All Questions
Tagged with matrices semigroups-and-monoids
7 questions
21
votes
2
answers
3k
views
Integer matrices with no integer eigenvalues
Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
- 1,045
12
votes
0
answers
321
views
Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
4
votes
1
answer
134
views
Joint spectral radius of $\{M,M^T\}$
Let $F$ be a bounded subset of ${\bf M}_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F_k$ the set of all products of $k$ ...
- 52.3k
2
votes
0
answers
119
views
The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
1
vote
0
answers
70
views
Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 883