a transformation by tensoring two things, yiou should be able to answer your own question. if not you need to clarify what you mean by the setup given in your question. i.e how does your T act?

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T is transformation matrix which defines a coordinates sytem transformation from a global to local normalized space like so:

[tex]

T= V \otimes W

[/tex]

where V =x

and W = x

x

defines a length along global coordinate system.

For this transformation matrix T what do their eigen values and eigen vectors mean, do they mean normalized spacing and directions respectively? I hope I am clear.

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