Metric on ℝ^2 Invariant under Matrix Transformations

[FrederikPhysics]

Hello, I am looking for some nontrivial metric on ℝ^2 invariant under the coordinate transformations defined by the 2x2 matrix
[1 a₁₂(θ)]
[a₂₁(θ) 1],
where a_ik is some real function of θ. In the same way that the Minkowski metric on ℝ^2 is invariant under Lorentz transformations.
Does this metric exist? If not does it exist for some related type of transformations? And why? Are there some other nice features about this kind of transformations/matrices?

 

[stevendaryl]

Hello, I am looking for some nontrivial metric on ℝ^2 invariant under the coordinate transformations defined by the 2x2 matrix
[1 a₁₂(θ)]
[a₂₁(θ) 1],
where a_ik is some real function of θ. In the same way that the Minkowski metric on ℝ^2 is invariant under Lorentz transformations.
Does this metric exist? If not does it exist for some related type of transformations? And why? Are there some other nice features about this kind of transformations/matrices?

Okay, this is something that can be completely solved using matrix algebra.

First of all, every nonsingular matrix can be diagonalized. So if we assume it's nonsingular, then there is a matrix $U$ such that $U g U^T = \tilde{g}$, where $\tilde{g}$ has the form:

$\tilde{g} = \left( \begin{array} \\ g_1 & 0 \\ 0 & g_2 \end{array} \right)$

and where $U^T U = 1$ ($U^T$ means the transpose of $U$).

So let's look for transformations $\tilde{T}$ that preserve $\tilde{g}$. That means that for any column matrices $u$ and $v$,

$(\tilde{T} v)^T \tilde{g} (\tilde{T} u) = v^T \tilde{g} u$

which means that $\tilde{T}^T \tilde{g} \tilde{T} = \tilde{g}$

You can find the form of $\tilde{T}$ by using matrix algebra, but I'll skip to the answer:

$\tilde{T} = \left( \begin{array} \\ cos(\theta) & \sqrt{\frac{g_2}{g_1}} sin(\theta) \\ \sqrt{\frac{g_1}{g_2}} sin(\theta) & cos(\theta) \end{array} \right)$

(This matrix has to be real, which means that if $\frac{g_2}{g_1} < 0$, then you have to choose $\theta$ to be imaginary, which means using $sinh$ and $cosh$ instead of $sin$ and $cos$).

Now, to get back to the original problem, if $\tilde{T}$ preserves $\tilde{g}$, then $T \equiv U^T \tilde{T} U$ is a transform preserving the original $g$.