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Math Help - Matricial equation problem

  1. #1
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    Matricial equation problem

    Well I hope I'm posting this in the right place.

    I have a problem that is written in the following form:
    (Ax - y)' * (Ax - y) where the symbol ( ' ) denotes the transpose of a matrix/vector.

    I have to arrange this equation and put in a form where I have:

    x'*Q*x

    The first statement is from a minimization problem in quadratic programming, so that's why I have to put it in the form of the second statement, so my problem here is really rearranging the first statement into the second statement.

    Thanks for your attention
    My regards
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  2. #2
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    Re: Matricial equation problem

    I don't know if there is a general rule for simplifying the original expression: ([A]x - y)T * ([A]x - y)

    However, if it is supposed to be capable of being rearranged into the form for Quadratic Programming, the [Q] matrix is symmetric. Not sure if that helps to work backward.
    Do you know ahead of time if the [A] matrix has any special properties?
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  3. #3
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    Re: Matricial equation problem

    Quote Originally Posted by DavidB View Post
    I don't know if there is a general rule for simplifying the original expression: ([A]x - y)T * ([A]x - y)

    However, if it is supposed to be capable of being rearranged into the form for Quadratic Programming, the [Q] matrix is symmetric. Not sure if that helps to work backward.
    Do you know ahead of time if the [A] matrix has any special properties?
    Thank you for your reply

    I know [A] from my problem. It is nonsquare and has generally more columns than rows.

    I am using the MATLAB routine quadprog to solve my quadratic problem, so that's why I need to know the matrix [Q] which is symmetric as you well said.
    Generally I have a much simple problem to solve which is:

    minimize 1/2*norm(x)^2 (I'm referring to the norm p=2 here).

    In this case I just have to: norm(x)^2 = x'*x, and so the [Q] matrix for this problem is the identity matrix: x'*[I]*x

    The problem I posted yesterday is a special case of the problem I just described. In this case I have the problem:
    min 1/2*norm(A*x-y)^2. Then I tried to follow the same strategy as before:

    norm(A*x-y)^2 = (Ax - x)' * (Ax - y)...

    Maybe there is a simpler strategy to solve this minimization problem...
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