I have some issues with a problem which is asking me to decompose the set of $2 \times 2$ complex matrices $\mathbb{C}^{2 \times 2}$ in orbits under the left multiplication operation on the group $GL_2 (\mathbb{C})$. I've proved that the subset of invertible matrices forms an orbit by using the fact that orbits are an equivalence relation and so are transitive. I've also shown (trivially) that the orbit of the zero matrix is of size one and contains only the zero matrix, but I'm not making any progress on decomposing the set of non-invertible, non-zero matrices – I don't understand what these orbits look like. Surely it isn't sufficient to say that any element of $g$ always carries a non-invertible matrix to a non-invertible matrix? There could be lots of smaller orbits contained in that subset, or am I misunderstanding/overthinking this?
Can someone help?
Best Answer
Hint. Elementary matrices (i.e. the matrices that perform elementary row operations) form a subset of invertible matrices. Do you recall any special form of matrix that is obtained by applying a series of elementary row operations to a matrix?