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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1answer
23 views

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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31 views

Show that a non-monic polynomial is not in the kernel of the evaluation map

Suppose $ P = x^n + a_{n-1}x^{n-1} + \cdots + a_0$, is a monic polynomial, $n \geq 1$ with coefficients in a field $K$ and that the $K$-vector space can be written as a direct sum $K = K e_1\bigoplus \...
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1answer
90 views

Determine the determinant of a companion matrix

Calculate for $ n \geq 2 $ and $ x, a_{0}, a_{1}, \ldots, a_{n-1} \in \mathbb{R} $ the determinant of the following matrix: $$\begin{bmatrix} {x} & {0} & {\cdots} & {\cdots} & {\cdots}...
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2answers
148 views

How do you rewrite a determinant of a matrix into a polynomial by induction?

$$\det\begin{bmatrix} {x} & {0} & {\cdots} & {\cdots} & {0} & {a_{1}} \\ {-1} & {x} & {0} & {\cdots} & {0} & {a_{2}} \\ {\ddots} & {\ddots} & {\ddots} &...
2
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0answers
56 views

Monic polynomial and companion matrix

Problem Let $p(T) := T^n-\alpha_{n-1}T^{n-1}-\alpha_{n-2}T^{n-2}-\cdots-\alpha_0 \in K[T]$. Additionally we have the companion matrix of $p$ $$A:= \begin{bmatrix} 0 & 1 & 0 & 0 &...
4
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1answer
106 views

Show the matrix commutes with companion matrix is a polynomial

Let $A$ be a linear transform on $n$-dimensional $V$ over a field $F$. Under a basis $\alpha_1, \cdots, \alpha_n$, the matrix representation of $A$ is as follows: $$A = \begin{bmatrix} 0 & 0 &...
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0answers
20 views

Counting the powers of a companion matrix that possess nonzero leading principal minors

Let $p$ be prime, let $A$ be the $n\times n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in ${\mathbb Z}_p$, and let $T=\{A^j\mid 1\leq j\leq p^n-1\}\...
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0answers
29 views

Characteristic polynomial of $r \times r$ companion matrix [duplicate]

I want to show that $a(x) = \det[C(a)-x I]$ where $a(x)=a_0+ax+...+a_{r-1}x^{r-1}+x^r$ and $C(a)$ is the companion matrix: $$\begin{vmatrix} 0&1&0&\dots& 0 \\ 0&0&1&\dots&...
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65 views

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)...
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1answer
241 views

How to compute the characteristic polynomial of a companion matrix to a polynomial with matrix-valued coefficients?

Consider we have a polynomial $p = z^m + b_{m-1}z^{m-1} + \dotsb + b_0$ with matrix coefficients $b_i \in M_n(\mathbb{C})$. Then we might consider the companion matrix $$T = \left[ \begin{matrix} 0_n &...
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69 views

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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0answers
40 views

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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2answers
76 views

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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1answer
194 views

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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2answers
212 views

Explicit formula of exponential of companion matrix

Let $$A=\begin{bmatrix} a_k & a_{k-1} & a_{k-2} & \dots & a_2 & a_1 \\ 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 \\ \dots & ...
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1answer
238 views

When are almost block companion matrices which yield a given characteristic polynomial connected?

This is motivated by this question Are matrices which yield a given characteristic polynomial and have specified structure connected? Let $\mathcal E \in M_n(\mathbb R)$ be a subset with following ...
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5answers
88 views

Finding the inverse of $A$

Find the inverse of $$A =\left[\begin{matrix}0 & 1 & 0 & 0\\ 0& 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ a & b & c & d\end{matrix}\right]$$ My attempts: $$A^{-1} ...
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412 views

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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2answers
730 views

Modulus of eigenvalues of a companion matrix

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&&\\ &...
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2answers
125 views

Does every polynomial with a Perron root have a primitive matrix representation?

Let $p(x)=x^6-13x^4-20x^3+x^2-x+2$ and $C$ be the companion matrix of $p(x)$. How can I find a primitive matrix similar to $C$ ? Is there a general method to transform the companion matrix with a ...
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1answer
150 views

How to check a special form of Companion Matrix is a Primitive Matrix or not?

Definition: A non-negative matrix square $A$ is called primitive if there is a $k$ such that all the entries of $A^k$ are positive.[1] Lemma: If the graph associated to $A$ is strongly connected and ...
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3answers
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Determinant of a companion matrix

I have to find determinant of $$A := \begin{bmatrix}0 & 0 & 0 & ... &0 & a_0 \\ -1 & 0 & 0 & ... &0 & a_1\\ 0 & -1 & 0 & ... &0 & a_2 \\ 0 &...
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0answers
35 views

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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1answer
695 views

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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1answer
118 views

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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1answer
130 views

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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0answers
275 views

Companion matrices for multivariate polynomials?

I am aware of companion matrices for single variable polynomials: $$p(x) = x^k+c_{k-1}x^{k-1}+\cdots +c_0$$ $${\bf C_p} = \left[\begin{array}{ll}{\bf 0}^T & -c_0 \\ {\bf I}_{k-1} & {\bf c}_{...
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0answers
214 views

Proof of FLOPS of matrix inverse by companion matrix

I was wondering if anyone could help me prove that the cost of matrix inversion by companion matrix is $2n^3$ + L.O.T. FLOPS.
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1answer
75 views

Are companion matrices useful for binary division?

The new find-first-set bit "ffs" CPU instruction found in the multi media extensions (MMX) 4 apparently made possible to start doing Newton-Raphson division (according to Wikipedia). Does someone ...
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1answer
112 views

Polynomials $\&$ Matrices

Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows $$ P(x)=\det (A-x\,I)\, . $$ Where $I$ is an identity matrix of order $n$. What ...
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0answers
706 views

Showing a matrix is non-derogatory [duplicate]

Prove that the matrix $$A = \begin{bmatrix} 0 & 1 & \\ \ddots & & \ddots \\ & \ddots & & \ddots \\ & & \...
1
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1answer
528 views

Compute the minimal polynomial of a companion matrix without using Cayley-Hamilton

I have $$A = \begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & 1 & -a_{n-1} \end{...
3
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1answer
82 views

Show an $n\times n$ matrix $A$ is sim. to the companion matrix for $p_A(t)\iff \exists$ a vector $x$ such that $x,Ax,\ldots,A^{n-1}x$ is a basis

Show that an $n\times n$ matrix $A$ is similar to the companion matrix for $p_A(t)$ if and only if there exists a vector $x$ such that $$x, Ax, \ldots, A^{n-1}x$$ is a basis for $\mathbb C^n$. The ...
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2answers
374 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
2
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2answers
196 views

Find the characteristic polynomial of this matrix

I've tried to find the characteristic polynomial of the following matrix $$A=\begin{pmatrix} 0 & 0 & \cdots & 0 & -a_n \\ 1 & \ddots & \ddots & \ddots & \...
3
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1answer
60 views

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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1answer
378 views

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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1answer
431 views

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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1answer
96 views

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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2answers
933 views

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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2answers
2k views

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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1answer
2k views

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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2answers
1k views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & \...
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2answers
181 views

Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $$A = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & a_1 \\ ...
4
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3answers
536 views

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 $A$ ...
4
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3answers
552 views

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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1answer
369 views

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 ...
3
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1answer
534 views

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,...
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1answer
1k views

When is a matrix similar to the companion matrix of its characteristic polynomial?

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 $...