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Let $V$ be a vector space over a field $F$ and let $A$ be in the endomorphisms of $V$. Then $A$ is called non derogatory if its minimal polynomial $m_A(x)$ is equal to its characteristic polynomial $p_A(x)$.

Assume that $A$ is non derogatory and that $K$ is an $A$-invariant subspace of $V$. Can we conclude that the restriction of $A$ to $K$ is non derogatory? How can we see that?

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up vote 2 down vote accepted

I think you should be able to prove this quite easily as soon as you prove the following characterization of "non-derogatory"

Call a basis "$A$-cyclic" if it is of the form $\{x,A(x),A^2(x),\cdots,A^n(x)\}$ (in an $n+1$-dimensional space). Then, a linear transformation $A$ is non-derogatory if and only if it admits an $A$-cyclic basis.

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