# A vector times its transpose gives a matrix???

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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!

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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

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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 :(

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Quote:
 Original post by gOOzeI 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.

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ok got it, thanks!

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