# Factor the matrix into the form QR where Q is orthogonal

1. Apr 8, 2010

### Dustinsfl

Factor the matrix into the form QR where Q is orthogonal and R is upper triangular.

$$\begin{bmatrix} a & b\\ c & d \end{bmatrix}*\begin{bmatrix} e & f\\ 0 & g \end{bmatrix}=\begin{bmatrix} -1 & 3\\ 1 & 5 \end{bmatrix}$$

$$\begin{bmatrix} a & c \end{bmatrix}*\begin{bmatrix} b\\ d \end{bmatrix}=0$$

$$ae=-1$$

$$af+bg=3$$

$$ce=1$$

$$cf+dg=5$$

Skipping some steps but I arrive at:$$\begin{bmatrix} 1 & \frac{4}{g}\\ -1 & \frac{4}{g} \end{bmatrix}*\begin{bmatrix} -1 & -1\\ 0 & g \end{bmatrix}=\begin{bmatrix} -1 & 3\\ 1 & 5 \end{bmatrix}$$

So as long as $$g \neq 0$$ it is all good?

2. Apr 8, 2010

### Staff: Mentor

Re: Factoring

Works for me. You have a matrix that is orthogonal and another that is upper triangular, and they multiply to make the matrix on the right. The moral of the story seems to be that such factorizations aren't unique.

3. Apr 8, 2010

### Hurkyl

Staff Emeritus
Re: Factoring

"Q is orthogonal" consists of three conditions, not one....

4. Apr 8, 2010

### Staff: Mentor

Re: Factoring

I forgot about the part where the columns have to be unit vectors...

5. Apr 8, 2010

### Dustinsfl

Re: Factoring

Ok so the column vectors also have to be unit vectors and what is the other stipulation?

6. Apr 8, 2010

### Hurkyl

Staff Emeritus
Re: Factoring

I was counting polynomial equations -- so what you just said counts as 2 conditions.