What is the most easy way to show that the $n$-dimensional standard normal distribution is invariant under rotations?

Question

I want to show that for an orthogonal matrix $O$, we have $O \cdot x\stackrel{d}{=}x$, where $x \sim N(0, \mathbb{I}_n)$. What is the most easy way to show that?

Answer 1

The characteristic function of $x$ is $$\phi_x(u) = e^{-u^\top u/2}.$$ The characteristic function of $Ox$ is $$\phi_{Ox}(u) = E[e^{iu^\top Ox}] = \phi_x(O^\top u) = e^{-u^\top OO^\top u/2} = \phi_x(u)$$ so $Ox \overset{d}{=} x$.

Answer 2

$$f(x_1, \ldots, x_n ) = (2\pi)^{-\frac{n}2}\exp({-\left\|x \right\|^2}/2)$$

and notice that $$\left\|Ox\right\|^2=\left\|x\right\|^2$$