# A = QR Decomposition

1. Jul 27, 2012

### g.lemaitre

1. The problem statement, all variables and given/known data
Decompose the following matrix using QR decomposition
\begin{bmatrix}
4 & 1 \\
3 & -1
\end{bmatrix}
\begin{bmatrix}
.8 & .6 \\
.6 & .8
\end{bmatrix}
The following matrix is supposed to be next to the previous but I can't figure out how to do that. Any help in that area would be appreciated.
\begin{bmatrix}
5 & .2 \\
0 & 1.4
\end{bmatrix}
2. Relevant equations
$$c_2 = (v_2 * u_1)q_1 + \parallel w_2 \parallel q_2$$
3. The attempt at a solution
I was able to get the first part of the answer
\begin{bmatrix}
.8 & \\
.6 &
\end{bmatrix}
It's the second part
\begin{bmatrix}
.6 & \\
-.8 &
\end{bmatrix}
that i'm having trouble with. I'm also not worried about the R part right now.
Ok, let's plug the numbers into this equation:
$$c_2 = (v_2 * u_1)q_1 + \parallel w_2 \parallel q_2$$
$$v_2 * u_1 = .2$$
$$c_2 = \begin{bmatrix} 1 & \\ -1 & \end{bmatrix}$$
$$\parallel w_2 \parallel = -49/25$$
Therefore,
$$\begin{bmatrix} 1 & \\ -1 & \end{bmatrix} = .2 \begin{bmatrix} .8 & \\ .6 & \end{bmatrix} - 49/25q_2$$
step two
$$\begin{bmatrix} 1 & \\ -1 & \end{bmatrix} = \begin{bmatrix} 4/25 & \\ 3/25 & \end{bmatrix} - 49/25q_2$$
step three
$$\begin{bmatrix} (25-4)/25 & \\ (-25-3)/25 & \end{bmatrix} = 49/25q_2$$
step four
$$\begin{bmatrix} 21/49 & \\ -28/49 & \end{bmatrix} = q_2$$
The answer is supposed to be
$$\begin{bmatrix} .6 & \\ -.8 & \end{bmatrix} = q_2$$
So I made an error somewhere.

Last edited: Jul 27, 2012
2. Jul 27, 2012

### fzero

It's tough to follow because you haven't defined every symbol, but $q_2$ should have unit norm. If you properly normalize your result, you'll find that it agrees with the solution.

3. Jul 27, 2012

### g.lemaitre

what do you mean by properly normalize.

4. Jul 28, 2012

### who_

To normalize a vector q is to find q / ||q||. Bascially, it means to scale the vector so that it lies on the unit sphere.

5. Jul 28, 2012

### g.lemaitre

amazing it worked! i was skeptical that it would but it did!