Math Is Fun Forum

Onyx

Member

Registered: 2009-02-24

Posts: 48

Difference between dyad and matrix?

Hi, this is a quick question about tensors. From what I've read a simple way to interpert dyads (2nd order tensors) is this: Suppose you have a vector H. By muliplying this vector by a scalar, a, we get a new vector V whose magnitude changes but whose direction remains the same:

If instead of the scalar, we use an order two tensor (a dyad),

we get a new vector U whose magnitude and direction is changed:

There are some special matrices that do similar things, for example for

the rotation matrix is given by:

which when muliplied by H gives a new vector equal in magnitude but rotated by

. I'm sure similar matrices can be made which also change the magnitude.

What I'm wondering is if purely abstractly dyads are simply matrices (2 dimensional arrays of numbers) and nth order tensors are simply n dimensional arrays of numbers. To me it seems like they're exactly the same thing, and while they're classified differently, I'm not sure if this is simply because of the specific application of tensors to vector transfomations, or whether there is some other more important difference I'm missing.

Can someone clear this up?

Thanks

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Difference between dyad and matrix?

It seems like you're going off a bad definition of tensor.  A tensor is a real-valued map (that it, the range is the real numbers) that is multi-linear.  Thus, if you have a two tensor T on a real vector space V:

T(u+v, w) = T(u,w) + T(v,w)
T(u, v+w) = T(u,v) + T(u,w)
T(k*u, v) = kT(u,v)
T(u, k*v) = kT(u,v)

Where u, v, w are in V.  In other words, each component acts as a linear transformation.  Indeed, 1-tensors are exactly linear transformations into the reals.

So all n x n matrices for n > 1 are not tensors.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Onyx

Member

Registered: 2009-02-24

Posts: 48

Re: Difference between dyad and matrix?

Thanks, that gives me a better idea.