91Ó°ÊÓ

Consider the matrix $M=\left[\begin{array}{cc}2& 1\\ 2& 1\\ 1& 5\end{array}\right]$where${\stackrel{→}{v}}_1=\left[\begin{array}{c}2\\ 2\\ 1\end{array}\right]$and${\stackrel{→}{v}}_2=\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]$

By the theorem of QR method, the value of${\stackrel{→}{u}}_1$and$r_{11}$is defined as follows:

$$\begin{gathered} r_{11}=\left|\left|\stackrel{→}{v_1}\right|\right| \\ {\stackrel{→}{u}}_1=\frac{1}{r_{11}}{\stackrel{→}{v}}_1 \end{gathered}$$.

Simplify the equation $r_{11}=\left|\right|{\stackrel{→}{v}}_1\left|\right|$as follows:

$$\begin{gathered} r_{11}=\left|\right|{\stackrel{→}{v}}_1\left|\right| \\ {r}_{11}=\left|\left|\left[\begin{array}{c}2\\ 2\\ 1\end{array}\right]\right|\right| \\ {r}_{11}=\sqrt{2^2+2^2+1^2} \\ {r}_{11}=3 \end{gathered}$$

Substitute the values 7 for $r_{11}$and $\left[\begin{array}{c}2\\ 2\\ 1\end{array}\right]$for ${\stackrel{→}{v}}_1$in the equation ${\stackrel{→}{u}}_1=\frac{1}{r_{11}}{\stackrel{→}{v}}_1$as follows:

$$\begin{gathered} {\stackrel{→}{u}}_1=\frac{1}{r_{11}}{\stackrel{→}{v}}_1 \\ {\stackrel{→}{u}}_1=\frac{1}{3}\left[\begin{array}{c}2\\ 2\\ 1\end{array}\right] \\ {\stackrel{→}{u}}_1=\left[\begin{array}{c}2/3\\ 2/3\\ 1/3\end{array}\right] \end{gathered}$$

Therefore, the values localid="1659437134701" ${\stackrel{→}{u}}_1=\left[\begin{array}{c}2/3\\ 2/3\\ 1/3\end{array}\right]\mathrm{and}{\mathrm{r}}_{11}=3$.

As $r_{12}=\stackrel{→}{u_1}·\stackrel{→}{v_2}$, substitute the values $\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]$for $\stackrel{→}{V_2}$and ${\left[\begin{array}{ccc}\frac{2}{3}& \frac{2}{3}& \frac{1}{3}\end{array}\right]}^T$for $\stackrel{→}{u_1}$in the equation

localid="1659437594586" $r_{11}=\stackrel{→}{u_1}.\stackrel{→}{V_2}$as follows:

$r_{11}=\stackrel{→}{u_1}.\stackrel{→}{V_2}$

$r_{12}={\left[\begin{array}{ccc}\frac{2}{3}& \frac{2}{3}& \frac{1}{3}\end{array}\right]}^T.\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]$

$$\begin{gathered} r_{12}=\frac{2}{3}+\frac{2}{3}+\frac{5}{3} \\ {r}_{12}=3 \end{gathered}$$

Substitute the values $\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]$for$\stackrel{→}{V_2}$,3 for $r_{12}$and $\left[\begin{array}{c}2/3\\ 2/3\\ 1/3\end{array}\right]$for $\stackrel{→}{u_1}$in the equation ${\stackrel{→}{v}}_2^{âŠ\yen }=\stackrel{→}{V_2}-r_{12}{\stackrel{→}{u}}_1$as follows:

${\stackrel{→}{v}}_2^{âŠ\yen }=\stackrel{→}{V_2}-r_{12}{\stackrel{→}{u}}_1$

$$\begin{gathered} {\stackrel{→}{v}}_2^{âŠ\yen }=\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]-3\left[\begin{array}{c}2/2\\ 2/3\\ 1/3\end{array}\right] \\ {\stackrel{→}{v}}_2^{âŠ\yen }=\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]-\left[\begin{array}{c}2\\ 2\\ 1\end{array}\right] \\ {\stackrel{→}{v}}_2^{âŠ\yen }=\left[\begin{array}{c}-1\\ -1\\ 4\end{array}\right] \\ \mathrm{Therefore},{\stackrel{→}{v}}_2^{âŠ\yen }=\left[\begin{array}{c}-1\\ -1\\ 4\end{array}\right]\mathrm{and}{r}_{12}=3 \end{gathered}$$

The value of $\stackrel{→}{u_2}$and $r_{22}$is defined as follows.

$$\begin{gathered} \begin{array}{ll}{r}_{22}=\left|\left|\stackrel{→}{V_2^{âŠ\yen }}\right|\right|& \end{array} \\ {\stackrel{→}{u}}_2=\frac{1}{r_{22}}\stackrel{→}{V_2^{âŠ\yen }} \end{gathered}$$

Simplify the equation localid="1659438712674" $\begin{array}{ll}{r}_{22}=\left|\left|\stackrel{→}{V_2^{âŠ\yen }}\right|\right|& \end{array}$as follows.

$r_{22}=\left|\left|\stackrel{→}{V_2^{âŠ\yen }}\right|\right|$

$r_{22}=\left|\left|\left[\begin{array}{c}-1\\ -1\\ 4\end{array}\right]\right|\right|$

$$\begin{gathered} r_{22}=\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2+{\left(4\right)}^2} \\ {r}_{22}=3\sqrt{2} \end{gathered}$$

Substitute the values $3\sqrt{2}$for $r_{22}$and $\left[\begin{array}{c}-1\\ -1\\ 4\end{array}\right]$for $\stackrel{→}{V_2^{âŠ\yen }}$in the equation $\stackrel{→}{u_2}=\frac{1}{r_{22}}\stackrel{→}{V_2^{âŠ\yen }}$as follows.

$$\begin{gathered} {\stackrel{→}{u}}_2=\frac{1}{r_{22}}{\stackrel{→}{v}}_2^{âŠ\yen } \\ {\stackrel{→}{u}}_2=\frac{1}{3\sqrt{2}}\left[\begin{array}{c}-1\\ -1\\ 4\end{array}\right] \\ {\stackrel{→}{u}}_2=\left[\begin{array}{c}-1/3\sqrt{2}\\ -1/3\sqrt{2}\\ 4/3\sqrt{2}\end{array}\right] \end{gathered}$$

The values localid="1659439047262" ${\stackrel{→}{u}}_2=\left[\begin{array}{c}-1/3\sqrt{2}\\ -1/3\sqrt{2}\\ 4/3\sqrt{2}\end{array}\right]$and $r_{22}=3\sqrt{2}$

Therefore, the matrices $Q=\left[\begin{array}{cc}2/3& -1/3\sqrt{2}\\ 2/3& -1/3\sqrt{2}\\ 1/3& 4/3\sqrt{2}\end{array}\right]$and $R=\left[\begin{array}{cc}3& 3\\ 0& 3\sqrt{2}\end{array}\right]$.

Hence, the QR factorization of the matrix is localid="1659439350112" $\left[\begin{array}{cc}2& 1\\ 2& 1\\ 1& 5\end{array}\right]=\frac{1}{3}\left[\begin{array}{cc}2& -1/\sqrt{2}\\ 2& -1/\sqrt{2}\\ 1& 4/\sqrt{2}\end{array}\right]\left[\begin{array}{cc}3& 3\\ 0& 3\sqrt{2}\end{array}\right]$.