12
votes
1answer
394 views

Does the Hasse principle hold for the square root problem on symmetric matrices?

Let $S$ be an $n \times n$ symmetric matrix with rational entries. It is known that the equation $XX^T=S$ has a solution in $\text{M}_n(\mathbb{Q})$ if and only if it has a solution in ...
3
votes
0answers
83 views

Large prime divisors in entries of matrix powers

Are any examples known of an integer matrix $A$, such that the largest prime divisor of some specified entry of $A^n$ grows exponentially in $n$? How about where it just grows strictly faster than any ...
3
votes
2answers
588 views

A problem about Determinant of sum of permutation matrices

Let $w_1$ and $w_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w_1(i)\neq w_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two ...
14
votes
1answer
379 views

Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem: What is the volume of the largest symmetric convex subset ...
13
votes
1answer
529 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, ...
6
votes
2answers
934 views

Eigenvalues of nonnegative integer matrices

Edit I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post: What are the possible eigenvalues of nonnegative integer matrices? Any answer ...
2
votes
3answers
1k views

How many matrices are possible for the given arrangement?

Given m & n, we have to find out the number of possible matrices of order m*n with the property that A(i,j) can be either 0 or 1 and that no contiguous sub-matrix of both length > 1 & breadth ...
8
votes
1answer
545 views

Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...
10
votes
3answers
757 views

Congruence subgroups as abstract groups

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup $$ \pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & ...
4
votes
1answer
315 views

Is that series-transformation known in the context of divergent summation?

Note: I asked this question in math.stackexchange but did not receive an answer Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular ...
3
votes
2answers
328 views

Rank of sum of Galois conjugates of a matrix

Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in ...
7
votes
4answers
493 views

If the series Σ pᵃ⁽ʷ⁾·xᴵʷᴵ is rational, is Σ a(w)·xᴵʷᴵ also rational (summation over words w in a regular language)?

Let $p$ be a prime number and let $a_i$ be a sequence of natural numbers such that the series $\sum_{i=1}^\infty p^{a_i} x^i$ is rational. A warm-up question: Question 1. Does it follow that the ...
1
vote
3answers
449 views

structure of singular matrices whose entries have modulus one

Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1. Suppose the matrix $A$ is singular. We will assume without loss of generality that all the entries in the first row and the ...
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 ...
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 ...
7
votes
1answer
461 views

What matrix groups can be embedded in $Sp_4$?

In a joint paper with Yifan Yang we constructed an "exotic" embedding of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$), namely, $$ ...
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. ...
17
votes
2answers
1k views

Lifting matrices mod 2 to integers.

The following question was motivated by my research. Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
4
votes
0answers
477 views

An inverse eigenvalue problem on Jacobi matrices

I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
18
votes
11answers
3k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
9
votes
1answer
290 views

An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...