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Results tagged with linear-algebra
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user 28104
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
1
vote
Accepted
Convergence on iterating a piecewise function
Let $f$ denote the function described in the question.
The assertion that every trajectory of $f$ except for the one starting at 0
ends in the cycle -1, 1, -1 is equivalent to the Collatz conjecture s …
- 19.6k
15
votes
1
answer
1k
views
Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, su …
- 63.9k
14
votes
Accepted
Identify one group of linear transformations
If I understand your question right, your group $G$ has order $5160960$,
and it has an elementary abelian normal subgroup $N$ of order $2^7$ such
that $G/N \cong {\rm S}_8$. This can be found with GA …
- 19.6k
16
votes
0
answers
779
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 …
CommunityBot
- 1
6
votes
Accepted
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's eve...
There are some quantifiers unclear in your question, but regardless of how
to read it, your assertion is false. -- The smallest counterexample with
blocks of pairwise distinct size all of whose eigen …
- 19.6k
5
votes
Counting matrices over finite fields of a given order
In GAP, you can find the number of elements of order $t$
in ${\rm GL}(2,q)$ by the following function:
NumberOfElementsOfGivenOrderInGL2q := function ( q, t )
return Sum(List(Filtered(ConjugacyClas …
- 19.6k
5
votes
Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$
As Geoff Robinson has already said, the answer to the question is no.
In dimension $4$, there are in total $120$ counterexamples, of which
$96$ have kernel of dimension $1$, and $24$ have kernel of di …
- 19.6k
6
votes
Algorithm for solving systems of linear Diophantine inequalities
GAP provides a function NullspaceIntMat which solves systems
of linear diophantine equations. The documentation says:
25.1-2 SolutionIntMat
* SolutionIntMat( mat, vec ) ───────────────────────────── …
- 19.6k
22
votes
Small-index subgroups of SL(3,Z)
In order to answer the question we need a finite presentation
of ${\rm SL}(3,\mathbb{Z})$ and a general method to find all subgroups of index
$\leq n$ of a finitely presented group:
A finite present …
- 19.6k