0
votes
1answer
13 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
2
votes
2answers
30 views

diagonalization of a matrix over finite fields

I'm having a problem with determine whether a matrix is diagonalizable over $\mathbb F_{2}$, over $\mathbb F_{3}$, etc. for example, for the following matrix: $$ \begin{bmatrix} 1 ...
1
vote
1answer
27 views

Creating a matrix such that all the sub-matrices are max rank

Let $A\odot B$ denote the elementwise multiplication of matrices $A$ and $B$. Given a binary matrix $B_{m \times n}=[b_{ij}]$, $b_{ij} \in \{0,1\}$, I want to find a matrix $A=[a_{ij}]$, $a_{ij}\in ...
0
votes
0answers
44 views

Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
0
votes
0answers
22 views

matirx of Frobenius map

Galois group of GF(8) is a cyclic group of order 3 . How is the matrix representation of generator of that? It is clear that its generator is Frobenius map? I think it is 3*3 matrices , one of them ...
0
votes
1answer
58 views

Calculating characteristic polynomials of matrices in GF(2)

How do you calculate a characteristic polynomial of a matrix in GF(2)? I understand the concept of characteristic polynomials in matrices using "regular" math with real numbers, but I'm a bit confused ...
0
votes
0answers
14 views

Univariate and Linear Representation Lemma

I'm trying to understand the proof of the lemma: Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{K}$ an extension field of degree $n$ of $\mathbb{F}$. Let $A$ be a linear mapping ...
0
votes
1answer
90 views

Count the number of rational canonical form&find similarity classess

For a finite field $F=F_q$ having $q=p^d$ elements ($p$ a prime integer), compute the number of similarity classes in the vector space $M_n(F)$ of $n\times n$ matrices over $F$. (Maybe count the ...
18
votes
1answer
169 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
2
votes
1answer
177 views

Subring of a matrix ring over a finite field?

I am trying to find a subring $S$ of the matrix ring $M_{p\times p}(\mathbb{F}_{q})$ over a finite field, where $p$ and $q$ are prime numbers . This subring must sastify the following three ...
0
votes
1answer
150 views

Matrix over a finite field?

I am trying to solve the following problem: Given is a $3\times 3$ matrix $M$ over $\mathbb{F}_{7}$, such that for every vectors $v,w\in \mathbb{F}_{7}^3\setminus \{0\}$ there exists an integer $n$ ...
1
vote
1answer
55 views

Matrix polynomial factorization

This is about exercise 1207 from the book "Problems and Solutions in Mathematics", 2nd edition, by Ta-Tsien. Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over the finite field ...
4
votes
1answer
435 views

Find the number of n by n matrices conjugate to a diagonal matrix

(a) Find the number of matrices of size $n$ by $n$ over the field of two elements which are conjugate to a diagonal matrix. What is the answer for $n = 4$? (b) What is the number of $n$ by $n$ ...
4
votes
1answer
107 views

Conjugacy classes and orders of matrices.

The following are prime decompositions in $\Bbb{Z}_7[x]$: $x^8+1= (x^2-x-1)(x^2+x-1)(x^2+3x-1)(x^2+4x-1)$ $x^4+1= (x^2+3x+1)(x^2+4x+1)$ (a) Give representatives for the conjugacy classes of ...
2
votes
0answers
48 views

counting symmetric nilpotent matrices

In a recent paper [ Counting symmetric nilpotent matrices , by A. Brouwer], the author states that the number of 3x3 symmetric nilpotent matrices over the field of q elements is given by the ...
4
votes
3answers
106 views

Order of matrices in $SL_2({\mathbb{F}_q})$

Could you tell me how to prove that in $SL_2({\mathbb{F}_q})$ the only element of even order is $-I$ ($ \ I$ - identity matrix)? I would really appreciate a thorough explanation, because I cannot ...
2
votes
1answer
449 views

Rank of matrices multiplication

Matrices $m_1$ and $m_2$ are over a finite field($GF(2^{8})$ for example). $m_1$ is a $m\times n$ matrix($n > m$) with $rank(m_1) = m$, and $m_2$ is a $n\times c$ ($c > n > m$) matrix ...
1
vote
0answers
44 views

root of binary matrix

There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication. Is there any way to find ...
3
votes
3answers
199 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
-1
votes
3answers
273 views

Why a full-rank matrix in a finite field is also full-rank in a expanded finite field?

For example, matrix MAT is full-rank in $GF(2^8)$, why MAT is also full-rank in $GF(2^{16})$ and $GF(2^{32})?$ Thanks
3
votes
2answers
105 views

Group does not contain any elements of order $p^2$?

I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$. ...
1
vote
1answer
261 views

program to find matrix group given generators (Matlab)?

Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group? The best I've ...
1
vote
1answer
808 views

hash function for matrices over finite field (Matlab)?

What is a good hash function for small nonsingular matrices over a field $\mathbb{F}_p$ for $p$ prime? I'm looking for an integer function which is close to being injective (but not necessarily ...
0
votes
1answer
616 views

Traceless matrices and commutators

Any traceless $n\times n$ matrix with coefficients in a field of caracteristic $0$ is a commutator (or Lie bracket) of two matrices. What happens when the field has positive caracteristic? When ...
3
votes
2answers
171 views

Example of matrix $M\in GL_3(\mathbb{Z}/7\mathbb{Z})$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is 3

I would want to make an example of a matrix $M \in GL_3(\mathbb{Z}_7)$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is $3$. Any hints how to do that ...
10
votes
1answer
373 views

Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...
0
votes
1answer
390 views

How many orthogonal matrices are there over a given finite ring or field?

I want to know how many $2\times 2$ orthogonal matrices exist over the ring $\mathbb{Z}_n$ or the field $\mathbf{F}_p$. And how many $2\times 1$ orthogonal vectors exist over the ring $\mathbb{Z}_n$ ...
18
votes
5answers
1k views

Every Function in a Finite Field is a Polynomial Function

From a bank of past master's exams I am going through: Let $F$ be a finite field. Show that any function from $F$ to $F$ is a polynomial function. I know that finite fields are fields of $p$ ...