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"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and $\mathbb{C}e_2$ where $e_1=(i,0)$, $e_2=(0,i)$ i.e standard basis element for $\mathbb{C}^2$",

could any one tell me what does it mean by preservation of eigen space?

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@EuYu: You could write that as an answer; there's nothing to add to it. – joriki Nov 8 '12 at 6:13
@joriki Thanks for telling me. It's done. – EuYu Nov 8 '12 at 6:18
up vote 2 down vote accepted

Normally it's taken to mean that the eigenspaces are invariant subspaces under $A$.

If this is not clear then consider the following. Suppose that $A$ and $B$ are commuting matrices. Let $\mathbb{v}$ be an eigenvector of $B$ under $\lambda$. Then $$BA\mathbb{v} = AB\mathbb{v} = \lambda A\mathbb{v}$$ Therefore it follows that $A\mathbb{v}$ remains an eigenvector of $B$ under the same eigenvalue so that the eigenspace $E_\lambda$ is invariant under $A$. In this case we say that $A$ preserves the eigenspaces of $B$.

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