All Questions
6,177 questions
4
votes
0
answers
155
views
Special linear group of a quotient
Let J be a non-trivial ideal of a commutative ring A. The canonical map from A to the quotient A/J induces a homomorphism $\varphi : SL_n(A) \to SL_n(A/J)$. In general $\varphi$ is not surjective (for ...
- 229
4
votes
1
answer
1k
views
An optimization problem in numerical linear algebra
Provided two diagonal real matrix which has positive entries, $V$ and $U$.
Find a real matrix $A$, satisfying $A^TA=a^2I$ for some scalar $a$, to minimise
$\left|A^TVA-U\right|\quad\quad(*)$ ...
- 52.3k
12
votes
0
answers
603
views
On a tentative generalization of the Schmidt decomposition
Background
I am a PhD student in Physics and I am currently developing quite refined computer codes that allow to simulate many-body quantum systems living on a lattice. The difficulty resides in ...
- 141
1
vote
2
answers
3k
views
Fast algorithms for computing nullspace of a positive semidefinite matrix over Z
Let $A \in \mathbb{Z}^{n \times n}$ be a positive semidefinite sparse matrix. I am looking for asymptotically fast algorithms for computing the nullspace basis of $A$ (or just random elements in the ...
- 379
2
votes
0
answers
292
views
Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices
I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:
$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
- 379
6
votes
4
answers
15k
views
how to find one column or one entry of the matrix inversion
Let $A$ be a square $n \times n$ matrix, which is invertible. Now we want to find the $i$-th column of $A^{-1}$ and one $(i,j)-$ entry of $A^{-1}$. Is there any way to compute only a small of portion ...
- 96.4k
3
votes
0
answers
311
views
what is the largest gap between rank and approximate rank
$\epsilon$-approximation rank of a matrix $M$ is the minimum rank of a real matrix $A$ which differs from $M$ at most $\epsilon$ in each entry. Associating any function $f:X\times Y\rightarrow${1,-1} ...
- 363
4
votes
1
answer
626
views
Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.
In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
- 59
2
votes
3
answers
3k
views
invertability of a matrix [closed]
Hi all,
I heard a claim that if I have a matrix $A\in\mathbb R^{n\times n}$ such that $A^n \to 0 \ (\text{for }n\to\infty )$
(that is, every entry of $A^n$ converges to $0$ where $n\to \infty$)
then ...
- 54.2k
0
votes
1
answer
152
views
"locally" factoring subgroups of Lie groups
I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).
I start with a subgroup ...
- 20.7k
2
votes
3
answers
3k
views
Is there any conclusions generalized Singular Value Decomposition into Hilbert Space
Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix
Singular ...
- 1,706
12
votes
4
answers
4k
views
Is any $(n-1)\times (n-1)$ submatrix of an $n \times n$ Vandermonde matrix invertible?
Given an $n \times n$ vandermonde matrix $V$ which is invertible, is any $(n-1) \times (n-1)$ submatrix of $V$ invertible also?
I think the answer is yes, but I don't know how to prove.
- 30.1k
8
votes
2
answers
462
views
Projecting the unit cube onto subspaces
Given a set $S$ of non-zero vectors in $\mathbb{R}^n,$ and a subspace $L,$ consider $f(S,L)=\max_{s \in S}\frac{\|Ps\|_2}{\|s\|_2}$ where $Ps$ is the orthogonal projection of $s$ onto $L.$
...
CommunityBot
- 1
2
votes
0
answers
200
views
Projecting the unit cube onto subspaces of dimension at least $2$
This is an updated revision of a recent question where I asked:
Given a set $S$ of non-zero vectors in $\mathbb{R}^n,$ and a subspace $L,$ consider $f(S,L)=\max_{s \in S}\frac{\|Ps\|_2}{\|s\|_2}$ ...
CommunityBot
- 1
1
vote
0
answers
77
views
r-locally linear functions of many variables
Suppose $m,n,r$ are positive integers. Suppose V is a $m$-dimensional vector space over a field F. Let $G(V,n,r)$ denote the space of $n$-tuples of elements of V with the property that the vector ...
- 7,320
3
votes
1
answer
346
views
enlarge the separation between two matrices
The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
\operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\...
- 28.6k
2
votes
2
answers
607
views
Projecting the unit cube onto a subspace [closed]
I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? ...
