Finite-dimensional normed vector space/Matrix powers/Eventually periodic/Characterizations/Exercise

Let ${}M$ be an ${}n\times n$-matrix over ${}{\mathbb {K} }$. Show that the following statements are equivalent.

a) In the sequence ${}M^{n}$, ${}n\in \mathbb {N}$, there is a repetion, that is,

${}M^{n}=M^{m}\,$

for some pair of numbers ${}n<m$.

b) In the sequence ${}M^{n}$, ${}n\in \mathbb {N}$, there only finitely many different matrices.

c) The sequence ${}M^{n}$, ${}n\in \mathbb {N}$, becomes eventually (starting at a certain position) periodic.

d) The Jordan blocks of ${}M$ over ${}\mathbb {C}$ have the form

${\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&0&1\\0&\cdots &\cdots &0&0\end{pmatrix}},$

or the form ${}(\lambda )$ with a

complex root of unity ${}\lambda$.

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