$SO(3)$ acting on the space of $3 \times 3$ matrices

Let $SO(3)$ act on the space of $3 \times 3$ real matrices by conjugation. How can I decompose the space of matrices into the sum on minimal invariant subspaces and figure out what they are isomorphic to?

I am familiar with the irreducible representations of $SU(2)$ and how they give the irreducible representations of $SO(3)$. I don't see how to relate this notion to the minimal invariant subspaces.

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Can you compute the character of this representation? – Qiaochu Yuan Jan 9 '12 at 22:14
Conjugation maps symmetric matrices into symmetric, anti-symmetric into anti-symmetric, preserves trace, determinant. This should be enough to get you going... – Sasha Jan 9 '12 at 22:47
Add to Sasaha's hint: check that conjugation (or transpose) commutes with the action of SO(3). – user22671 Jan 9 '12 at 22:55

Hints:

Some simple observations:

1) Due to the cyclic property of the trace $\mathop{\rm tr} M = \mathop{\rm tr} (O M O^T)$ for any matrix $M$ and $O \in SO(3)$.

2) Given a symmetric matrix $S=S^T$, the conjugated matrix $S' = O S O^T$ is also symmetric.

3) Statement 2) also holds when replacing symmetric with antisymmetric.

4) $1+3 + 5 = 9$

I hope this helps... (alternatively you can calculate the character as Qiaochu suggested)

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