Questions tagged [companion-matrices]

On the companion matrix to a certain monic polynomial.

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Clarification on Companion matrix

I am studying Jordan forms. I see that Hoffman & Kunze define the companion matrix of a polynomial $a_0+a_1x+\dots+a_{k-1}x^{k-1}+x^k$ as $$ \begin{pmatrix} 0 & 0 & 0 & \dots &0 &...
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polynomial root finding with companion matrix

I understand the logic behind the polynomial rootfinding algorithms that use companion matrices. I checked the numpy implementation and it's pretty straightforward but I don't understand this comment: ...
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Solution of $Ax = x$ where $A$ is a companion matrix

I am looking for conditions under which the only solution to the equation $Ax = x$, where the matrix $A$ is a companion matrix \begin{equation} A = \begin{pmatrix} 0 & 1 & 0 & \cdots &...
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Reading off module properties from the companion matrix

Let $P\in \mathbb{F}[x]$ be a monic polynomial of degree $n$ over a field $\mathbb{F}$, and $M_P$ its companion matrix. The matrix $M_P$ gives a module of $\mathbb{F}[x]$ on $\mathbb{F}^n$, by letting ...
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Matrix with invariant subspace

Find the 5×5 real matrix $ A$ such that: $ A(1\ 1\ 1\ 1\ 1)^T =(1\ 1\ 1\ 1\ 1)^T $ $ A^{-1} = A^T $ $ A^6 = A $ $ A $ has distinct eigenvalues. If $ W $ is the subspace generated by $ (1\ -1\ 0\ 0\ 0)...
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Is every subspace can be represented as T-cyclic for some T? [closed]

Definition of $T$-cyclic subspace: Let $T$ be an linear operator on $V$. Take $v \in V$. The subspace generated by $\text{span}(\{v,T(v),T^2(v),\cdots\})$ is called $T$-cyclic subspace generated by $v$...
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An interesting way to calculate the powers of a companion matrix

Let $m\geq 3$ be a positive odd number and let $M$ be the $m\times m$ matrix defined by $$M=\begin{bmatrix}0&1&0&0&\cdots&0\\ 0&0&1&0&\cdots &0\\ 0&0&0&...
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How to verify this matrix identity?

Let $m\geq 3$ be a positive odd number and let $M$ be the $m\times m$ matrix defined by $$M=\begin{bmatrix}0&1&0&0&\cdots&0\\ 0&0&1&0&\cdots &0\\ 0&0&0&...
Zuriel's user avatar
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Cube roots of the identity matrix

Describe the set of cube roots of the identity matrix in $M_{n \times n}(\mathbb{R})$. My attempt: Let $I_n$ denote the identity matrix of dimension $n \times n$. To find the cube roots of $I_n$, one ...
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Theorem 2, Section 7.1 of Hoffman’s Linear Algebra

Definition: Suppose $p=x^k+\sum_{i=0}^{k-1}a_i\cdot x^i\in F[x]$ is a monic polynomial. Then companion matrix of $p$ is $$\begin{bmatrix} & & & -a_0\\ 1& & &-a_1\\ &\ddots &...
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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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How to calculate the eigenvalues of $A^\ast A$ where $A$ is a companion matrix?

Let $$A=\left[\begin{array}{cc}0&-a_0\\I_{n-1}&\xi\end{array}\right]$$ be a companion matrix where $\xi=[-a_1\ \cdots\ -a_{n-1}]^T\in \mathbb{C}_{n-1}$. Hence, $$A^\ast A=\left[\begin{array}{...
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Companion matrix must have one Jordan block per eigenvalue: recurrence sequence perspective

Please let me know if anything is unclear in my post, as it is not recieving any responses yet, I would provide additional explanation if needed. Also if you have any idea for my problem below please ...
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Characteristic polynomial of A and -A (where A is a companion matrix)

Is there something we can say about the characteristic polynomial of $A$ and $-A$ where $A$ is a $n \times n$-matrix; $A$ is a companion matrix? I have found an example where $$A = \begin{pmatrix} ...
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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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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}...
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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 ...
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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 ...
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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)=...
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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 ...
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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 ...
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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 &...
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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\...
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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 ...
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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 ...
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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? ...
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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 ...
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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 \,\, \...
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Companion Matrix of first degree monomial $x$

What is the Companion matrix of $x$? Is it $[0]$ or $[1]$?
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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 & \...
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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 ...
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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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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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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} &...
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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 &...
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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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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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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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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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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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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 $$C=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -2 & 1 & 2\end{pmatrix}$$ ...
Undergrad2019's user avatar
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2 answers
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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 & ...
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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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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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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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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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