0
votes
1answer
83 views

Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation. I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
3
votes
0answers
102 views

Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows: For a ...
0
votes
0answers
86 views

Linear system with many solutions from a finite set

Basically I am looking for a linear system with many solutions from a finite set. Choose a finite set of rationals $S$ and fix positive integer $k$. Let $A$ be a linear system with $n$ variables ...
1
vote
0answers
52 views

A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...
7
votes
2answers
875 views

Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...
2
votes
2answers
175 views

Obstructions to Smith normal form of a special type

Suppose I have a Smith normal form $S,$ and I want to have an $M \in SL(n, \mathbb{Z}),$ such that $M - I$ has SNF $S.$ Is this always possible? For a (potentially) somewhat harder question, what if I ...
8
votes
1answer
523 views

How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
7
votes
1answer
776 views

A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity: Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
3
votes
2answers
200 views

Minkowski successive minima inequality for a lattice base?

Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors. The Minkowski successive minima inequality says ...
4
votes
3answers
299 views

Algebraically Independent Numbers and Affine Linear Maps

Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in ...
2
votes
0answers
68 views

decomposition according to embeddings

Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and $L/\mathbb{Q}$ a finite extension. Assume that $V$ is equipped with a structure of $L$-vector space. Then there is a decomposition ...
12
votes
0answers
515 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
3
votes
1answer
140 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M=\lbrace (m_{ij})\rbrace_{i=1,...,n,\ j=1,...,k}$, where $m_{ij}$ is the greatest common ...
2
votes
1answer
228 views

Galois deformations with Panchiskin condition

Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
3
votes
1answer
304 views

Fields whose embeddings into the complex numbers are invariant under complex conjugation

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...
3
votes
2answers
427 views

Average size of determinants of integer matrices?

I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are ...
3
votes
3answers
785 views

Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$ Is it true or false that a combination of two (or more, in general) of the ...
19
votes
4answers
1k views

The sum of same powers of all matrices modulo p

The following is a problem from our department algebra competition for students: Non-question. An experimental-math geek was trying to raise all matrices $17\times17$ over the field with 17 ...
13
votes
1answer
530 views

Free subgroups of GL(2,Z)

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle < {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, ...
12
votes
1answer
310 views

Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
3
votes
1answer
208 views

Counting integer matrices with specific values of minors

Are there any results concerning the following problem: Count all $n \cdot (n + m)$ integer matrices ($m \ge 1$) with norm ($\|A\| = \max{|a_{ij}|}$ or $\|A\| = \sqrt{\sum{a_{ij}^{2}}}$ ) less than ...
4
votes
1answer
210 views

power log distance between matrices

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
3
votes
3answers
472 views

Eigenvectors of the Fourier matrix

Let $n$ be a positive integer. The $n$ by $n$ Fourier matrix may be defined as follows: $$ F^{*} = (1/\sqrt{n}) (w^{(i-1)(j-1)}) $$ where $$ w = e^{2 i \pi /n} $$ is the complex $n$-th root of ...
5
votes
1answer
388 views

determining symplecticity (if that's a word)

Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the ...
3
votes
3answers
677 views

Canonical form of symmetric integer matrix M

Let $M$, $N$ be a symmetric matrix over a ring $R$. $M$ and $N$ are said to be equivalent if there exist an invertible matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose ...
5
votes
2answers
394 views

Stable Conjugacy for Integer Matrices

Let $F$ be a field, and $E$ an extension field. Then two matrices in $GL_n(F)$ are conjugate if and only if they are conjugate in $GL_n(E)$. I'm curious whether the analogous fact holds for rings of ...
7
votes
3answers
541 views

the largest eigenvalue of the matrix A with A_{ij}=(i \times j) mod p for p is a prime.

For a prime p, consider the $(p-1) \times (p-1)$ matrix A with entry to be $A_{ij}=(i \times j) mod$ $p$. every row (column) is permutation of 1 to p-1, such a permutation is useful in one version of ...
1
vote
2answers
351 views

Singular matrices with integer entries

I am motivated by the following paper by Greg Martin and Erick B. Wong: http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf Here the authors prove that assuming that the entries of an $n ...
3
votes
0answers
319 views

Linear independence over Q of logarithmic powers of prime numbers

I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...) Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$. My question concerns a ...
5
votes
6answers
811 views

How many 0, 1 solutions would this system of underdetermined linear equations have?

The problem: I have a system of N linear equations, with K unknowns; and K > N. Although the equations are over $\mathbb Z$, the unknowns can only take the values 0 or 1. Here's an example with N=11 ...
0
votes
1answer
177 views

Conditions to the existence of a solution of a system of congruences [closed]

Let $p$ be a prime. Consider the following congruences: $$ \begin{array}{lcl} a_1 x & = & c_1 (\text{mod } p) \\\\ \vdots & & \vdots\\\\ a_n x & = & c_n (\text{mod } p) \\ ...
4
votes
0answers
194 views

Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
14
votes
3answers
988 views

symmetric integer matrices

Suppose I have a symmetric positive definite matrix $M$ with integer entries. I want to decide whether $M = A A^t,$ with $A$ likewise integral. I assume that decision problem is NP-complete, as is the ...
8
votes
0answers
402 views

integral matrix of order p

Hi everyone Let $p$ be a prime number. I am interested to classify $\{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}$ up to conjugacy. One reason to consider this problem is its relation to ...
9
votes
6answers
2k views

Status of the Hadamard Circulant conjecture

The following feels like a community wiki question, so I do it here: Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser (a long standing difficult conjecture): ...
6
votes
0answers
645 views

Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
1
vote
0answers
155 views

Sum of two free o-submodules in a vector space over a local field

Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$. Given two free ...
4
votes
0answers
329 views

Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$. Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
11
votes
0answers
565 views

Regular languages of matrices and their generating functions

My question is somewhat related to this question. Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
4
votes
2answers
396 views

A mapping from a lattice to itself

Consider $\mathbb{Z}^{n}$ for $n = 2^r$ where $r \geq 1$ . Look at the iterates of the following function $T$ from $\mathbb{Z}^n$ to itself. $T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, ...
0
votes
1answer
446 views

Rational solutions of homogeneous equations

Can every solution of a homogeneous linear system be approximated by a solution in rational numbers? In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an ...
8
votes
0answers
892 views

roots of quadratic forms

This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
27
votes
3answers
2k views

Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
-3
votes
2answers
2k views

Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + ...
10
votes
4answers
1k views

Rational congruence of binomial coefficient matrices

Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on. Say that two matrices $A$ and $B$ over the ...
11
votes
2answers
2k views

Prime/undecomposable matrices

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
6
votes
4answers
3k views

Positive solutions of linear Diophantine equations

Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...
6
votes
1answer
1k views

Is there good intution of the trace map?

I have never understood the trace map,not even after reading Geometric Interpretation of Trace. The problem with many answers in the above discussion is the geometric intuition does not apply to other ...
53
votes
6answers
5k views

Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes: Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...
12
votes
2answers
804 views

Semi-linear operators

If $V_1$ and $V_2$ are finite-dimensional vector spaces over a field $E$, each equipped with an $E$-linear operator $\phi$, we can tell if $V_1$ and $V_2$ are isomorphic as $\phi$-modules by comparing ...