# Metric on ℝ^2 Invariant under Matrix Transformations

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• FrederikPhysics
In summary, the conversation discussed a search for a nontrivial metric on ℝ^2 that is invariant under coordinate transformations defined by a 2x2 matrix. It was found that the metric can be solved using matrix algebra and the form of the transformation was determined. It was also noted that if the ratio of the diagonal elements of the metric is negative, the transformation requires the use of sinh and cosh instead of sin and cos.
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 aik 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?

FrederikPhysics said:
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 aik 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 $g$.

## 1. What is a metric on ℝ^2?

A metric on ℝ^2 is a mathematical concept that defines the distance between any two points in the two-dimensional Euclidean space ℝ^2. It is a function that takes in two points as inputs and outputs a non-negative real number representing the distance between them.

## 2. How is a metric on ℝ^2 calculated?

A metric on ℝ^2 can be calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In other words, the distance between two points (x1, y1) and (x2, y2) in ℝ^2 is given by √((x2-x1)^2 + (y2-y1)^2).

## 3. What does it mean for a metric on ℝ^2 to be invariant under matrix transformations?

A metric on ℝ^2 is invariant under matrix transformations if it remains unchanged when the points in the two-dimensional space are transformed using a matrix. This means that the distance between any two points in the transformed space will be the same as the distance between the two points in the original space.

## 4. Why is it important for a metric on ℝ^2 to be invariant under matrix transformations?

It is important for a metric on ℝ^2 to be invariant under matrix transformations because it allows for consistent measurements and comparisons between points in the two-dimensional space, regardless of how they are transformed. This is especially useful in fields such as geometry, physics, and computer graphics.

## 5. How can I determine if a metric on ℝ^2 is invariant under matrix transformations?

A metric on ℝ^2 is invariant under matrix transformations if it satisfies the properties of a metric and remains unchanged when the points in the space are transformed using a matrix. This can be checked by applying the matrix transformation to the points and calculating the distance between them using the metric. If the distance remains the same, then the metric is invariant under matrix transformations.

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