- 30.1k
4
votes
3
answers
422
views
probability that a random element of Z/NZ can be written as a subset sum of others
How could one calculate the probability that any element in $\mathbb{Z}/N\mathbb{Z}$ can be written as a subset sum of $n$ random elements in $\mathbb{Z}/N\mathbb{Z}$?
In other words, say I pick $n$...
- 30.1k
3
votes
1
answer
218
views
decompositions of matrices over $\mathbb{Q}$
Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_{n}(\mathbb{Z}_{(p)})$, where $ \mathbb{Z_p}$ denotes the ...
- 20.8k
1
vote
1
answer
2k
views
Linear combination of orthogonal projection matrices
Hello!
Let $v_i\in R^d$ $(i=1,...,n,n>d)$ be unit-length vectors ($v_i^Tv_i=1$). Then $v_iv_i^T$ is an orthogonal projection matrix, which has many elegant properties. Now consider a linear ...
- 61
3
votes
2
answers
5k
views
Elementary proof that the Hilbert Matrix is invertible with integer entries [duplicate]
Possible Duplicate:
Deriving Inverse of Hilbert Matrix
The inverse of the Hilbert Matrix is made up entirely of integer entries, but I can't seem to
find an elementary proof for that though, any ...
CommunityBot
- 1
1
vote
1
answer
2k
views
Can a complex non-skew Hermitian matrix have purely imaginary eigenvalues?
I am trying to determine if a certain matrix can have purely imaginary eigenvalues. My question in its most general form is weather a complex matrix that is not skew-Hermitian and irreducible can ...
- 6,342
0
votes
2
answers
2k
views
How to accelerate/avoid multiplication for large matrices in Matlab? [closed]
The setting is here.
X: 6000x8000 non-sparse matrix
B: 8000x1 sparse vector with only tens of non-zeros
d: positive number
M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
- 96.4k
9
votes
5
answers
5k
views
Use of traces in physics
This is borderline physics, but I'll post here first as it relates to a mathematical concept.
I was wondering if anyone could clarify what the physical significance is of traces, whether of matrices ...
CommunityBot
- 1
0
votes
0
answers
1k
views
Determinant of special generalized Vandermonde matrix
Good evening!
I have a generalized Vandermonde matrix of special form:
$\left( \begin{array}{ccccc} a_{0,0} & a_{0,1} \cdot x_0 & a_{0,2} \cdot x_0^2 & \ldots & a_{0,m-1} \cdot x_0^{m-...
CommunityBot
- 1
0
votes
1
answer
249
views
Going from individual elements back to to matrix/vector notation [closed]
Note: Moved to math.stackexchange.com. Sorry for the off-topic question!
[Context: I'm working through problem set one of Andrew Ng's Machine Learning course, question 2, trying to derive the Hessian ...
CommunityBot
- 1
9
votes
2
answers
1k
views
Question on eigenvalue square root subadditivity
ORIGINAL QUESTION
Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a
$2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller
eigenvalue of a $2\times2$ matrix. Is it true ...
- 9,486
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
2
votes
1
answer
211
views
Nonsingularity of certain block matrices
I am completely stuck in the following linear algebra problem.
Consider a finite group $H$ and $N\times N$-matrices $M_{g,h}$ with entries in $\mathbb{Z}$ for all $g,h\in H$. Assume $\sum_{h\in H} ...
CommunityBot
- 1
7
votes
2
answers
715
views
Bimodules over division rings
Inspired by other questions i have two questions about modules over division rings: given a division ring $D$ with center $Z(D)=K$. One has the notion of dimension for left modules (vector spaces) $V$ ...
- 2,941
3
votes
2
answers
1k
views
Hadamard product of matrices over finite field
Hello, all!
I consider Hadamard product $A \circ B$ of matrices $A$, $B$ over finite field. I know $\det{A}$ and $\det{B}$ and want to know about $\det{(A \circ B)}$. Wikipedia and Google let me know ...
- 96.4k
3
votes
2
answers
344
views
Pseudo-idempotent matrix generating a free module
Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
- 6,919
2
votes
2
answers
861
views
Spectral gap of a product of Markov processes
For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...
- 23.2k
5
votes
2
answers
680
views
Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).
Is there an efficient algorithm for finding the solution $x$ of
$b = Ax$
that minimizes the Hamming weight of $x$, where
$A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo 2")...
