Determining decomposition into orbits

Asked Oct 23 '13 at 13:17

Active Oct 23 '13 at 13:17

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Decompose the set $\mathcal{M}_2(\mathbb{C})$ of $2 \times 2$ complex matrices into orbits for the following operations of $GL_2(\mathbb{C})$: left multiplication.

Could someone give me a complete answer to this question? I have another "similar" question relation to conjugation. In addition to that I am practicing for an exam, and have other related problems and am kind of having difficulty understanding them. Would someone therefore have a link for similar type of problems (decomposition into orbits, determining stabilisers)

asked Oct 23 '13 at 13:17

amir

1,311

are you trying for classes of similar matrices?? – Oct 23 '13 at 13:19
@PraphullaKoushik I don't understand your question. Could you rephrase it ? – amir Oct 23 '13 at 13:22
You said, you want to see for orbits corresponding to conjugacy action.... $A$ is related to $B$ by conjugation iff $A$ and $B$ are similar.. Do you see why? – Oct 23 '13 at 13:24
@PraphullaKoushik Ok but I first asked regarding the left multiplication, not conjugation. – amir Oct 23 '13 at 13:26
I realized proof of similar matrices.. I am trying to see for left multiplication.... – Oct 23 '13 at 13:27
@PraphullaKoushik I have a general question though. How would you generally proceed in order to determine the decomposition into orbits ? – amir Oct 23 '13 at 13:40
let us continue this discussion in chat – amir Oct 23 '13 at 13:44
@user43418 Did you ever figure this out? – Jeff Nov 21 '13 at 19:08

Determining decomposition into orbits

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