- #1
ELB27
- 117
- 15
Homework Statement
Let ##U## be a ##2\times 2## orthogonal matrix with ##\det U = 1##. Prove that ##U## is a rotation matrix.
Homework Equations
The Attempt at a Solution
Well, my strategy was to simply write the matrix as
$$U = \begin{pmatrix}
a & b\\
c & d
\end{pmatrix}$$
and using the given properties to solve for ##a,b,c## and ##d##. I have 4 equations:
1. Determinant = 1
2. ##U^TU = I## where ##I## is identity
where the last property gives me 3 equations (one of the entries is redundant). Thus, I have 4 equations in 4 unknowns. When I solve them, I get 1 free variable, and my matrix turns out to be of the form:
$$U = \begin{pmatrix}
x & \mp \sqrt{1-x^2}\\
\pm \sqrt{1-x^2} & x
\end{pmatrix}$$
and since all entries must be real (orthogonal matrix) we have the constraint on ##x##:
$$-1≤x≤1$$
Clearly, these properties are satisfied if we let ##x=\cos\theta## or ##x=\sin\theta##, thus obtaining a rotation matrix in its standard notation. However, I do not see how to prove that these trigonometric functions are the only possible solutions. Also, how does one define formally and rigorously a rotation matrix? Only as a matrix of cosines and sines (with the appropriate values of determinant etc.)?
Any suggestions/comments will be greatly appreciated!