Math 6610 • Analysis of Numerical Methods I

{Least Squares Problems}

\partitem{a} For a full rank real \(m\times n\) matrix \(A\), show that
\(X=A^\dagger\), the pseudoinverse of \(A\), minimizes \(\|AX-I\|_F\) over all
\(n\times m\) matrices \(X\). What is the value of the minimum? (Hint: Relate the
problem to a set of least-squares problems).

\partitem{b} For a real full rank \(m\times n\) matrix \(A\) and vector
\(b\in\R^m\), explain how to solve the least-squares problem of finding
\(x\in\R^n\) that minimizes \(\|Ax-b\|_2\) using \textbf{i)} the normal equations,
and \textbf{b)} a QR factorization of the matrix \(A\). What are the advantages
and disadvantages of each of these methods?

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