All Questions
5,883 questions
18
votes
3
answers
2k
views
Torsion in GL_n(Z)
Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map ...
- 68.9k
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 16
0
votes
0
answers
608
views
Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
- 157
3
votes
1
answer
572
views
When is a finite matrix a "good" approximate representation of an operator?
I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions
$\rho(r) = \sum_{i=1}^N q_i ...
CommunityBot
- 1
6
votes
3
answers
757
views
How many products specify a sum?
Suppose that I have $n$ unknown variables $x_1,\ldots,x_n$. I wish to compute their sum:
$$Sum(x) = \sum_{i=1}^nx_i$$
However, the only access to these variables is through products: that is, for any ...
- 9,756
1
vote
1
answer
369
views
A matrix with trace entries.
This question is related to On a positivity of a matrix with trace entries.
Let $A_1, \cdots, A_m$ be strictly contractive $n\times n$ complex matrices .Is it true that
$$\left(\begin{array}{cccc}Tr\...
CommunityBot
- 1
4
votes
2
answers
442
views
A mapping from a lattice to itself
Consider $\mathbb{Z}^{n}$ for $n = 2^r$ where $r \geq 1$ . Look at the iterates of the following function $T$ from $\mathbb{Z}^n$ to itself.
$T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, |...
- 32.4k
13
votes
0
answers
713
views
Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
CommunityBot
- 1
4
votes
1
answer
1k
views
Integer vectors in the kernel of an integer matrix
Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
Question: How long is the shortest non-zero integer vector in the kernel of $A$?
Example: If ...
CommunityBot
- 1
1
vote
2
answers
156
views
How to study the behavior of a particular function on a Vector Space.
Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
$T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
$T(k + X) = T(X)$ for all $k \in K$ ...
- 18.7k
2
votes
3
answers
1k
views
how to get nonzero eigenvalues of a large symmetric matrix with lots of duplicate rows
Is there a nice trick for this? I would like to compute the eigenvalues more efficiently.
7
votes
3
answers
2k
views
Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
- 39.9k
6
votes
1
answer
520
views
Bisymmetric Matrix, solving set of linear equations.
A bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals.
If $A$ is a bisymmetric matrix and I'm interested in solving $Ax=b$.
Are there techniques used to ...
- 25.1k
-2
votes
1
answer
470
views
Little conjecture about sums of reciprocals
Given a finite list $x_i$ of $N$ positive reals, it seems that $\sum_{i=1}^N x_i = \sum_{i=1}^N x_i {}^{-1} \Rightarrow \sum_{i=1}^N x_i \geq N$. Can anyone give me a proof?
- 39.9k
4
votes
0
answers
453
views
Convergence of the relaxation method for every parameter in the relevant disk
For large size matrices, the resolution of linear systems $Ax=b$ is often done iteratively. The matrix $A$ is split as $A=M-N$, with $M$ invertible, and one performs
$$x^{k+1}=M^{-1}(Nx^k+b).$$
The ...
- 52.3k
3
votes
1
answer
1k
views
problems of subspace of M_n(C)
let $M_n(c)$ denote the n times n matrices over the complex number field. $N$ be a subspace of
$M_n(C)$.
1 If there is no unitary lies in $N$, what is the maximum of the dimension of $N$ can be?
...
- 52.3k
1
vote
1
answer
383
views
Relaxation Scheme for $Au=f$ error analysis
Hello
I'm trying to answer this question, but am completely stuck.
Argue that in analyzing the error in a stationery linear relaxation scheme applied to $Au=f$, it is sufficient to consider $Au=0$ ...
- 52.3k
6
votes
0
answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
2
votes
1
answer
4k
views
Bidiagonalization and SVD of matrix
I can't find a single solid explanation of how to implement this -- whitepapers too detailed/confusing. Closest I came to an answer was this:
http://www.hep.ucl.ac.uk/~bino/libbpm/doc/pro/html/...
