We have to stochastic variables X and Y, and we define
$ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} $
for $a,b,c,d \in \mathbb{R}$, i.e. $\tilde{X}=aX+bY$ and $\tilde{Y}=cX+dY$. We know that both $X$ and Y, and $\tilde{X}$ and $\tilde{Y}$ are uncorrelated, and the distribution of (X,Y) has density $g(x,y)=f(x)f(y)$. Here $f$ is a density with respect to the lebesgue measure for a probability measure on $\mathbb{R}$, and f is continous and strict positive.
Furthermore, we say that the distribution of (X,Y) is invariant to orthogonal matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ if $(X,Y)$ and $(\tilde{X},\tilde{Y})$, have equal distributions. In general, a matrix O is orthogonal if $OO^T=O^TO=I$, where I is the identity matrix.
Now I have to show that if the distribution of $(X,Y)$ is invariant to orthogonal transformations, then it holds that
$f(x)f(y)=f(0)f(\sqrt{x^2+y^2})$ for alle $x,y \in \mathbb{R}^2$.
Can anyone help me with this problem?
Consider orthogonal matrix
$$ A = \left( \begin{array}{cc} \frac{x}{\sqrt{x^2+y^2}}& \frac{y}{\sqrt{x^2+y^2}} \\ -\frac{y}{\sqrt{x^2+y^2}} & \frac{x}{\sqrt{x^2+y^2}} \end{array} \right). $$