Find the Inverse of a Vector: Solve for x in Terms of b

[OhMyMarkov]

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]

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]

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.