Questions tagged [companion-matrices]

For questions about companion-matrices

Filter by
Sorted by
Tagged with
2 votes
1 answer
44 views

minimal and characteristic polynomial of this operator [duplicate]

The following is Problem 18 from Chap8.C of Axler's Linear Algebra Done Right. Edited to add a transcription of the original problem(in the image) P18. Suppose $a_0, a_1, ...., a_{n-1} \in \mathbb{C}...
user avatar
0 votes
0 answers
32 views

The unipotent companion matrix over a division ring

Let $D$ be a division ring and let $C(f)$ be a $n\times n$ companion matrix of the monic polynomial $f(x)$ in which the coefficients belong to $D$. I wonder whether $f(x)=(x-1)^n$ when $(C(f)-I)^n=0$. ...
user avatar
2 votes
1 answer
96 views

If $b$ is a cyclic vector, what is significant about $\begin{bmatrix} A - \lambda I & b \end{bmatrix}$?

Problem. Let $A$ be a complex $n \times n$ matrix and $b$ be a complex $n \times 1$ vector. Prove Rank$\begin{bmatrix} b & Ab & A^2b & \cdots & A^{n-1} b \end{bmatrix} = n$ if and ...
user avatar
-1 votes
1 answer
45 views

Converting a monic polynomial of degree $n$ to its companion matrix

Given a polynomial of form $$p_A = t^n + a_{n-1} t^{n-1} + \cdots + a_0$$ How can construct the companion matrix $A$ such that $\det(A-tI) = p_A$ assuming that you don't already know how $A$ should ...
user avatar
  • 13
1 vote
0 answers
90 views

Companion matrices over the Galois field with two elements

Let us denote $\mathbb{F}_2$ by the Galois field with two elements. The companion matrix of the monic polynomial $f(x)=a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}+x^n$ is the square matrix defined as $$C(f)=...
user avatar
  • 11
1 vote
1 answer
30 views

Let $f \in \mathbb{F}[x]$ and $A_f=c(f) $ be the companion matrix of $f$. Given $ g \in \mathbb{F}[x] $ show that $ \dim \ker g(A_f) \leq \deg g $.

Theorem: Let $ f \in \mathbb{F}[x] $ and $ A_f = c(f) $ be the companion matrix of $ f $. Given $ g \in \mathbb{F}[x] $ show that $ \dim \ker g(A_f) \leq \deg g $. ( Hint: show that $ \deg f - \deg g ...
user avatar
  • 1,585
1 vote
0 answers
24 views

Generating synthetic multivariate time series with stable VAR model

I am trying to generate stable multivariate time series (MTS) using a VAR model. Here I don't try to fit a VAR model on existing data, but to create the data from a VAR process by manually setting the ...
user avatar
  • 11
0 votes
1 answer
122 views

Determinant of sort of companion matrix

Note: I'm trying no EROs and no induction. This is the matrix I'm given: $$A = \begin{bmatrix} b & 0_F & 0_F & \cdots & 0_F & 0_F & a_1\\ -1_F & b & 0_F & \cdots &...
user avatar
  • 1
1 vote
1 answer
46 views

Solving a general ODE

$ \newcommand{\der}[1]{\frac{d^#1 u}{dx^#1}} $ I'm trying to solve $$\der n =\sum^{n-1}_{k=0}\alpha_k\der k +e^{-x}.$$ It can be transformed to an set of equations: $$\begin{cases} u_0'=u_1\\ u_1'=u_2\...
user avatar
  • 1,664
2 votes
2 answers
57 views

Existence of a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$.

Does there exist a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$? I have tried using determinants to see if I can arrive at some sort of contradiction, but it doesn't quite help. Also, I ...
user avatar
1 vote
0 answers
29 views

Companion matrix of matrix valued polynomial coefficients

For a given scalar monic polynomial of the form $p: \mathbb{C} \rightarrow \mathbb{C}, p(x) = c_0 + c_1 x + \cdots + x^n$, one can compute the roots of this polynomial by computing the eigenvalues of ...
user avatar
  • 175
0 votes
0 answers
11 views

Find the first intersection of a vector with a polynomial of order $n$ (only in direction of that vector)

For a point $Q$ and unit vector $\vec u$, I would like to find the first intersection $B$ of the line $\ell$ through $Q$ along $u$ with a polynomial of order $n$ $$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... ...
user avatar
  • 101
0 votes
1 answer
44 views

Characteristic polynomials of powers of a companion matrix in terms of coefficients.

