Cartesian Metric is Rotation Invariant

Theorem
The cartesian metric does not change under rotation.

Proof
Let the cartesian metric be $$\delta _{ij}=$$.

Also, let $$\delta_{ij}^\prime$$ be the metric of the coordinate system of $$\delta _{ij}$$ rotated by a rotation matrix $$A$$.

Then, $$\delta_{ij}^\prime=$$.

$$=$$, so

$$\delta_{ij}^\prime==$$.

For rotation matrices, we have $$A^T=A^{-1}$$, so $$\delta_{ij}^\prime$$ reduces to

$$\delta_{ij}^\prime====$$, where $$I$$ is the Identity Matrix.

Thus, $$\delta_{ij}^\prime=\delta_{ij}$$

The result follows.