Let $C(f)$ be the companion matrix of a monic polynomial $f(t)\in \mathbb{F}[t]$. I need to show that the Smith Normal Form of $tI - C(f)$ is equal to the diagonal matrix $\,diag(1,1,1,...,f(t))$.

A little bit baffled on how to begin. I've constructed the companion matrix and written out my monic polynomial. I think there are some relationships between equivalent/similar matrices of the form $tI-A$ that might help. But I'm definitely scratching my head on understanding how to link the Smith Normal Form to all of this.


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