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2 votes
0 answers
137 views

Decompose a rational matrix as an integer matrix and an inverse of integer matrix

Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
ghc1997's user avatar
  • 823
3 votes
2 answers
270 views

Generating $\mathbf{PGL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Q})$

It is well known that $\mathbf{PGL}_2(\mathbb{Z})$ is finitely generated, and that $\mathbf{PGL}_2(\mathbb{Q})$ isn't. My question is: what is a fast, natural way to see these properties without ...
THC's user avatar
  • 4,547
2 votes
0 answers
102 views

Decomposition of a 4D rotation into a particular sequence of simple rotations

I asked this question in math.stackexchange two days ago, but no one has answered yet. I suspect it is "hard enough" that it is appropriate to post it here as well. I am new to stackexchage, ...
3Brown1Blue's user avatar
0 votes
0 answers
92 views

Classification of elements $GL(d, \mathbb{R})$

Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here. Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
Adam's user avatar
  • 1,043
17 votes
2 answers
1k views

Are unitarily equivalent permutation matrices permutation similar?

Two matrices $A, B \in \mathbb R^{n \times n}$ are called unitarily equivalent if there exists an unitary matrix $U \in \mathbb C^{n \times n}$ such that $A = U B U^{\ast}$. If in addition $U$ is a ...
shuhalo's user avatar
  • 5,327
13 votes
2 answers
414 views

Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?

Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
Qfwfq's user avatar
  • 23.3k
3 votes
1 answer
869 views

Regular semisimple elements in $SL(n,q)$

Consider $G=GL(n,q)$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such ...
Tree's user avatar
  • 117
4 votes
0 answers
1k views

Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
wonderich's user avatar
  • 10.5k
2 votes
2 answers
565 views

Regarding minimal elementary generators for $GL(n, \mathbb{Z})$

I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
Bogdan Luchian's user avatar
8 votes
2 answers
336 views

What is the most efficient way to factor a matrix into a given set of generators?

I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general ...
j0equ1nn's user avatar
  • 2,436
7 votes
2 answers
1k views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More ...
expmat's user avatar
  • 1,271