# Tagged Questions

**3**

votes

**0**answers

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

**0**answers

85 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

**0**answers

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

**2**answers

857 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

**2**answers

174 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

**1**answer

498 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

**1**answer

772 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

**2**answers

193 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

**3**answers

296 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

**0**answers

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

**0**answers

511 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

**1**answer

136 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

**1**answer

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

**1**answer

298 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

**2**answers

422 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

**3**answers

762 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

**4**answers

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

**1**answer

524 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

**1**answer

307 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

**1**answer

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

**1**answer

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

**3**answers

459 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

**1**answer

379 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

**3**answers

665 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

**2**answers

390 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

**3**answers

540 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

**2**answers

345 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

**0**answers

309 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

**6**answers

798 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

**1**answer

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

**0**answers

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

**3**answers

984 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

**0**answers

401 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

**6**answers

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

**0**answers

638 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

**0**answers

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

**0**answers

328 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

**0**answers

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

**2**answers

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

**1**answer

444 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

**0**answers

885 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

**3**answers

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

**2**answers

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

**4**answers

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

**2**answers

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

**4**answers

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

**1**answer

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

**6**answers

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

**2**answers

802 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 ...

**1**

vote

**1**answer

114 views

### an exercise on integrality of characteristic polynomials

Suppose A is a matrix with coefficient in $Q_{\ell}$, and all the coefficients of its char. polynomial are in $Z$ (thus an integral polynomial). Prove that the char. polynomial of $A^n$ is also ...