# Questions tagged [companion-matrices]

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### A question in Corollary Section 7.1 of Hoffman Kunze Linear Algebra

I am self studying Chapter -7 of Linear Algebra from Hoffman Kunze and I have a question in 1st section in last corollary whose image I am adding . Image of Theorem 1: I have a question in 1st ...
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### Similarity transformation and representation of matrices

I'm trying to understand this passage of a book: Why this last expressions shows that the $i$th column of $\bar{A}$ is the representation of $Aq_i$ with respect to the basis $\{q_1,\ldots q_n\}$? I ...
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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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### Worst case scenarios for companion matrix method

In what case is the companion matrix method for computing the roots of a polynomial expected to suffer accuracy loss. Are nearby roots the main issue?
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### Finding similarity transformation matrix for the companion matrix.

Given that $$A:=\begin{bmatrix}3 & 5 & -1 & 8\\ 1 & 2 & 4 & 9\\ 5 & -4 & 7 & 6\\2 & 4 & -3 & -1\end{bmatrix}, \;\;\;b = \pmatrix{1\\ 5 \\ -1 \\ -8},$$ I ...
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### Characterization of companion matrices

Let $K$ be an arbitrary field and let $A\in\mathbb{M}_n(K)$. Let $m(X)$ and $p(X)$ be the minimal polynomial of $A$ over $K$ and the characteristic polynomial of $A$ over $K$, respectively. According ...
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### Symmetric part of a companion matrix

I hope this question is sufficiently interesting for you to try to answer it. Sorry if its too trivial, but, working with a few similarty transformations I've arrived to the following conclusion, and ...
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### Prove there is a basis of $V$ w.r.t. which $T$ is the companion matrix of $a(x)$

Given a linear transformation $T$ on a finite-dimensional vector space $V$ over a field $K$, and giving $V$ the $K[x]$-module structure determined by $f(x)v = F(T)(v)$, Frederick Goodman shows on page ...
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### Is there a general way to get the QR decomposition of a companion matrix?

Is there a general way to get the QR decomposition of a companion matrix? Is it considered a sparse matrix? Is shifting always required in this case?
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### Finding roots of polynomial using companion matrix

The standard method for finding roots of a polynomial is to form the companion matrix, balance it, then compute the eigenvalues by double shift QR algorithm. This method is used by Matlab ROOTS ...
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### Expression for polynomial of companion matrix

I am rather stuck on an exercise concerning the companion/controllability matrix (the exercise stems from a course in control theory). Given the companion matrix \begin{equation} A=\left(\begin{...
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### Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~,$$ I can ...
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### Eigenvalues of a companion matrix

I've been tasked with the following: Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$. Show that if $\lambda$ is an ...
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### Eigenvalues of an upper Hessenberg matrix

I'm interested in calculating the roots of an 11th degree polynom. To do so, I calculated the $10 \times 10$ companion matrix which eigenvalues are the roots of the polynomial. Now, the eigenvalues ...
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### Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ ...
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### Applications of companion matrices

I'm looking for interesting applications of companion matrices. I can also use the Frobenius Normal Form. I already covered the Cayley-Hamilton Theorem and the application to linearly recursive ...
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### Matrix similar to a companion matrix

I am currently intensively reading my linear algebra notes under dim light and was wondering whether it is true, that a an endomorphism whose minimal polynomial has the same degree as the dimension of ...
### Jordan basis of $A$ when $A$ is the companion matrix?
The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is:  M_i^{h,...
Let $A$ be a complex matrix and $A_c$ the companion matrix of its characteristic polynomial. From what I have read, I believe the following two statements to be true: not every $A$ is similar to \$...