- #1
Dustinsfl
- 2,281
- 5
Factor the matrix into the form QR where Q is orthogonal and R is upper triangular.
[tex]\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}*\begin{bmatrix}
e & f\\
0 & g
\end{bmatrix}=\begin{bmatrix}
-1 & 3\\
1 & 5
\end{bmatrix}[/tex]
[tex]\begin{bmatrix}
a & c
\end{bmatrix}*\begin{bmatrix}
b\\
d
\end{bmatrix}=0[/tex]
[tex]ae=-1[/tex]
[tex]af+bg=3[/tex]
[tex]ce=1[/tex]
[tex]cf+dg=5[/tex]
Skipping some steps but I arrive at:[tex]\begin{bmatrix}
1 & \frac{4}{g}\\
-1 & \frac{4}{g}
\end{bmatrix}*\begin{bmatrix}
-1 & -1\\
0 & g
\end{bmatrix}=\begin{bmatrix}
-1 & 3\\
1 & 5
\end{bmatrix}[/tex]
So as long as [tex]g \neq 0[/tex] it is all good?
[tex]\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}*\begin{bmatrix}
e & f\\
0 & g
\end{bmatrix}=\begin{bmatrix}
-1 & 3\\
1 & 5
\end{bmatrix}[/tex]
[tex]\begin{bmatrix}
a & c
\end{bmatrix}*\begin{bmatrix}
b\\
d
\end{bmatrix}=0[/tex]
[tex]ae=-1[/tex]
[tex]af+bg=3[/tex]
[tex]ce=1[/tex]
[tex]cf+dg=5[/tex]
Skipping some steps but I arrive at:[tex]\begin{bmatrix}
1 & \frac{4}{g}\\
-1 & \frac{4}{g}
\end{bmatrix}*\begin{bmatrix}
-1 & -1\\
0 & g
\end{bmatrix}=\begin{bmatrix}
-1 & 3\\
1 & 5
\end{bmatrix}[/tex]
So as long as [tex]g \neq 0[/tex] it is all good?