All Questions
6,548 questions
12
votes
5
answers
2k
views
Is this formulation of the Singular Value Decomposition standard?
In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (...
- 52.3k
7
votes
4
answers
2k
views
commuting matrices
I have a set $\{A_1, A_2, .. A_k\}$ of $n$ by $n$ real matrices and I know that they are 'perturbated versions' of a set of commuting matrices : $\{P_1,..,P_k\}$, by perturbated versions I mean that I ...
- 28.6k
5
votes
1
answer
250
views
An equation in the free associative ring
Let $X$ be an alphabet, $u,v,p,q,r,s$ be words in the alphabet $X$. I am looking for four elements in the free associative ring $R$ (i.e. four linear combinations of words in $X$) $x,y,z,t$ such that $...
- 18.7k
2
votes
1
answer
220
views
Central extension and direct sum
Let $R$ be an associative ring with 1 and suppose that $Q$ is a central extension of $R.$ I'd like to know how the ring structure of $Q$ and $R$ are related. For example, it's easy to see that if $Q$ ...
- 3,702
2
votes
1
answer
331
views
Symmetric polynomials preserving $-1,1$ matrices
If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$
where $t$ is ``transpose", are both symmetric.
Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the ...
- 13.5k
3
votes
2
answers
278
views
$n$-forms representing zero (versus division rings)
Let me start with two observations.
In the classification of quadratic forms with rational coefficients, one has the following statement: a quadratic form in five indeterminates represents $0$ over $...
- 65.4k
7
votes
1
answer
389
views
Is this Negativstellensatz with uniform denominators known?
A theorem of Reznick states that if $f>0$ is a real homogeneous polynomial in several polynomials that is positive away from the origin of ${\mathbb{R}}^n$, then for large $N$, the form $(\sum x_i^...
- 39.9k
6
votes
1
answer
806
views
Radicals of binomial ideals
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,.....
CommunityBot
- 1
3
votes
0
answers
528
views
A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede
In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...
- 52.3k
9
votes
2
answers
3k
views
Jacobson radical = intersection of all maximal two-sided ideals
I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
- 3,702
8
votes
2
answers
2k
views
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
- 44.7k
3
votes
3
answers
11k
views
Eigenvalues of Matrix Product
Hi Everyone,
Assume that we have a real symmetric matrix $H$, which can be written in the form $H=D \cdot B$, where $D$ is a positive diagonal matrix, and $B$ is a diagonally dominant matrix.
All ...
- 1
14
votes
2
answers
926
views
"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
CommunityBot
- 1
3
votes
1
answer
566
views
Is every polynomial a limit of polynomials in quadratic variables?
Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$.
Call a homogeneous polynomial $f$ of degree $d$ a polynomial in quadratic variables if it is of the ...
- 25.5k
3
votes
1
answer
670
views
Is the set of polynomial sum of squares closed under limits?
A real polynomial $f(x_1,\ldots, x_n)$ in several variables is a sum of squares if there are polynomials $g_1,\ldots, g_k$ such that $f=g_1^2+\cdots +g_k^2$.
Fix a positive number $d>0$. The ...
- 25.5k
11
votes
1
answer
1k
views
Can ⨁_I A be isomorphic to ∏_I A for infinite I?
Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$?
The obvious ...
- 20.8k
4
votes
1
answer
2k
views
How to determine the kernel of a Vandermonde matrix?
Given a Vandermonde matrix
$
V=
\begin{bmatrix}
1 & 1 & 1 & \ldots & 1 \\\\
x_1 & x_2 & x_3 & \ldots & x_n \\\\
x_1^2 & x_2^2 & x_3^2 & \ldots & x_n^2 ...
- 30.1k
2
votes
1
answer
403
views
Finding a 5-cycle in a sparse graph efficiently.
Hi,
I was reading this thread: Finding a cycle of fixed length
I want to find a 5-cycle in a graph. Actually, what I really want is a shortest odd cycle of length at least 5, but maybe that is a ...
CommunityBot
- 1
2
votes
1
answer
234
views
Norms over some subspaces
Let $C$ be the set of all vectors of dimension $n$ such that each of its entries are one of $-1$, $0$ and $1$ and also that the every $v \in C$ has at least $\frac{n}{100}$ $1$'s and at least $\frac{...
- 1,302
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
CommunityBot
- 1
5
votes
1
answer
353
views
Infinite subfields of division algebras with finite center
Assume we are given a non-commutative division algebra $D$ over a finite field $\mathbb F_q$, with the center of $D$ equal to $\mathbb F_q$. Clearly $D$ must be infinite dimensional over its center.
...
- 25.5k
3
votes
2
answers
755
views
What is the characteristic of the module over Jacobson semisimple ring？
We know a ring R is semisimple ring iff every module over R is semisimple，a ring R is von-Neumann regular ring iff every module over R is flat，What about the Jacobson semisimple ring？
- 12.3k
5
votes
1
answer
693
views
Infinite dimensional division algebras with finite center, and their involutions
Let $q$ be a prime power, and $D$ a non-commutative division algebra (skew field) over $\mathbb{F}_q$ (the finite field with $q$ elements) such that the center $C(D)$ equals $\mathbb{F}_q$.
