All Questions
Tagged with characteristic-polynomial abstract-algebra
18 questions
0
votes
1
answer
88
views
A $4\times 4$ matrix counterexample. [duplicate]
A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then ...
- 105
1
vote
0
answers
82
views
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
5
votes
0
answers
291
views
Conjecture: Any two matrices of size $n×n$ with same characteristic and minimal polynomial are similar implies $n\le 3$.
Notations:
$\mathcal{M}_n(\Bbb{R}) $: the set of all $n×n$ matrices over $\Bbb{R}$
$\chi_A(x)$: Characteristic polynomial of $A$
$m_A(x)$ : Minimal polynomial of $A$
$A\sim B$ : $\exists P\in Gl_n(\...
- 13.2k
5
votes
0
answers
115
views
Does an algebra automorphism of $M_n(R)$ preserves the characteristic polynomial?
These days, I am playing around Skolem Noether for matrix algebras, and a question arises.
If $F$ is a field , a by-product of Skolem Noether implies that any $F$-algebra automorphism $\rho$ of $M_n(F)...
- 15.7k
0
votes
0
answers
162
views
permutation representation of the symmetric group $𝑆_𝑛$ and its trace
I have this algebra task which I have encountered problems with proving a specific identity for,
Consider the permutation representation of the symmetric group $𝑆_𝑛$, which gives a group ...
- 63
1
vote
1
answer
1k
views
Non-conjugate matrices having the same minimal & characteristic polynomials & the same dimension of the eigenspace
What is the smallest value of $n$ for which there are two non-conjugate $n\times n$ matrices which have the same minimal and characteristic polynomials and eigenspaces of equal dimension?
I first ...
- 187
2
votes
2
answers
184
views
Find characteristic polynomial of $A^2+I$.
Let $A$ be a $3\times3$ matrix with $A^3+A+I_3=0$ with coefficients in
$\mathbb{Q}$. Find the rational canonical form of $A$ and the
characteristic polynomial of $A^2+I_3$.
Let $p(x)=x^3+x+1$. Since $...
- 163
2
votes
0
answers
213
views
Module characteristic polynomial.
I know that if I have a matrix $A,$ the characteristic polynomial is determinant of the matrix $(A-\lambda I)$ where, $\lambda$ is an eigenvalue and $I$ is an identity matrix, and the characteristic ...
user889696
0
votes
1
answer
43
views
How to prove that the characteristic polynomial of this specific matrix is not the power of a linear polynomial?
I was reading an algebra paper, and the problem that appeared to me is the following:
The authors defined a group $G = A \rtimes \left<x\right>$, where $A$ is a finitely generated free abelian ...
- 747
2
votes
0
answers
201
views
Given a 4x4 Singular Matrix. find characteristic polynomial
I'm trying to solve this issue:
Given A, a 4x4 singular Matrix. It is known that $\rho(A+2I)=2$ and $|A-2I| =0$.
Find the characteristic polynomial of A, is A similar to a diagonal matrix?
I've ...
- 411
1
vote
1
answer
338
views
Dummit and Foote 12.2.16: Determining all $2 \times 2$ matrices with entries from $\mathbb F _{19}$ of order $2$
This is exercise 12.2.16 of Abstract Algebra by Dummit and Foote.
Show that $x^5-1 =(x-1)(x^2-4x+1)(x^2+5x+1)$ in $\mathbb F_{19}[x]$. Use this to determine, up to similarity, all $2 \times 2$ ...
user459879
1
vote
1
answer
40
views
The minimal poly of $\sqrt[3]{2}$ over $\Bbb{Q}$ is equal to $\det(T_a - xI)$ where $T_a$ is a matrix over $\Bbb{Q}$ that represents mult. by $a$.
Let $K/F$ be a field extension of degree $n \in \Bbb{N}$ and for each $a \in K$ define $L_a(x) = a x$. Then $L_a(x)$ is an $F$-linear transformation of $K$ as a vector space of dimension $n$. So ...
- 21.6k
4
votes
3
answers
609
views
Prove that every permutation matrix satisfies its characteristic polynomial.
Let $P$ is a permutation matrix which represents the permutation $\sigma\in S_{n}$. Let $\sigma_{1}$,$\sigma_{2}$,...$\sigma_{k}$ denote the disjoint permutations in the cycle form of $\sigma$.
Let $...
- 378
8
votes
2
answers
570
views
Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial.
I was told that
Two permutation matrices represent conjugate permutations iff they have same characteristic polynomial (where the conjugacy is considered only in $S_{n}$).
The first implication is ...
- 378
4
votes
1
answer
55
views
Question involving characteristic polynomial of a linear transformation
I was wanting some hints on a question and I have no idea how to approach this:
Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ ...
- 343
6
votes
3
answers
2k
views
Cyclic Modules, Characteristic Polynomial and Minimal Polynomial
Suppose that $\mathrm{dim}_{F}M<\infty$ for $F$ a field and $M$ an $F$ vector space. Let $T$ be a linear transformation on $M$. Show that $M$ is cyclic (as an $F[x]$ module) if and only if $m(x)$ ...
- 2,082
6
votes
2
answers
715
views
A function from a matrix to its characteristic polynomial.
Maybe it's a known fact, but I would like to know whether the function sending a square matrix of order $n$ to its characteristic polynomial is continuous. If this is true, there are some resources ...
- 23.8k
21
votes
1
answer
982
views
What do characteristic polynomials characterize?
Let $R$ be an integral domain and $F$ a finitely generated free module over $R$. For a linear transformation $\alpha\in\operatorname{End}_R(F)$, the characteristic polynomial is \begin{equation}
p_\...
- 15.2k