# Thread: Matrix group help

1. ## Matrix group help

Hiya,
I've been puzzling over this for ages and I cant find the appropriate proof anywhere... can anyone help me?

I need to prove that the unitary 1x1 matrix U(1) is isomorphic to the special orthogonal 2x2 matrix SO(2) and that they can be thought of as a circle.

thanks alot

2. A unitary 1×1 matrix is a complex number z such that $\bar{z}z = 1$, in other words a complex number of modulus 1. This represents a point on the unit circle in the Argand diagram (that's where the circle comes in), and it is of the form $z=e^{i\theta}$.

An element of SO(2) is a 2×2 real orthogonal matrix with determinant 1. Any such matrix must be of the form $\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$, and it represents a rotation through an angle $\theta$.

The isomorphism from U(1) to SO(2) is the map that takes $e^{i\theta}$ to $\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$.

3. Thankyou,
But there should be a proof for this. That answers too basic.

4. Originally Posted by Loonywoody
Thankyou,
But there should be a proof for this.
That's your job, not mine.

You only have to google "SO(2)" to find most of the proofs that you want here (along with a whole lot of links associated with sulphur dioxide).