- 17.9k
3
votes
1
answer
358
views
Feasibility of a matrix equation
I am working on a problem where the following equation came up
$${\bf X}_1{\bf A}{\bf X}_2{\bf A}^T{\bf X}_3{\bf A}-{\bf X}_4={\bf A}$$
where ${\bf A}$ is an arbitrary $n\times n$ and ${\bf X}_i$s ...
- 54.2k
6
votes
1
answer
11k
views
Number of n-th roots of unity over finite fields [closed]
How many $n$-th roots of unity does a finite field $\mathbb{GF}\left(p^k\right)$ have? ($p$ is prime). And then kind of related to that, is there a finite field with exactly $n_1,\ldots,n_N$ $n_1$-th,....
- 449
0
votes
2
answers
273
views
Nonexistence of projection
Following is a doubt I got when I was reading Csiszar's Annals of Statistics paper "Why least squares and maximum entropy: An axiomatic approach to inference for linear inverse problems".
He comes up ...
- 779
3
votes
1
answer
1k
views
Recurrence Equations for Matrix Determinant
Context: I'm reading this paper http://portal.acm.org/citation.cfm?id=1382468
Definitions:
$B_1 = I$
$B_{k+1} = AB_k - \frac{1}{k} tr (AB_k)I$
$det(A) = \frac{(-1)^n}{n} tr(AB_n)$
Question: How ...
- 663
12
votes
1
answer
3k
views
invariant polynomials on 3 by 3 matrices
Hi there:
A freshman level question here. A polynomial p on the entries of n by n matrices is said to be invariant if p(A)=p(sAs^{-1}) for every invertible matrix s. For example, for 3 by 3 matrices, ...
- 118k
7
votes
1
answer
784
views
AX=XB and the Cecioni--Frobenius theorem
The Frobenius--Cecioni theorem states that if $A$ and $B$ are square matrices with entries in a field $k$ then the dimension of the $k$ vector space of solutions of
$$
AX=XB
$$
is given by the sum
$$
\...
- 33.8k
2
votes
2
answers
547
views
The product of non-commuting semisimple matrices need not be semisimple
In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. ...
- 5,654
6
votes
2
answers
1k
views
Minimizing determinant(Ztranspose.A.Z)
Let $A$ be a fixed positive semi-definite symmetric $m\times m$ matrix, and let $p$ be a fixed positive integer. Let $Z$ vary over all $m\times p$ matrices with orthonormal columns, and denote the ...
CommunityBot
- 1
8
votes
3
answers
5k
views
Exact computation of the null-space basis of an integer matrix
Hi all,
Let $\mathbf{A} \in \mathbb{Z}^{M \times N}$. Suppose that $\mathbf{A} \cdot \vec{x} = \vec{0}$, where $\vec{x} \in \mathbb{N}^{N \times 1}$. Does anyone know about a C/C++/Java program that ...
- 2,368
4
votes
2
answers
633
views
A Question about Orthogonal Operators
Let $X$ be a finite-dimensional inner product space, and $T$ a linear operator on $X$. Let $W$ be a subset of $X$ with the following property:
If $T$ preserves norms on $W$, then $T$ is orthogonal on ...
- 76
0
votes
2
answers
2k
views
decomposition of an orthogonal matrix
Hi,
I have a matrix : $W=I+U^TV$
$dim(W)=(D,D)$
$dim(U)=dim(V)=(N,D)$ with $N < < D$
I need it to be orthogonal ie $W^TW=I$
which gives me : $V^TU+U^TV+V^TUU^TV=0$
From that point, i ...
CommunityBot
- 1
0
votes
1
answer
879
views
identity for matrices whose determinant is 1.
For any matrices $A$,$B$ in $SL(2,\mathbb{C})$, let $M(A,B)$ be the 4 by 4 matrix whose columns are matrices $I$,$A$,$B$,$AB$. Then it is not hard to verify that $det M(A,B)= 2- tr[A,B]$.
Is there an ...
- 361
4
votes
4
answers
596
views
Generalization of Jordan Decomposition for Several Commuting Operators
Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...
- 9,132
2
votes
3
answers
285
views
is there any efficient way to compute the follow matrix equations easily
Let $A$ and $D$ are $n\times n$ diagnal matrices, and $B$ is an $n\times n$ orthogonal matrix. Is there any efficient way to compute the follow matrix equations easily?
$\sum_{i=0}^{k} A^i \cdot B^T \...
- 391
13
votes
1
answer
329
views
Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
- 11.4k
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111