Finite-dimensional K-vector spaces/Stability concepts/Determinant/Exercise

Let ${}\varphi \colon V\rightarrow V$ be an endomorphism of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Show the following statements.

a) If ${}\varphi$ is asymptotically stable, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, converges zo ${}0$.

b) If ${}\varphi$ is stable, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, is bounded.

c) If the sequence ${}\varphi ^{n}$, ${}n\in \mathbb {N}$, converges, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, converges to ${}0$ or to ${}1$.

Create a solution