- #1

CGandC

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- TL;DR Summary
- Does minimal polynomial zero out the linear operator restricted to any subspace?

The usual theorem is talking about the linear operator being restricted to an invariant subspace:

However, I had difficulty understanding why we needed the assumption that ## W ## is ##T##-invariant, I mean - If ## m_T(x) ## is the minimal polynomial of ##T## so ## m_T(T)=0 ## and thus for any subspace ## W \subseteq V ## (

I had no problem understanding its proof, it appears here for example: https://math.stackexchange.com/ques...-t-w-is-diagonalizable-if-t-is-diagonalizableLet ##T## be a diagonalizable linear operator on the ##n##-dimensional vector space ##V##, and let ##W## be a subspace of ##V## which is invariant under ##T##. Prove that the restriction operator ##T_W## is diagonalizable.

However, I had difficulty understanding why we needed the assumption that ## W ## is ##T##-invariant, I mean - If ## m_T(x) ## is the minimal polynomial of ##T## so ## m_T(T)=0 ## and thus for any subspace ## W \subseteq V ## (

**not necessarily ## T##-invariant**) ## m_T(T_W) =0 ##; so why in the above theorem it was necessary for ## W \subseteq V ## to be ## T ##-invariant?