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Math Help - Lagrange's Method

  1. #1
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    Lagrange's Method



    Please help me with this question.

    Does anyone know how it ends up with 2 lambda*x transpose*A equal to zero?? I don't understand how the number 2 came up!! Thanks for any help.
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  2. #2
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    Quote Originally Posted by kinkojun View Post


    Please help me with this question.

    Does anyone know how it ends up with 2 lambda*x transpose*A equal to zero?? I don't understand how the number 2 came up!! Thanks for any help.
    Well, it doesn't "end up with 2 lambda*x transpose*a equal to zero"! It ends up with m plus that equal to 0.

    As to where the "2" came from, it is essentially from differentiating a square. Or you might prefer to think of it as using the product rule.

    Your Lagrangian is L= mx+ \lambda(x^TAx- 1). Differentiating x^T A x- 1, with respect to x, gives Ax+ x^TA and, since Ax= x^TA, that is the same as 2x^TA.
    Last edited by Jester; January 24th 2010 at 06:06 AM. Reason: fixed latex
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    Quote Originally Posted by HallsofIvy View Post
    Well, it doesn't "end up with 2 lambda*x transpose*a equal to zero"! It ends up with m plus that equal to 0.

    As to where the "2" came from, it is essentially from differentiating a square. Or you might prefer to think of it as using the product rule.

    Your Lagrangian is L= mx+ \lambda(x^TAx- 1). Differentiating x^T A x- 1, with respect to x, gives Ax+ x^TA and, since Ax= x^TA, that is the same as 2x^TA.
    I was actually doubting whether differentiating x^T is actually same as differentiating x. It seems that they are the same in term of differentiation even though it is transposed. Thank you.

    As a further question in this particular case, I have uploaded an example from my text book which i could not understand the last part of it.





    1st pic is connected with the 2nd picture and follow up by the 3rd picture. In particular, I could not understand the whole part of the example in the 3rd pic after finish proving the equation in the 2nd pic. Could you please explain how it works in the 3rd pic. Thank you very much.
    Last edited by kinkojun; January 24th 2010 at 06:34 AM. Reason: Unclear Picture
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