All Questions
Tagged with ac.commutative-algebra matrices
55 questions
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
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
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 1,446
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 (...
- 12.9k
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
- 16.2k
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
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 ...
- 63.9k
6
votes
1
answer
260
views
Vanishing linear combinations of minors
Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
- 7,300
5
votes
1
answer
265
views
Is every matrix involution over a UFD diagonalisable?
Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}_n(A)$ diagonalisable?
This is of ...
- 2,928
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
198
views
What are the properties of this set of infinite matrices and operations on them?
Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & . \\
0 & a_0 & a_1 & a_2 & . \\
0 & 0 & a_0 & a_1 & . \\
...
- 11.2k
3
votes
1
answer
102
views
Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)
Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $...
- 6,262
1
vote
1
answer
209
views
Primary decomposition of huge ideals using M2/Singular
I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
- 71
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ 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
1
vote
1
answer
227
views
If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses?
Let $L$ and $M$ be matrices over a commutative ring $R$ equipped with an involution "$*$". Define $L \oplus M$ (the "direct sum" of $L$ and $M$) to be $\begin{bmatrix}L & 0 \\ ...
- 1
6
votes
1
answer
208
views
Cayley-Hamilton over super rings
If $R$ is a commutative ring, then the Cayley-Hamilton theorem states that any endomorphism $\phi: R^{n} \rightarrow R^{n}$ of a rank $n$ free module satisfies its own characteristic polynomial, in ...
- 63.9k
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
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
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
0
votes
1
answer
453
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 $x$ ...
- 3,009
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
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 ...
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 ...
- 63.9k
0
votes
1
answer
243
views
A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]
For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
CommunityBot
- 1
4
votes
1
answer
127
views
On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
- 2,784
2
votes
1
answer
244
views
radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices
Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set
$$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
- 5,039
1
vote
0
answers
67
views
Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
- 553
2
votes
0
answers
87
views
How to prove this matrix is idempotent and that it obeys a telescoping identity
Let $P_k$ be the size $k$ leading principal submatrix of the real-valued $N \times N$, invertible matrix $M$, and let $Q_k$ be the size $N$ matrix having $P_k^{-1}$ in the top left corner, and zeroes ...
- 63.9k
11
votes
1
answer
949
views
Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
CommunityBot
- 1
1
vote
0
answers
95
views
Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
3
votes
1
answer
236
views
For $T$ the $2\times 2$ triangular matrices over $R$, can we write $GL_2(T)=U(T)E_2(T)$?
Let $R$ be a commutative ring with identity, and let $T = T_2(R)$ be the ring of $2\times 2$ upper triangular matrices over $R$. Is it true that the following identity holds?
$$GL_2(T)=U(T)E_2(T)$$
...
- 7,893
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
3
votes
1
answer
629
views
Determinant of block matrix
I expect this to be true and proven, but I can't find any proofs of this. So anyone can confirm or deny this?
Let $R$ be a commutative ring, and let $M$ be a $kn\times kn$ matrix, which can be split ...
- 4,991
1
vote
1
answer
152
views
Pairs of matrices
Consider two matrices $A, B\in\mathcal{M}_n(\mathbb{C})$, such that $A, B$ has no common eigenvectors. Is it true that for some nonzero $t\in\mathbb{C}$, matrix $A+tB$ is similar to diagonal matrix $\...
- 291
3
votes
0
answers
319
views
a generalization of the annihilator of cokernel ideal (some new invariants of modules?) [closed]
Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$.
The basic invariants of $A$ ...
- 2,232
3
votes
1
answer
402
views
the pfaffian-adjugate and its counterparts for matrices odd size
Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
- 2,232
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
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
5
votes
1
answer
231
views
What is the criterion for a matrix containing vectors and their permutations being invertible?
Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix $...
- 23.7k
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
CommunityBot
- 1
9
votes
2
answers
900
views
Compute adjugate matrix over commutative ring
Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix.
My current approach is to use the Cayley-Hamilton theorem:
$$\text{adj}(A) = -(A^...
- 3,741
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 ...
- 121
5
votes
1
answer
786
views
Vector Spaces of Symmetric Matrices of Low Rank
Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where $\textrm{Sym}_n(K)...
10
votes
2
answers
1k
views
When the determinant of a 2x2 polynomial matrix is a square?
Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
CommunityBot
- 1
1
vote
1
answer
324
views
Simultaneous triangularizability over a commutative ring
Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $...
- 4,111
2
votes
3
answers
755
views
On matrices in linear forms with vanishing determinant
This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
CommunityBot
- 1
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 ...
- 6,210
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
2
votes
2
answers
798
views
Determinantal rings are Cohen-Macaulay
Consider a $n\times n$ matrix $M$ with entries in $R=\mathbb{C}[x_1,\dots,x_n]$. Let $I$ be the ideal of $(n-1)\times(n-1)$ minors of $M$. Is $\mathcal{O}_{\mathbb{C}^n}/I$ Cohen-Macaulay?If not, ...
- 672