# Thread: Orthogonal Matrix

1. ## Orthogonal Matrix

Can anyone help me with this question:

Find a 3 by 3 orthogonal matrix which has 5/13 as its (2,2)-element..

2. Originally Posted by yakuut
Find a 3 by 3 orthogonal matrix which has 5/13 as its (2,2)-element..
The columns of the orthogonal matrix have to form an orthonormal basis for the space. So start by finding a unit vector whose middle coordinate is 5/13. Then find two other vectors so that the three vectors form an orthonormal basis for $\mathbb{R}^3$.

3. I know I didn't post this question but I haven't got a clue how to do it either.

After reading your post, here's what I tried:

$\left( \frac{5}{13} \right)^2+a^2=1 \Rightarrow \ a=\frac{12}{13}$

Therefore I just need to arrange $0, \frac{12}{13}$ and $\frac{5}{13}$ in the matrix so that each vector is linearly independent from the other ones. I got this:

$\begin{pmatrix}
{\frac{12}{13}}&{\frac{12}{13}}&{0}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{\frac{5}{13}}&{0}&{\frac{5}{13}}
\end{pmatrix}=\frac{1}{13}\begin{pmatrix}
{12}&{12}&{0}\\
{0}&{5}&{12}\\
{5}&{0}&{5}
\end{pmatrix}$
?

4. Originally Posted by Showcase_22
I know I didn't post this question but I haven't got a clue how to do it either.

After reading your post, here's what I tried:

$\left( \frac{5}{13} \right)^2+a^2=1 \Rightarrow \ a=\frac{12}{13}$

Therefore I just need to arrange $0, \frac{12}{13}$ and $\frac{5}{13}$ in the matrix so that each vector is linearly independent from the other ones. I got this:

$\begin{pmatrix}
{\frac{12}{13}}&{\frac{12}{13}}&{0}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{\frac{5}{13}}&{0}&{\frac{5}{13}}
\end{pmatrix}=\frac{1}{13}\begin{pmatrix}
{12}&{12}&{0}\\
{0}&{5}&{12}\\
{5}&{0}&{5}
\end{pmatrix}$
?
$\begin{bmatrix}12/13\\5/13\\0\end{bmatrix}$ is fine for the middle column, but the other two columns are not orthogonal to it. You could for example use $\begin{bmatrix}-5/13\\12/13\\0\end{bmatrix}$ for one of them. Can you see a vector (hint: without any fractions in its entries) that is orthogonal to both of these?

Just to emphasise the point: linear independence is not enough, the columns must be orthogonal to each other.

5. Originally Posted by Opalg
Can you see a vector (hint: without any fractions in its entries) that is orthogonal to both of these?
Ah, it's just $\begin{pmatrix}
{0}\\
{0}\\
{1}
\end{pmatrix}$
since those two vectors don't have a k component.

So ultimately we have:

$\begin{pmatrix}
{0}&{\frac{12}{13}}&{-\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$

Can you also have:

$\begin{pmatrix}
{0}&{\frac{12}{13}}&{\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
or $\begin{pmatrix}
{0}&{-\frac{12}{13}}&{-\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
?

(ie. any situation in which the magnitude of the rows and columns is 1?)

6. Originally Posted by Showcase_22
Ah, it's just $\begin{pmatrix}
{0}\\
{0}\\
{1}
\end{pmatrix}$
since those two vectors don't have a k component.

So ultimately we have:

$\begin{pmatrix}
{0}&{\frac{12}{13}}&{-\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
Yes.

Originally Posted by Showcase_22
Can you also have:

$\begin{pmatrix}
{0}&{\frac{12}{13}}&{\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
or $\begin{pmatrix}
{0}&{-\frac{12}{13}}&{-\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
?

(ie. any situation in which the magnitude of the rows and columns is 1?)
No, because those columns are not orthogonal. The scalar product of $\bigl[\tfrac{12}{13},\tfrac5{13},0\bigr]^{\mathsf{T}}$ and $\bigl[\tfrac5{13},\tfrac{12}{13},0\bigr]^{\mathsf{T}}$ is $\tfrac{60+60}{169}$. You need exactly one coordinate to have a negative sign, so as to make the scalar product $\tfrac{60-60}{169} = 0$.

7. Originally Posted by Opalg
Yes.
YAY!

No, because those columns are not orthogonal. The scalar product of $\bigl[\tfrac{12}{13},\tfrac5{13},0\bigr]^{\mathsf{T}}$ and $\bigl[\tfrac5{13},\tfrac{12}{13},0\bigr]^{\mathsf{T}}$ is $\tfrac{60+60}{169}$. You need exactly one coordinate to have a negative sign, so as to make the scalar product $\tfrac{60-60}{169} = 0$.
I get it. So you can also have $
\begin{pmatrix}
{0}&{-\frac{12}{13}}&{\frac{5}{13}}\\
{0}&{\frac{5}{13}}&{\frac{12}{13}}\\
{1}&{0}&{0}
\end{pmatrix}
$
since the scalar product of $\begin{pmatrix}
{-\frac{12}{13}}\\
{\frac{5}{13}}\\
{0}
\end{pmatrix}$
and $\begin{pmatrix}
{\frac{5}{13}}\\
{\frac{12}{13}}\\
{0}
\end{pmatrix}$
is $-\frac{12(5)}{13^2}+\frac{5(12)}{13^2}=0$.

Thanks Opalg.