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Math Help - Examining minimizing and solutions

  1. #1
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    Examining minimizing and solutions

    I am supposed to work out to show that it has a solution where x minimizes . I guess my question is where to even begin. When I do the block matrix multiplication, I get:

    Which tells me r + Ax = b and A*r = 0... but I don't know how to complete it or bring in the minimization?
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  2. #2
    Senior Member Shanks's Avatar
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    on one hand, r + Ax = b and A*r = 0, that is, A*b-A*Ax=0 or
    A*Ax-A*b=0.
    on the other hand,we have
    the square of = (Ax-b)*(Ax-b)=(x*A*-b*)(Ax-b)=x*A*Ax-2b*Ax+b*b
    taking the derivative, x minimizes if and only if
    2A*Ax-2A*b=0 that is A*Ax-A*b=0.
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