# Questions tagged [companion-matrices]

On the companion matrix to a certain monic polynomial.

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### How do I prove that for every monic polynomial $P$ there exists a linear operator $f$ such that $P$ is the minimal polynomial of $f$?

Let $\mathbb{F}$ be a field and $P(T)=T^{k}+a_{k-1}T^{k-1}+\dots+a_{1}T+a_{0}\in\mathbb{F}[T]$ a monic polynomial. Prove: There exist a vector space $V$ over $\mathbb{F}$, a linear operator $f:V\to V$ ...
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### Comparing values and signs of determinant of companion matrices.

Suppose $A,B \in M_{n}(\Bbb{R})$ such that $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C_{2}\middle|\frac{I}{0\dots0}\right]$ , where $A$ and $B$ have different first columns (...
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### Companion matrix for $(x^2+1)^2$

I know how to find campanion matrix from polynomial.but in Kurtis linear algebra book I found In that$(a_{14})$entries is 1 .from my calculation s I got everything correct except that . Where I ...
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### Companion matrix, polynomial

I have the following polynomial $$p(t) = a_0 + a_1 t + a_2 t^2 + t^3$$ and the following information Let $p$ be indicated as above. Suppose $\lambda$ is a real root in the polynomials p, in other ...
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### Is the cyclic vector for companion matrix independent of the specific companion matrix under consideration?

Suppose $C_1, C_2 \in M_n(\mathbb R)$ are two matrices in companion form. If $v$ is a cyclic vector for $C_1$, i.e., $\{v, C_1 v, C_1^2 v, \dots, C_1^{n-1}v\}$ is a basis for $\mathbb R^n$, does this ...
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### Solving differential equations given a companion matrix?

So i'm given a differential equation $$f'''(t)-2f''(t)-f'(t)+2f(t)=0$$ where $$t\in \mathbb{R}$$ and $$C=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -2 & 1 & 2\end{pmatrix}$$ ...
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### Eigenvectors of companion matrix

In Matrix form the determination of eigenvectors of a companion matrix has been appeared in different questions on this site; my question is a little bit different, I was unable to do an ...
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I have the companion matrix \mathbf{M}:=\left(\begin{array}{ccccc} 1-p+pa_{1}&pa_{2}&pa_{3}&\cdots&pa_{m}\\ 1&&&&\\ &1&&&\\ &&1&&\\ &...