Math 6420 • PDE

Let \(A\in\mathbb{R}^{n\times m}\), \(B\in\mathbb{R}^{r\times m}\), and
\(b\in\mathbb{R}^n\) be given. Assume \(R(B^*)=N(A)\). For fixed \(\mu\ge 0\), consider
the problem
\[
  \min_{x\in\mathbb{R}^m} \|Ax-b\|_2^2+\mu\|Bx\|_2^2.
\]
\begin{enumerate}
\item[(a)] Find the necessary conditions (analogous to the normal equations for the standard
least-squares problem) and prove that the necessary conditions have a solution when \(\mu=0\).
\item[(b)] Prove that the necessary conditions have a unique solution if \(\mu>0\).
\end{enumerate}