Re: Matrix notation (was Re: The Hodge dual...)

grubb_at_math.niu.edu
Date: 11/30/04 

Date: Tue, 30 Nov 2004 18:47:28 +0000 (UTC)

Igor Khavkine <k_igor_k@lycos.com> wrote in message news:<pan.2004.11.27.16.46.28.612973@lycos.com>...

>
> I wonder if there are similar tricks that can be played with the tensor
> product of vector spaces. Can the tensor product be represented as a
> coproduct in another category? If so, is there an analogous "block
> decomposition" of maps between tensor products of spaces in diagramatic
> notation?

Yes, there is, at least for the block description. But it's a bit trickier.

First let's do the tensor product of two vectors, say
[a_1]
[a_2]
[...]
[a_m]

and

[b_1]
[b_2]
[...]
[b_n]

Notice that these vectors are not of the same size. The tensor
product is then the vector
[a_1 b_1]
[a_1 b_2]
[...]
[a_1 b_n]
[a_2 b_1]
[...]
[a_2 b_n]
[...]
[...]
[a_m b_1]
[...]
[a_m b_n].

So the dimension of the tensor product is obtained by multiplying
the dimensions of the individual vectors. The components of the
tensor product are obtained by multiplying the components of the
individual vectors.

If one wants the tensor product of two row vectors, you get
a similar row vector. But if you want to take the tensor product
of the row vector [a_1 a_2 ...a_m] and the column
vector
[b_1]
[...]
[b_n]

you get the matrix
[a_1 b_1 a_2 b_1 .... a_m b_1]
[a_1 b_2 a_2 b_2 .... a_m b_2]
[.................................]
[a_1 b_n a_2 b_n .... a_m b_n].

As for the tensor product of two linear transformations, let's
stick to column vectors and suppose we have an m by n matrix

[a_11 ... a_1n]
[...............]
[a_m1 ... a_mn]

and a p by q matrix

[b_11 ... b_1q]
[..............]
[b_p1 ... b_pq]

and we want to take the tensor rpoduct of these two
matrices. The result will be a mp by nq matrix whose entries
are products of the entries of the original matrices:

[a_11 b_11 a_11 b_12 ...a_11 b_1q a_12 b_11 ...a_12 b_1q ...a_1n b_1q]
[a_11 b_21 .....................................................a_1n b_2q]
[........................................................................]
[a_11 b_p1 .................................................... a_1n b_pq]
[...]
[...]
[a_m1 b_11 a_m1 b_12..........................................a_m1 b_1q]
[...]
[a_mn b_p1 a_mn b_p2 .........................................a_mn b_pq]

Using this, it is also easy to see several of the relationships between
tensor products and direct sums, but I think I'll pass on that for right now :).

--Dan Grubb

Relevant Pages

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