All Questions
31 questions
21
votes
1
answer
2k
views
Almost commuting unitary matrices
Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
- 901
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, ...
- 19.6k
12
votes
2
answers
983
views
Common basis for permutation matrices
How can I check whether there exists a common basis with respect to which two matrices 饾惔 and 饾惖 are permutation matrices?
More explicitly, let $A$ and $B$ be two unitary matrices whose eigenvalues ...
- 295
10
votes
3
answers
680
views
Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$
Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
- 1,769
9
votes
4
answers
1k
views
Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$
For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...
- 7,722
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{麓}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{麓}=\mathbb{Z}^{麓}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
7
votes
2
answers
200
views
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space.
Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
- 431
6
votes
1
answer
205
views
Can a projective solvable group be transitive?
Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
- 11.3k
5
votes
4
answers
3k
views
Parametrization of O(3)
Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
- 95
5
votes
1
answer
503
views
does this set of permutations form a group? And more
Consider the group of $mn\times mn$ permutation matrices $\mathfrak{S}_{mn}$ and partition each such matrix $P$ into $n^2$ blocks of $m\times m$ matrices $Q_{i,j}$. Now, transpose each $Q_{i,j}$ (...
- 43.2k
5
votes
2
answers
335
views
Determinant of the "quantum" version of the group $\mathbb{Z}_n$
Let $[0]_q:=0$ and $[n]_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.
Question. Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries
$$[i+j\bmod n]...
- 43.2k
5
votes
0
answers
203
views
Number of elements in $\mathrm{GL}(n,p)$ with maximal order
I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
5
votes
0
answers
148
views
Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
- 6,351
4
votes
1
answer
204
views
Making Hermitian matrices almost commute
Consider two Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$. I'm interested in finding another Hermitian matrix $A'$ that is close to $A$ and almost commutes with $B$. More precisely, I'd like ...
- 209
4
votes
1
answer
776
views
Finding isomorphism between $\mathbb{Z}^2\ltimes_{A,B} \mathbb{Z}^4$ and $\mathbb{Z}^2\ltimes_{A,C} \mathbb{Z}^4$
Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2a-...
4
votes
0
answers
149
views
Zero diagonal nonsymmetric block checkerboard matrix: orbits and numerical ranges
Let $A \in \mathbb{R}^{m \times m}$ be a nonsymmetric zero diagonal matrix with a zero/non-zero pattern which is symmetric and persymmetric (i.e. symmetric in the northeast-to-southwest diagonal).
If ...
- 323
4
votes
0
answers
163
views
Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring
Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$.
For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
- 7,527
2
votes
2
answers
265
views
Commuting nilpotent matrices and conjugation isomorphisms
Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices.
Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
- 375
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}(\...
- 823
2
votes
0
answers
90
views
decidability special case of column generation problem
I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
- 55
2
votes
0
answers
212
views
Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
2
votes
0
answers
259
views
Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices
How does the spectra of $DU$ change when $D$ runs over all diagonal unitary matrices? Here $U$ is a fixed unitary matrix. Precisely, let spec$(X)$ be a set of eigenvalues of $X$.
For a unitary matrix $...
- 723
2
votes
1
answer
512
views
Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
- 823
1
vote
1
answer
171
views
transposing "unrimmed" permutations
Denote the set of $n\times n$ permutation matrices by $\mathfrak{S}_n$. The ordinary transpose preserves this group.
Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to ...
- 43.2k
1
vote
1
answer
252
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
1
vote
0
answers
213
views
When is the product of regular matrices regular
$\DeclareMathOperator{\GL}{\operatorname{GL}}$We say that a matrix $g \in \GL_n(F)$ is regular if it has a centraliser of minimal dimension, or equivalently, if the minimal and characteristic ...
- 121
1
vote
1
answer
386
views
Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product? [closed]
I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$.
What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product
$$\...
- 131
0
votes
1
answer
450
views
Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
- 1,703
0
votes
1
answer
427
views
SO(3) transformation that produces a reflection [closed]
This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
$H=I_{3}-2v\...
- 3
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
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 ...
- 1,043