Questions tagged [companion-matrices]
For questions about companion-matrices
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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
$$A = \begin{pmatrix}
...
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Comparing values and signs of determinant of companion matrices.
Suppose $A,B \in M_{n}(\Bbb{R})$ such that $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C_{2}\middle|\frac{I}{0\dots0}\right]$ , where $A$ and $B$ have different first columns (...
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minimal and characteristic polynomial of this operator [duplicate]
The following is Problem 18 from Chap8.C of Axler's Linear Algebra Done Right.
Edited to add a transcription of the original problem(in the image)
P18. Suppose $a_0, a_1, ...., a_{n-1} \in \mathbb{C}...
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The unipotent companion matrix over a division ring
Let $D$ be a division ring and let $C(f)$ be a $n\times n$ companion matrix of the monic polynomial $f(x)$ in which the coefficients belong to $D$. I wonder whether $f(x)=(x-1)^n$ when $(C(f)-I)^n=0$. ...
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If $b$ is a cyclic vector, what is significant about $\begin{bmatrix} A - \lambda I & b \end{bmatrix}$?
Problem.
Let $A$ be a complex $n \times n$ matrix and $b$ be a complex $n \times 1$ vector. Prove Rank$\begin{bmatrix} b & Ab & A^2b & \cdots & A^{n-1} b \end{bmatrix} = n$ if and ...
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Converting a monic polynomial of degree $n$ to its companion matrix
Given a polynomial of form $$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$$
and the following information
Let $p$ be indicated as above. Suppose $\lambda$ is a real root in the polynomials p, in other ...
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2
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Is the cyclic vector for companion matrix independent of the specific companion matrix under consideration?
Suppose $C_1, C_2 \in M_n(\mathbb R)$ are two matrices in companion form. If $v$ is a cyclic vector for $C_1$, i.e., $\{v, C_1 v, C_1^2 v, \dots, C_1^{n-1}v\}$ is a basis for $\mathbb R^n$, does this ...
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Solving differential equations given a companion matrix?
So i'm given a differential equation
$$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}$$
...
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Explicit formula of exponential of companion matrix
Let $$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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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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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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Determinant of a companion matrix
I have to find determinant of $$A := \begin{bmatrix}0 & 0 & 0 & ... &0 & a_0 \\ -1 & 0 & 0 & ... &0 & a_1\\ 0 & -1 & 0 & ... &0 & a_2 \\ 0 &...
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Worst case scenarios for companion matrix method
In what case is the companion matrix method for computing the roots of a polynomial expected to suffer accuracy loss. Are nearby roots the main issue?
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Finding similarity transformation matrix for the companion matrix.
Given that $$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},$$ I ...
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Characterization of companion matrices
Let $K$ be an arbitrary field and let $A\in\mathbb{M}_n(K)$. Let $m(X)$ and $p(X)$ be the minimal polynomial of $A$ over $K$ and the characteristic polynomial of $A$ over $K$, respectively.
According ...
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Symmetric part of a companion matrix
I hope this question is sufficiently interesting for you to try to answer it. Sorry if its too trivial, but, working with a few similarty transformations I've arrived to the following conclusion, and ...
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Companion matrices for multivariate polynomials?
I am aware of companion matrices for single variable polynomials:
$$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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