Prove that every polynomial $p \in P_n(\mathbb{C})$ is the characteristic polynomial of some linear map.

Question

Prove that every polynomial $p \in P_n(\mathbb{C})$ is the characteristic polynomial of some linear map.

First attempt. If $p \in P_n(\mathbb{C})$, then $p(x) = (x - a_1)\cdots(x - a_m)$ where $n \geq m = \deg(p)$. So, take some matrix having $a_1,...a_m$ as complex eigenvalues and the correspondent linear map.

After that, I found this question: Characteristic polynomials exhaust all monic polynomials? . So I think my answer is not correct. I have two questions.

1 - Can someone explain where my "proof" fails?

2 - I'm not interested in construct such matrix, so is there any simpler proof?

The companion matrix is pretty much as simple as you can get: It's the polynomial $p$ acting on the quotient ring $\mathbb C\left[t\right]/\left(p\right)$, represented as a matrix with respect to the obvious basis. — darij grinberg, Sep 18 '19 at 06:02

score 1 · Accepted Answer · answered Sep 18 '19 at 05:58

Your answer is correct, but it has no practical use, since it assumes that you know how to factor your polynomial as the product of first degree polynomials. And the usual proof actually proves that if the coefficients of the polynomial belong to a subfield $k$ of $\mathbb C$, then you can take a linear map from $k^n$ into itself whose characteristic poynomial is the one that you are interested in.
Not that I am aware of.

Prove that every polynomial $p \in P_n(\mathbb{C})$ is the characteristic polynomial of some linear map.

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