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Magazine Problem 1
Solve the normal equations to find the least squares solution and 2-norm error for the following inconsistent systems: (a) \(\left[\begin{array}{ll}1 & 2 \\ 0 & 1 \\ 2 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{l}3 \\ 1 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{ll}1 & 1 \\ 2 & 1 \\ 3 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]\) (c) \(\left[\begin{array}{ll}1 & 2 \\ 1 & 1 \\ 2 & 1 \\ 2 & 2\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{l}3 \\ 3 \\ 3 \\\ 2\end{array}\right]\)
Problem 2
Find the least squares solutions and RMSE of the following systems: (a) \(\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 0 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3}\end{array}\right]=\left[\begin{array}{l}2 \\ 2 \\ 3 \\\ 4\end{array}\right]\) (b) \(\left[\begin{array}{lll}1 & 0 & 1 \\ 1 & 0 & 2 \\\ 1 & 1 & 1 \\ 2 & 1 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3}\end{array}\right]=\left[\begin{array}{l}2 \\ 3 \\ 1 \\\ 2\end{array}\right]\)
Problem 3
Find the least squares solution of the inconsistent system $$ \left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \\ 1 & 0 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{l} 1 \\ 5 \\ 6 \end{array}\right] $$
Problem 3
Let \(A=\left[\begin{array}{ccc}1 & 0 & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1\end{array}\right]\). Prove that for any \(x_{0}\) and \(b\), GMRES converges to the exact solution after two steps.
Problem 4
Prove that the Gauss-Newton Method applied to the linear system \(A x=b\) converges in one step to the solution of the normal equations.
Problem 5
Prove that the 2-norm is a vector norm. You will need to use the Cauchy- Schwarz inequality \(|u \cdot v| \leq\|u\|_{2}\|v\|_{2}\).
Problem 5
Apply Householder reflectors to find the full QR factorization of the matrices in Exercise \(1 .\)
Problem 9
Prove that a square matrix is orthogonal if and only if its columns are pairwise orthogonal unit vectors.
