Group Action on GL(2,R) by Conjugation: Orbit and Isotropy Group of a Matrix

[Oxymoron]

Question

Let $G=GL(2,\mathbb{R})$ be the group of invertible $2\times 2$ martrices with real entries. Consider the action of $G$ on itself by conjugation. For the element

$$A= \left(\begin{array}{cc}<br /> 2 & 1 \\ <br /> 0 & 3<br /> \end{array}\right)$$

of $G$, describe (i) the orbit and (ii) the isotropy group of $A$

Sorry, I have no working out because I am completely stumped. Can anyone give me some helpful hints or pointers. Thanks

 

[Oxymoron]

The orbit is going to be of the form

$$O_A = \{gA\,|\, g\in GL(2,\mathbb{R})\}$$

Here $g$ can be found by the following way

$$gA = Ag$$ where $$g \in GL(2,\mathbb{R})$$

$$\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{cc}2 & 1 \\ 0 & 3\end{array}\right) = \left(\begin{array}{cc}2 & 1 \\ 0 & 3\end{array}\right)\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)$$

$$\left(\begin{array}{cc}2a & a+3b \\ 2c & c+3d\end{array}\right) = \left(\begin{array}{cc}2a+c & 2b+d \\ 3c & 3d\end{array}\right)$$

Which implies that $2c = 3c = 0 \Leftrightarrow c = 0$. Hence

$$\left(\begin{array}{cc}2a & a+3b \\ 0 & 3d\end{array}\right) = \left(\begin{array}{cc}2a & 2b+d \\ 0 & 3d\end{array}\right)$$

And we can write

$$g = \left(\begin{array}{cc}a & d-a \\ 0 & d\end{array}\right)$$

Therefore the orbit can be described as

$$O_A = \{gA\,|\, g\in GL(2,\mathbb{R})\}$$

So

$$O_A = \left(\begin{array}{cc}2a & -2a + 3d \\ 0 & 3d\end{array}\right)\quad \forall a,b \in \mathbb{R}$$

how does this look?

 

[Oxymoron]

The isotropy group is the subgroup of $GL(2,\mathbb{R})$ consisting of the elements that do not move $A$. That is

$$G_A = \{gA = A\,|\,g\in GL(2,\mathbb{R})\}$$

Therefore we have

$$\left(\begin{array}{cc}2a & -2a+3d \\ 0 & 3d\end{array}\right) = \left(\begin{array}{cc}2 & 1 \\ 0 & 3\end{array}\right)$$

So $a = 1, \, d = 1$. Therefore

$$G_A = \left(\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right) = e$$

the isotropy subgroup consists of the identity element.

 

[Oxymoron]

Anyone know if I have done this correctly? Anyone?