...
- 5,654
3
votes
1
answer
407
views
A silly technical question on Albert algebras
Apologies in advance for this spectacularly uninteresting question, but it has just come up in my work. (Okay, not in a truly important way, but I am trying to gauge the scope of a certain ...
- 12.3k
5
votes
0
answers
517
views
Monomial-type ideals in polynomial rings
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
- 805
7
votes
2
answers
566
views
Rational powers of ideals in Noetherian rings
Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, ...
- 805
3
votes
0
answers
359
views
Do Isometry Groups Tell Us How Difficult Norms are to Compute?
The question: Consider two norms N1 and N2 on the space of n-by-n complex matrices. N1 and N2 have the same isometry group and computing N1 is NP-HARD. Does it follow that computing N2 is NP-HARD as ...
- 6,351
6
votes
0
answers
940
views
inverse eigenvalue problem on graph laplacian
I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody ...
- 61
2
votes
1
answer
337
views
Question about the square of the Jacobson radical
Let $A$ be an associative ring, and $e\in A$ be an idempotent i.e. $e^2=e.$ It
is well-known that $J(eAe)=eJ(A)e,$ where $J(-)$ denotes the Jacobson radical.
It seems natural to try to compare $J(eAe)^...
- 9,132
26
votes
1
answer
998
views
Idempotents in Rings of Differential Operators
Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator on ...
CommunityBot
- 1
38
votes
3
answers
3k
views
What is the current status of Agrawal's conjecture?
In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture:
If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in $\mathbb{Z}_n[X]/(X^r-...
- 71
2
votes
4
answers
1k
views
An inequality question
Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$
with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$
with dimension $2$ of $\mathbb{R}^{3}$, such that $\left\...
- 45.7k
1
vote
1
answer
399
views
Which linear transformations between f.d. Hilbert spaces contract the inner product?
Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$
...
- 10.3k
3
votes
1
answer
594
views
Analogies between orthogonal/unitary groups and their indefinite counterparts
Suppose I have $A\in U(n)$ such that $A^t=A$ (which is a bit un-natural, as usually you'd consider the hermitian transpose, not the transpose).
Well, then $A=X+iY$ say, for $X$ and $Y$ real matrices. ...
- 373
4
votes
0
answers
256
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 39k
7
votes
0
answers
460
views
Quantum polynomial rings and singularities
Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
- 3,758
2
votes
1
answer
1k
views
Condition number for Ellipsoid method matrix
Hello,
When using the ellipsoid method (for solving a linear program for example), the volume of the ellipsoid at each iteration is proven to decrease, and do so by at least a factor of $e^{1/2n}$.
...
CommunityBot
- 1
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
15
votes
2
answers
2k
views
Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?
I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...
- 22.6k
7
votes
2
answers
1k
views
Upper bound to the number of generators
When defining noetherian ring/module there's no condition on the number of generators of ideals/submodules (apart from being finite).
However, in some cases we can do better:
-A noetherian module ...
- 30.5k
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 ...
- 538
3
votes
3
answers
1k
views
Structure theorem for finitely generated Z[G] modules
For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?
- 57.6k
5
votes
4
answers
2k
views
Diagonalization of Infinite Hermitian matrices
We know that $n\times n$ square Hermitian matrices can be diagonalized and have real eigenvalues.
Suppose I have a countable sized Hermitian matrix $A=(a_{ij})$ where the indices $i$ and $j$ run ...
- 60.5k
28
votes
1
answer
1k
views
Formally real Jordan algebras
In 1934, Jordan, von Neumann and Wigner gave a nice classification of finite-dimensional simple Jordan algebras that are 'formally real', meaning that a sum of squares is zero only if each term in the ...
- 9,366
6
votes
2
answers
2k
views
Tight bound for sum of entries of the inverse of a nonnegative matrix
While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
- 52.3k
5
votes
2
answers
537
views
If $k[S]$ is noetherian, is S finitely generated?
Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...
CommunityBot
- 1
6
votes
1
answer
632
views
Norm of commutators (bis)
This question is slightly related to a popular one with the same title (see here).
Let $k$ be a field with characteristic zero. It is known (see Exercise 310) that a matrix $A\in M_n(k)$ is nilpotent ...
CommunityBot
- 1
7
votes
1
answer
2k
views
"Linear algebra" over Z/nZ - reference please!
Let A be a matrix with entries in Z/nZ. (n is not assumed to be prime.) Then the size of the row span is the size of the column span. All computations are mod n, so both these numbers are finite.
I ...
- 25.8k
0
votes
0
answers
289
views
Good and/or standard notation for the abelianization of a Lie algebra
I'd like to solicit good notations for the abelianization of a Lie algebra $\mathfrak g$. One could write $\mathfrak g/[\mathfrak g,\mathfrak g]$, or even $H_1(\mathfrak g)$ but I'd like something ...
- 4,898