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Symmetric vector to tensor representation?

  1. Apr 9, 2012 #1
    Hi all!

    I have a discrete 2D vector field with a particular characteristic: At every point, instead of having a single vector, I have two vectors which are in the opposite direction. For example, at point p(x,y)=p(0,0) I have two vectors: v1(1,1) and v2(-1,-1). And so on for all points.

    I understand this becomes an "eigenvector field" situation, or a "tensor field", or "symmetric tensor field" situation.

    At the end, I wish to find the so called "degenerate points" (refer to Delmarcelle, Hasselink 1993).

    But first, I wish to translate these two vectors on a single point to a 2x2 tensor representation, such that T(point)=[T11(x,y) T12(x,y) ; T12(x,y) T22(x,y)].

    Summarizing,

    Given, two symmetric vectors at a point, v1(point)=[x1;y1] and v2(point)=[x2;y2], such that v2(point)=(-1)*v1(point) represent them in tensor form T(point)=[T11(x,y) T12(x,y) ; T12(x,y) T22(x,y)].

    Much appreciated!!!
     
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 10, 2012 #2
    Just realized that I can use the dyadic product of two vectors to generate my tensor.

    Thus, v1(point)=[1; 2]; v2(point)=[-1;-2] can give T(point)=[(1)(-1) (1)(-2); (2)(-1) (2)(-2)]
    T(point)=[-1 -2;-2 -4].

    Thus I can have the same eigenvectors if [dyadic product(v1,v2)] or [dyadic product(v2,v1)] since they give me the same tensor.

    If anyone is interested.
     
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