All Questions
10
questions
0
votes
1
answer
30
views
Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for all $ j $.
Let $ f(x) = (x - x_1) \cdots (x - x_n) $ be an $ n $-degree monic irreducible polynomial with integer coefficients. Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for ...
- 11
1
vote
0
answers
45
views
Show that $A=cI$ for c non-zero real number.
We have A 2×2 matrix with real numbers. We know that there is an odd n such that $A^{n}=I$. Show that $A=cI$ for c non-zero real number.
We can find easily that $det(A)=1$. I took the polynomial $P(x)=...
- 1,311
2
votes
1
answer
128
views
Prove that the determinant is irreducible
Lemma. Let $Y=(y_{ij})$ be a $n\times n$ matrix. Then $\det Y$ is an irreducible polynomial of $y_{ij}.$
Proof. Consider a special case $Y=Y_1=tE_n+\sum\limits_{j=1}^nx_je_{j,j+1}$ where $e_{j,j+1}$ ...
- 1,313
0
votes
1
answer
46
views
Is there a matrix $A \in \mathbb{Q}^{n \times n}$ such that $P(A)=\mathbf{0}$, where $P$ is a monic polynomial with rational coefficients?
Given a monic polynomial of degree $k \leq n$ with rational coefficents, can I always find a matrix $A \in \mathbb{Q}^{n \times n}$ that is a root of $P$?
What I have already tried:
Using the Cayley-...
- 524
2
votes
1
answer
47
views
Does a matrix exist whose eigenpolynomial equals to any polynomial $f(x)$?
For any polynomial $f(x) = \sum _{i = 0} ^{n} a_i x_i $ $(n \ge 1, a_n = 1)$, does a matrix $A$ exist so that $\left \vert \lambda I - A \right \vert = f(\lambda)$ ?
I verified some cases:
For $n = ...
- 137
4
votes
1
answer
184
views
On reducibility over $\mathbb{Z}$ of a special class of polynomials .
Definitions
Let positive integer number $n$ be referred to as $p$-composite if there exist such positive integer numbers $k_1$ and $k_2$ that
$$
n=k_1+k_2+2k_1k_2\equiv k_1*k_2.
$$
Let $\mathbb{K}_n$...
- 26.7k
1
vote
1
answer
59
views
Polynomials over $k$ and over $M_n(k)$
For any field $k$, let $M_n(k)$ denote the ring of $n\times n$ matrices over $k$. Out of curiosity, do we know a way to relate polynomials (especially irreducible polynomials) in $k[x]$ with ...
- 2,965
1
vote
0
answers
40
views
Symmetric matrices of indeterminates and an irreducible polynomial
Let $W =[w_{ij}] $ be a symmetric matrix of indeterminates. I wonder if the polynomial $p(W) := \mathrm{tr}(W^3)-r_0$, $r_0 \in \mathbb{R}$ is irreducible over reals.
- 965
1
vote
0
answers
24
views
$A \times B^{-1}$ has irreducible characteristic polynomial when $A,B$ are random integer matrices -- simple proof?
Let $A,B$ be $d\times d$ integer matrices with each entry drawn uniformly from $[0,2^n)$, and define the rational matrix $C = A \times B^{-1}$. Is there a simple way to prove that $C$'s characteristic ...
- 105
2
votes
1
answer
107
views
Nondiagonalizable Matrix and Polynomials
I got the following problem:
If $A$ is a nondiagonalizable square matrix of order $n$ over field $\mathbb{F}$ then there exists a polynomial $P$ of degree $n-1$ over $\mathbb{F}$ such that $(P(A))^2=...
- 2,517