Visualising a tensor product

lethe

05-31-03, 06:32 PM

Originally posted by EI_Sparks
Hmm. Fast question (I hope!) :

Take a curved surface S in R3, a point x in S and a tangent plane to S at point x, TxS. Now here's the question: How do you visualise the tensor product of TxS with itself?

(And how the heck do you write that out in vBulletin when [sup] isn't a valid tag? :confused: )

well, a tensor product is nothing more than an ordered pair of vectors. one way you might try to visualise it is by imagining parallelograms in the tangent plane. every parallelogram is represented by 2 vectors. there are some inaccuracies with this picture however, so i don t want to stress it too much.

a better picture would be to construct a 4 dimensional "double tangent space" so to speak by taking two copies of the tangent space and putting them together perpendicular. it s hard to imagine a 4 dimensional tangent space. it certainly won t fit in R<sup>3</sup>, so instead imagine that your tangent space is 1-dimensional (like if instead of a surface, it is the tangent space to a curve). then take two copies of this tangent line, and make one of them the x-axis, and one the y-axis, and the resulting plane is the space of tensor products of the two tangent spaces.

you can do the same thing with your tangent planes, take two copies of the tangent plane and arrange them so that all the vectors in one are perpendicular to all the vectors in the other. you need 4 dimensions to be able to do this, but once you have done it, you ve got the tensor product of the tangent spaces.

there are a few other ways to think about tensor products. especially when it comes to tensor products of operators. you nkow what a linear operator is, right? it takes a vector from a vector space, and gives you a new one. suppose you have two vector spaces, and two linear operators, one for each vector space. the tensor product of the linear operators is the unique linear operator that acts on the tensor product of the vector spaces formed from the two starting linear operators.

one way to think of that is: if your linear operators are two matrices, then to get the tensor product you multiply the two matrices, but not in the usual matrix way. instead you take one matrix and multiply it into every element of the second matrix, yielding a matrix of matrices. that is a matrix with a matrix as entries.

to make subscripts and superscripts, use the < sup > and < sub > html tags, instead of the vBulletin codes.