• Advertisement
Sign in to follow this  

A vector times its transpose gives a matrix???

This topic is 2741 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Recommended Posts

Hi,

As I was working on this paper:
http://hal.inria.fr/docs/00/09/96/24/PDF/acdld-apr-03.pdf

and implementing the 3D curvature tensor estimator given in part 2.1 (titles "Robust 3D curvature tensor field"), I got stuck on the last term of the formula (e times its transpose where e is a unit vector). Since the result T(v) of this formula is a matrix and all the other terms are scalars, this vector multiplication has to give me a matrix.

A row vector multiplied by its transpose (column) gives a scalar if I'm right (dot product).

I guessed that e times its transpose (e*e') in my case is not the usual dot product but the multiplication of a column vector by a row vector. Assuming that e=(x,y,z)'

e*e' would give me the matrix:
[x*x x*y x*z]
[y*x y*y y*z]
[z*x z*y z*z]

I'd like to know if that seems correct to you and I would like to understand the geometric meaning of this matrix. If I understand what this matrix actually represents in my 3D space, I'd understand what the last term does in my tensor's formula (I have estimated normal and curvature vector sign problems, some of them are flipped, that could be the answer)

Thanks a lot!

Share this post


Link to post
Share on other sites
Advertisement
AxB * BxC = AxC matrix.

If v is a column vector, that is Nx1:

v' * v = 1xN * Nx1 = 1x1 // inner product (dot product)
v * v' = Nx1 * 1xN = NxN // outer product

Share this post


Link to post
Share on other sites
Outer product, so that has a name. Thanks! :)

I found many abstract definitions of it, but that's still pretty hard to figure out what that actually represents in 3D space. Anyway, my flipped curvature vector problem (result of my eigenvectors not having the correct signs) seems to come from something else :(

Share this post


Link to post
Share on other sites
Quote:
Original post by gOOze
I found many abstract definitions of it, but that's still pretty hard to figure out what that actually represents in 3D space.(


Think of what the matrix (e e') does when it operates on some other vector v. By associativity,

(e e') v = e (e' v) .

e' v is just the usual inner product; it's a scalar.

Does this look familiar? When e is a unit vector, this is just the projection of v onto e.

Hence, when e is a unit vector, the matrix e e' is the "take a vector and project it onto e" operator.

Share this post


Link to post
Share on other sites
Sign in to follow this  

  • Advertisement