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Reducing to a matrix

  1. Feb 9, 2010 #1
    I'm trying to find a way of reducing a rank-3 tensor field into a matrix, but am having trouble finding a good way to do it. The situation set up is as follows:

    Let's say that I have a [itex] 3\times 3[/itex] matrix and a [itex] 5 \times 1 [/itex] vector as follows

    [tex] A(x) = \begin{pmatrix} a_{11}(x) & a_{12}(x) & a_{13}(x) \\ a_{21}(x) & a_{22}(x) & a_{23}(x) \\ a_{31}(x) & a_{32}(x) & a_{33}(x) \end{pmatrix}, \qquad \qquad x = \begin{pmatrix} x_1 \\ x_2\\ x_3 \\ x_4 \\ x_5 \end{pmatrix} [/tex]

    Now I want to find a tangent plane to a surface defined by A(x) at some point [itex] \bar x [/itex] to create some constraints, namely, I want to do something of the form

    [tex] \nabla A(x) (x - \bar x) = 0 [/tex]

    Now the problem here is that [itex] \nabla A(x) [/itex] is a rank 3 tensor. I need to find some what of ''matricizing'' this tensor into a matrix so that I can solve a linear programming problem with it. Any ideas?
     
  2. jcsd
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