What is the alternative representation of SO(2) and its significance?

[spaceofwaste]

The usual representation I see of an element of SO(2) is:

\left( \begin{array}{ c c } cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \right)

and it is easy to show that if you make a passive rotation of a cartesian frame by \theta then this matrix will take the comps of an arbitrary vec to those in the new rotated frame.

However this matrix:

\left( \begin{array}{ c c } sin(\theta) & cos(\theta) \\ -cos(\theta) & sin(\theta) \end{array} \right)

is also a valid representation of SO(2), since it has det=1, and transpose equal to inverse. I have played about with a few drawings but just don't see what this actually represents.

 

[D H]

In your second matrix, theta is the angle between the original y-axis and the transformed x axis, with positive theta denoting a clockwise rotation. Alternatively, it is the angle from the transformed x-axis to the original y axis, with positive theta denoting a counterclockwise rotation. Neither interpretation is particularly useful or intuitive, which is why it isn't used.