# Lagrange's Method

• January 24th 2010, 04:37 AM
kinkojun
Lagrange's Method
http://i277.photobucket.com/albums/k...8/DSC00004.jpg

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.
• January 24th 2010, 05:51 AM
HallsofIvy
Quote:

Originally Posted by kinkojun
http://i277.photobucket.com/albums/k...8/DSC00004.jpg

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 $2x^TA$.
• January 24th 2010, 06:29 AM
kinkojun
Quote:

Originally Posted by HallsofIvy
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.

http://i277.photobucket.com/albums/k...8/DSC00007.jpg
http://i277.photobucket.com/albums/k...8/DSC00006.jpg
http://i277.photobucket.com/albums/k...DSC00009-1.jpg

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. (Wink)