Minimize the various norms given a set of linear equations.

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 $x$? A is $3 \times 2$. B is $3 \times 1$.

Answer 1

Given the problem $Ax = b$, let $A^\dagger$ be the Moore-Penrose pseudo-inverse of $A$. Setting $x$ as the vector $$ z = A^\dagger b $$ Will minimize both $\|Ax - b\|_2$ and $\|x\|_2$.