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Math Help - Optimization Problem

  1. #1
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    Optimization Problem

    Firstly I dont know if this belongs in this forum but its my first post!

    Now down to bussiness:
    I have a 56x4 matrix and another 56x4 matrix which and i want to make a 4x4 transformation matrix.
    I can simply make the matrix by doing MatrixA/MatrixB
    However, I would like to try to optimize two variables:
    - the accuracy of the transformation matrix converting MatrixA to MatrixB
    and
    - minimize the Sum(TransformationMatrix.^2)

    How would I go about doing this?
    The only way i can think of doing this is by getting MatrixA/MatrixB and randomly varying it, testing the accuracy and the Sum(TransformationMatrix.^2)

    Is there a more elegant or better way to do this? I'm coding it so its fine if its iterative.
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  2. #2
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    i don't think there's enough information there to really understand your problem, but from what little i can gather, linear least squares may be relevant to your interests

    Linear least squares - Wikipedia, the free encyclopedia
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  3. #3
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    This is probably just a repeat of what you will find on the wiki page.

    Let A be your first 54x4 matrix, B the second and X be the 4x4 solution. I think you are trying to find a solution to:

    AX=B
    \therefore A^TAX=A^TB
    \therefore (A^TA)^{-1}(A^TA)X=(A^TA)^{-1}A^TB
    \therefore X=(A^TA)^{-1}A^TB

    You will need to look more closely at the wiki page to learn why this is a least squares solution.
    (A^TA)^{-1}A^T is known as a pseudo inverse of A (you may want to google pseudo inverse).
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