Math 6420 • PDE

Let \(A\) be a real \(m\times n\) matrix, and let \(b\in\mathbb{R}^m\) be a column vector.
Consider the problem
\[
  A^*Ax=A^*b.
  \tag{*}
\]
\begin{enumerate}
\item[(a)] Explain the relationship between \((*)\) and the problem
\[
  \min_{x\in\mathbb{R}^n}\lVert Ax-b\rVert^2.
\]

\item[(b)] Under what conditions on \(A\), \(A^*\), and \(b\) is it guaranteed that \((*)\) has a
solution \(x\)? Under what conditions is it guaranteed that \((*)\) has a unique solution?

\item[(c)] Given \(\varepsilon>0\), consider the problem
\[
  (A^*A+\varepsilon I)x_\varepsilon=A^*b.
  \tag{**}
\]
Formulate an associated minimization problem, describe existence and uniqueness, and explain
how \(\lim_{\varepsilon\to0+}x_\varepsilon\) is related to solutions of \((*)\).
\end{enumerate}