Minimize norm - Least Squares - Linear Algebra

Question

Given $Ax = b$, I know how to use least squares to minimize $||Ax-b||^2$. How do I minimize the 2-norm ($||x||_2$) and the Frobenius norm of $x$?

Answer 1

You should explain more about the problem you have. The answer so far is that you can just take $x=0$ That way, the norms are $0$ which is a minimum.

The thing is that if you want to minimize $||Ax-b||$, that implies that a value of $x$ for which $Ax=b$ does not exist (otherwise, the $x$ that minimises $||Ax-b||$ is exactly the $x$ for which $Ax=b$). This usually occurs in the case when $Ax=b$ is a predetermined system (meaning $A$ has more rows than columns).

On the other hand, if you want to minimize $x$, I assume you want to find the minimum $x$ for which $Ax=b$. This can only happen if there exist many values of $x$ for which $Ax=b$ (and you look for the smallest one). The only way that can happen is if $A$ has more columns than rows.