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Let $ A = (\alpha_ {1}, ..., \alpha_ {n}) $ be a basis of a vector space $ V $ and let $ \varphi: V \rightarrow V $ be the endomorphism given by the conditions $\varphi(\alpha _{i})=\alpha _{i+1}$ for $i=1,...,n-1$ and $\varphi(\alpha _{n})=a_{0} \alpha_{1}+a_{1} \alpha_{2}+...+a_{n-1} \alpha_{n}$. Find a characteristic polynomial of endomorphism $\varphi$.

In this task I have a matrix $M^{A}_{A}$ which has $\alpha_{i+1}$ in subsequent columns and this $\alpha_{i+1}$. However I completely don't know how to do this task because this fact is insufficient for me.

Can you help me?

Bernard
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MP3129
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  • Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, \ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see https://math.stackexchange.com/questions/10216/ (although the asker of that question has excluded the simplest answer). – darij grinberg Mar 07 '19 at 21:22

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Hint: The matrix of this endomorphism is $$\begin{bmatrix} 0&0&0&\dots&0&a_0\\ 1&0&0&\dots&0&a_1\\ 0&1&0&\dots &0&a_2 \\[-1ex] \vdots&\vdots&\ddots&&\vdots&\vdots\\ 0&0&0&\dots&0&a_{n-2}\\ 0&0&0&\dots&1&a_{n-1} \end{bmatrix},$$ so its characteristic polynomial is the determinant $$\begin{vmatrix} -x&0&0&\dots&0&a_0\\ 1&-x&0&\dots&0&a_1\\ 0&1&-x&\dots &0&a_2 \\[-1ex] \vdots&\vdots&\ddots&&\vdots&\vdots\\ 0&0&0&\dots&-x&a_{n-2}\\ 0&0&0&\dots&1&a_{n-1}-x \end{vmatrix}.$$ Denote it as $D(a_0,a_1,\dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation: $$D(a_0,a_1,\dots,a_{n-1})=-xD(a_1,\dots,a_{n-1})+(-1)^{n-1}a_0.$$ Deduce from this relation that $$D(a_0,a_1,\dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-\dots-a_1x-a_0).$$

Bernard
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