- Thread starter
- #1

- Feb 5, 2012

- 1,621

Here's a question I encountered recently and did partway. I need your advice on how to proceed.

**Question:**

What can be said about the Jordan normal form of a linear transformation \(f:V\rightarrow V\) where \(V\) is a vector space over \(\mathbb{C}\), if we know that \(f^3=f^2\) ?

**My Attempt:**

Let \(x\) be a eigenvector of \(f\) and let \(\lambda\) be the corresponding eigenvalue. Than,

\[f(x)=\lambda x\]

\[\Rightarrow f^{2}(x)=\lambda f(x)=\lambda^2 x\Rightarrow f^{3}(x)=\lambda^2 f(x)=\lambda^3 x\]

Since \(f^{3}=f^2\) we have,

\[\lambda^2=\lambda ^3\Rightarrow \lambda =0 \mbox{ or }\lambda =1\]

Now this is where I get stuck. Could you help me out?