# Smith Normal Form of a companion matrix of monic polynomial

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 $$\mbox{diag}(1,1,1,\dots,f(t))$$.

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.

• Surely the usual "reduce to Smith Normal Form" algorithm gives this at once? Jan 25, 2020 at 13:54
• Perhaps? What is that algorithm? Mar 22, 2022 at 20:03
• If the Smith NF of a matrix $M$ is $d_1,d_2,\dots, d_n$ then $d_1d_2\dots d_k$ is the hcf of all the $k\times k$ minors of $M$. Apply this to $M=tI-C(f)$, it's clear that there's always a $k\times k$ minor $1$ for $k<n$. Mar 23, 2022 at 7:40