All Questions
Tagged with characteristic-polynomial companion-matrices
16 questions
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0
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82
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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 ...
- 1,809
4
votes
1
answer
93
views
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
$$A = \begin{pmatrix}
...
- 55
2
votes
1
answer
753
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minimal and characteristic polynomial of this operator [duplicate]
The following is Problem 18 from Chap8.C of Axler's Linear Algebra Done Right.
Edited to add a transcription of the original problem(in the image)
P18. Suppose $a_0, a_1, ...., a_{n-1} \in \mathbb{C}...
-1
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1
answer
145
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Converting a monic polynomial of degree $n$ to its companion matrix
Given a polynomial of form $$p_A = t^n + a_{n-1} t^{n-1} + \cdots + a_0$$ How can construct the companion matrix $A$ such that $\det(A-tI) = p_A$ assuming that you don't already know how $A$ should ...
- 13
1
vote
1
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210
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Is the auxiliary equation of a differential equation related to characteristic polynomial for matrix eigenvalues?
I am taking a course on differential equations and one of the topics is solving second order differentials with the help of an auxillary equation.
However one thing that's been bugging me alot is that ...
- 612
0
votes
1
answer
31
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Characteristic polynomial $p(x) = (2 + x) (-x) (1-x)$ and possible matrices problem
I'm asked to find, if possible, a non triangular matrix which has $p(x) = (2 + x) (-x) (1-x)$ as its characteristic polynomial.
The book doesn't describe any method to do this, and after a while ...
- 327
2
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0
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156
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Monic polynomial and companion matrix
Problem
Let $p(T) := T^n-\alpha_{n-1}T^{n-1}-\alpha_{n-2}T^{n-2}-\cdots-\alpha_0 \in K[T]$.
Additionally we have the companion matrix of $p$
$$A:= \begin{bmatrix}
0 & 1 & 0 & 0 &...
- 25
3
votes
1
answer
899
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
$$T = \left[
\begin{matrix}
0_n &...
- 10.4k
1
vote
1
answer
201
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Polynomials $\&$ Matrices
Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows
$$
P(x)=\det (A-x\,I)\, .
$$
Where $I$ is an identity matrix of order $n$. What ...
- 1,887
2
votes
0
answers
899
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Showing a matrix is non-derogatory [duplicate]
Prove that the matrix
$$A = \begin{bmatrix}
0 & 1 & \\
\ddots & & \ddots \\
& \ddots & & \ddots \\
& & \...
- 6,728
2
votes
2
answers
414
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Find the characteristic polynomial of this matrix
I've tried to find the characteristic polynomial of the following matrix
$$A=\begin{pmatrix}
0 & 0 & \cdots & 0 & -a_n \\
1 & \ddots & \ddots & \ddots & \...
- 65
2
votes
2
answers
4k
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Eigenvalues of a companion matrix
I've been tasked with the following:
Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$.
Show that if $\lambda$ is an ...
- 411
2
votes
2
answers
3k
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How do I find the characteristic polynomial and eigenvalues?
For the following matrix, compute
its characteristic polynomial
its eigenvalues
$$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$
So I think I know ...
- 357
1
vote
2
answers
233
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Eigenvalues of negative companion matrix
Here's a homework question I've been stuck on for a while.
Given
$$A = \begin{bmatrix}
0 & 0 & 0 & \cdots & 0 & a_0 \\
-1 & 0 & 0 & \cdots & 0 & a_1 \\
...
- 754
3
votes
1
answer
2k
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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 $...
- 185
10
votes
1
answer
20k
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Characteristic polynomial of companion matrix [duplicate]
I have a matrix in companion form,
$$A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots &...
- 3,518