Are there interesting relations between the coefficients of the characteristic polynomial of a companion matrix and the coefficients of (some of) the characteristic polynomials of its (matrix) powers? ...
user avatar
  • 172
1 vote
1 answer
104 views

Is the auxiliary equation of a differential equation related to characteristic polynomial for matrix eigenvalues?

I am taking a course on differential equations and one of the topics is solving second order differentials with the help of an auxillary equation. However one thing that's been bugging me alot is that ...
user avatar
1 vote
1 answer
86 views

Similarity to companion matrix, uniqueness

Let $A \in \mathcal{M}_{n\times n}(\mathbf{R})$ and $b \in \mathbf{R}^n$ with $n\geqslant2$ be given. Now should $(A,b)$ satisfy the Kalman rank condition \begin{equation} \text{rank}[b \,\, Ab \,\, \...
user avatar
  • 607
1 vote
1 answer
31 views

Companion Matrix of first degree monomial $x$

What is the Companion matrix of $x$? Is it $[0]$ or $[1]$?
user avatar
  • 7,072
0 votes
1 answer
35 views

Trying to determine the determinant of an abstract matrix

I'm trying to write about linear homogeneous recurrence relations and I've come up on the following matrix :$$A=\begin{pmatrix} c_1 & c_2 & \cdots & c_{k-1} & c_k\\ 1 & 0 & \...
user avatar
0 votes
1 answer
26 views

Characteristic polynomial $p(x) = (2 + x) (-x) (1-x)$ and possible matrices problem

I'm asked to find, if possible, a non triangular matrix which has $p(x) = (2 + x) (-x) (1-x)$ as its characteristic polynomial. The book doesn't describe any method to do this, and after a while ...
user avatar
  • 147
0 votes
1 answer
62 views

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 ...
user avatar
  • 2,150
0 votes
1 answer
30 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 ...
user avatar
2 votes
1 answer
257 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}...
user avatar
3 votes
2 answers
194 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} &...
user avatar
2 votes
0 answers
114 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 &...
user avatar
  • 25
4 votes
1 answer
257 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 &...
user avatar
  • 855
1 vote
0 answers
32 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\}\...
user avatar
  • 11
1 vote
0 answers
34 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&...
user avatar
  • 470
2 votes
0 answers
105 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)...
user avatar
2 votes
1 answer
550 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 &...
user avatar
  • 4,422
1 vote
0 answers
273 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 ...
user avatar
1 vote
0 answers
52 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 ...
user avatar
  • 11
1 vote
2 answers
117 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 ...
user avatar
  • 3,296
1 vote
1 answer
362 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 $$C=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -2 & 1 & 2\end{pmatrix}$$ ...
user avatar
3 votes
2 answers
430 views

Explicit formula of exponential of companion matrix

Let $$A=\begin{bmatrix} a_k & a_{k-1} & a_{k-2} & \cdots & a_2 & a_1 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & ...
user avatar
  • 159
6 votes
1 answer
339 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 ...
user avatar
  • 3,296
2 votes
5 answers
162 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} ...
user avatar
3 votes
0 answers
621 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 ...
user avatar
  • 10.2k
1 vote
2 answers
1k 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&&\\ &...
user avatar
  • 793
4 votes
2 answers
164 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 ...
user avatar
  • 41
0 votes
1 answer
183 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 ...
user avatar
  • 1,799
11 votes
3 answers
2k views

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 &...
user avatar
  • 2,548
1 vote
0 answers
39 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?
user avatar
  • 280
1 vote
1 answer
992 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 ...
user avatar
  • 4,432
0 votes
1 answer
196 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 ...
user avatar
1 vote
1 answer
178 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 ...
user avatar
  • 489
3 votes
0 answers
359 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}_{...
user avatar
  • 24.1k
0 votes
0 answers
237 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.
user avatar
2 votes
1 answer
80 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 ...
user avatar
  • 24.1k
1 vote
1 answer
161 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 ...
user avatar
  • 1,799
2 votes
0 answers
895 views

Showing a matrix is non-derogatory [duplicate]

Prove that the matrix $$A = \begin{bmatrix} 0 & 1 & \\ \ddots & & \ddots \\ & \ddots & & \ddots \\ & & \...
user avatar
0 votes
1 answer
818 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{...
user avatar