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Inverse of a vector

OhMyMarkov

Member
Mar 5, 2012
83
Hello Everyone!

I've been trying to solve an equation and got to this place: $\sum _{j=1} ^K (x_j - b_j) = 0$ which gives $e^T x = e^T b$. Now I need to solve for $x$ i.e. find $x$ in terms of the b. How can I do that?

Thank you for the help!
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Hello Everyone!

I've been trying to solve an equation and got to this place: $\sum _{j=1} ^K (x_j - b_j) = 0$ which gives $e^T x = e^T b$. Now I need to solve for $x$ i.e. find $x$ in terms of the b. How can I do that?

Thank you for the help!
Can we have some more context, or perhaps the original problem?

CB
 

OhMyMarkov

Member
Mar 5, 2012
83
Definitely,

The original problem is to find the minimum of the following function: $T(x) = ||y-Ax||^2 + ||x-b||^2$ where the system y = Ax is overdetermined. Of course we already know the closed form solution of the first norm (linear least squares). My question was regarding the second norm. What I want to do is to try to find a closed form solution for the minimum of the second norm, and add the two solutions up. What I did was, similar to the approach of finding the linear least squares solution, differentiate the second norm with respect to $x_j$s and set to zero. I got what is there in my first post, and now I'm stuck there.

Thank you.