All Questions
Tagged with ac.commutative-algebra matrices
21 questions with no upvoted or accepted answers
7
votes
0
answers
658
views
Row rank and column rank of matrix with entries in a commutative ring
Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is ...
- 356
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
4
votes
0
answers
62
views
Is there a $3\times 3$ matrix over a Dedekind domain not similar to a matrix with zero top right entry?
Let $R=\mathbb{Z}[\sqrt{-5}]$, which is well known to be a Dedekind
domain but not a PID. Let $\mathrm{M}_{3}(R)$ be the set of $3\times3$
matrices over $R$. Does there exist a matrix $A\in\mathrm{M}_{...
- 3,813
4
votes
0
answers
208
views
An operator derived from the divided difference operator $\partial_{w_0}$
Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
- 1,997
3
votes
0
answers
43
views
Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
- 213
3
votes
0
answers
97
views
Minimal localization need it to "diagonalize" a matrix
Let $A$ be an $n\times n$-matrix over $\mathbb Z[t^\pm]$. In general doesn't exist $P,Q\in GL(n,\mathbb Z[t^\pm])$ such that $PAQ$ is a diagonal matrix (this happens cause $\mathbb Z[t^\pm]$ is not a ...
3
votes
0
answers
210
views
the annihilator of cokernel in a particular case
Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...
- 2,232
3
votes
0
answers
197
views
A similarity problem over matrices over Gaussian integers
Let $R = \mathbb Z[\sqrt{-1}]$ and
$$\Omega = \{X \in GL_4(R) : X \overline X = I_4 \text{ or } -I_4 \},$$
where $\overline X$ is the complex conjugate matrix of $X$.
Two matrices $A, B \in \Omega ...
- 1,841
2
votes
0
answers
117
views
A very specific quotient of a determinantal variety
I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
- 185
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
2
votes
0
answers
97
views
How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices
Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
- 131
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
2
votes
0
answers
87
views
How to prove this matrix is idempotent and that it obeys a telescoping identity
Let $P_k$ be the size $k$ leading principal submatrix of the real-valued $N \times N$, invertible matrix $M$, and let $Q_k$ be the size $N$ matrix having $P_k^{-1}$ in the top left corner, and zeroes ...
- 51
2
votes
0
answers
113
views
compute conjugacy classes of matrices over $\mathcal{Z}$
Given an irreducible polynomial $f(X)\in\mathbb{Z}[X]$, do you know an efficient algorithm to compute the number of conjugacy classes of matrices $A\in M_n(\mathbb{Z})$ with characteristic polynomial $...
- 389
2
votes
0
answers
193
views
Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)
Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\...
- 169
1
vote
0
answers
77
views
$M^2=0$ defines a Koszul algebra ? What if $M$ is Manin's endomorphism's of Koszul $A$ ? (Here $M^2=\sum_k M_{ik}M_{kj}$ - resembles $d^2=0$).)
Consider a matrix $M$ which elements $M_{ij}$ are generators of some algebra $K$,
impose new relations: $M^2=0$ and get a new algebra $K_{2}$.
Question 1: Is it true that $K_2$ is Koszul algebra when ...
- 24.9k
1
vote
0
answers
147
views
Gelfand's representation on matrices: construct maximal ideal in matrix algebra
I would like to see a constructive proof (some algorithm?) of the following statement:
Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
- 807
1
vote
0
answers
179
views
Matrix factorizations over $GL_2$ of a real quadratic ring of integers
tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...
- 1,090
1
vote
0
answers
67
views
Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
- 553
1
vote
0
answers
95
views
Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
1
vote
0
answers
221
views
Centralizer in a matrix algebra over commutative polynomials
Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative
polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$.
I would like to know what is the ...
- 179