Rotation invariance of higher than 2 dimensions

Question

According to this $f_2(x_1,x_2) = x_1^2 + x_2^2$ is invariant under rotation. I wanted to ask if a function $f_n(x_1,x_2,...,x_n) = x_1^2 + x_2^2 + ...+ x_n^2$ is also rotation invariant. In other words, is this feature preserved as we go to higher dimensions?

Answer 1

Sure, if $Q$ is a rotation matrix ($Q^T Q = I$), then with the Euclidean norm we have $\|x\|^2 = \langle x , x \rangle = \langle x , Q^T Qx \rangle = \langle Qx , Qx \rangle = \|Qx\|^2$.

Answer 2

Yes, this is preserved as we go to higher dimensions as long as the number of terms is the same as the dimension of the space. The Pythagorean theorem (iterated) says this sum of squares is the square of the distance from the origin to the point.