Math 6610 • Analysis of Numerical Methods I

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