Linear Algebra with Applications, 10th Edition

Section 7.7 Exercises

1. Find the solution x to the least squares problem, given that $A=\mathit{\text{QR}}$ in each of the following:

   1. $\begin{array}{r}Q=\left[\begin{array}{cc}\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\\ \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\\ 0& \hfill 0\end{array}\right],\hfill \\ \begin{array}{ll}\begin{array}{c}R=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\end{array}$

   2. $\begin{array}{r}Q=\left[\begin{array}{ccc}1& 0& \hfill 0\\ 0& \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\\ 0& \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\\ 0& 0& \hfill 0\end{array}\right],\hfill \\ \begin{array}{ll}\begin{array}{c}R=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{c}1\\ 3\\ 1\\ 2\end{array}\right]\end{array}$

   3. $\begin{array}{r}Q=\left[\begin{array}{ccc}1& 0& \hfill 0\\ 0& \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\\ 0& \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}\end{array}\right],\hfill \\ \begin{array}{ll}\begin{array}{c}R=\left[\begin{array}{cc}1& 1\\ 0& 1\\ 0& 0\end{array}\right],\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{r}\begin{array}{c}1\\ \sqrt{2}\end{array}\\ -\sqrt{2}\end{array}\right]\end{array}$

   4. $\begin{array}{r}Q=\left[\begin{array}{cccc}\frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}& \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& 0& \hfill \frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}\\ \frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}& 0& \hfill \frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill -\frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}\\ \frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}& 0& -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& \hfill -\frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}\\ \frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}& -\frac{{\displaystyle 1}}{{\displaystyle \sqrt{2}}}& 0& \hfill \frac{{\displaystyle \begin{array}{c}1\end{array}}}{{\displaystyle 2}}\end{array}\right],\hfill \\ \begin{array}{ll}\begin{array}{c}R=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{r}2\\ -2\\ 0\\ 2\end{array}\right]\end{array}$

2. Let

   $$A=\left[\begin{array}{c}D\\ E\end{array}\right]-\left[\frac{\begin{array}{c}{d}_1\\ d_1\\ \ddots \\ d_n\end{array}}{\begin{array}{c}{e}_1\\ e_1\\ \ddots \\ e_n\end{array}}\right]$$

   and

   $$\mathbf{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_{2n}\end{array}\right]$$

   Use the normal equations to find the solution x to the least squares problem.

3. Let

   $$\begin{array}{cc}\begin{array}{c}A=\left[\begin{array}{cc}1& 0\\ 1& 3\\ 1& 3\\ 1& 0\end{array}\right],\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{r}-4\\ 2\\ 2\\ 2\end{array}\right]$$

   1. Use Householder transformations to reduce A to the form

      $$\left[\begin{array}{c}{R}_1\\ O\end{array}\right]=\left[\begin{array}{cc}\times & \times \\ 0& \times \\ 0& 0\\ 0& 0\end{array}\right]$$

      and apply the same transformations to b.

   2. Use the results from part (a) to find the least squares solution of $A\mathbf{x}=\mathbf{b}$.

4. Given

   $$\begin{array}{cc}\begin{array}{cc}A=\left[\begin{array}{cc}1& \hfill 5\\ 1& \hfill 3\\ 1& \hfill 11\\ 1& \hfill 5\end{array}\right]& \text{and}\end{array}& \mathbf{b}\end{array}=\left[\begin{array}{r}1\\ -1\\ 3\\ 5\end{array}\right]$$

   1. Use Algorithm 5.6.1 to compute the factors Q and R of the modified ...