- 3,217
3
votes
2
answers
4k
views
Matrix products under which the determinant behaves multiplicatively
The determinant behaves multiplicatively with respect to the usual matrix product
$$
\det(AB) = \det(A)\det(B),
$$
and also with respect to the Kronecker (or tensor) product of square matrices
$$
\...
- 758
2
votes
0
answers
241
views
subspace separation and M-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
$$
\operatorname{sep}(A,B)=\min_{X\neq 0}\frac{\left\...
- 20.2k
13
votes
1
answer
2k
views
Banach-Mazur distance between $\ell^p$-norms
Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$
is an operator norm ...
- 31.5k
10
votes
5
answers
990
views
Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
CommunityBot
- 1
1
vote
1
answer
2k
views
Principal Minors of Matrix Product
Suppose $A$ is a positive definite matrix and $B$ is a non-symmetric
matrix with all positive principal minors.
Is their product $AB$ a matrix with all positive principal minors?
I believe the ...
- 14.5k
1
vote
1
answer
940
views
maximal number of mutually orthogonal vectors
Let $F$ be a field, $n$ be a positive integer. Denote by $h_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, ...
- 20.8k
0
votes
1
answer
637
views
Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\...
- 60.5k
3
votes
0
answers
328
views
Integer relation detection for Subset Sum or NPP?
Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
- 513
5
votes
2
answers
3k
views
Closedness of finite-dimensional subspaces
Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed?
I suspect yes, but I can't come up with a proof, and it seems like locally ...
- 4,416
8
votes
2
answers
679
views
To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?
I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For example,...
CommunityBot
- 1
4
votes
1
answer
254
views
Embedding into Permutation Representation
Let $\rho$ be irreducible representation of group $G$.
How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
- 52.9k
2
votes
0
answers
658
views
Alternating bilinear forms over local rings
Suppose k is a field and V a vector space over k. If b is an alternating nondegenerate bilinear form in V, it has a symplectic basis. A symplectic basis is a basis where the basis vectors come in ...
- 33.8k
7
votes
3
answers
744
views
Looking for applications of a nice result in linear algebra
Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M_n(k)$ which commutes with $...
- 14.5k
8
votes
5
answers
15k
views
Eigenvalues of A+B where A is symmetric positive definite and B is diagonal
If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any ...
CommunityBot
- 1
7
votes
1
answer
372
views
Simultaneously orthogonally transform two SPD matrices to tridiagonal form?
Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...
-1
votes
1
answer
1k
views
Sum of two unitary matrix is equal to every matrix? [closed]
Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?
- 18.7k
12
votes
3
answers
3k
views
How to combine linear constraints on a matrix and its inverse?
Suppose there exists a $(n \times n)$ matrix $A$ that is real and invertible (nothing unusual or special about $A$). We do not know the entries of $A$. However, we do have linear constraints, some of ...
- 1,959
8
votes
0
answers
1k
views
roots of quadratic forms
This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
- 118k
2
votes
0
answers
299
views
Uniqueness of dimension for topological vector spaces
Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ satisfying
.
Linearly Independent: For all functions $f$ in $\mathbb{...
CommunityBot
- 1
1
vote
3
answers
206
views
Linear space of translatable functions.
What are the functions $f$ so that a set $\{a \cdot f(x+b) : a \in \mathbb{R}, b \in \mathbb{R}\}$ is a finite dimensional linear vector space ?
Is there a complete characterization of such functions?...
- 14.5k
1
vote
2
answers
3k
views
matrices self-adjoint with respect to some inner product
Is it possible to give a nice characterization of matrices $A \in R^{n \times n}$ which are self-adjoint with respect to some inner product?
These matrices include all symmetric matrices (of course) ...
- 20.2k
5
votes
2
answers
687
views
Dependence of trace norm on matrix size for smooth vs. random matrices.
Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...
- 846
1
vote
1
answer
715
views
Find a matrix's nullspace from submatrix nullspace
This is probably a basic question, but my linear algebra is weak.
Suppose I want to compute the nullspace of a matrix A using some iterative method (e.g. Lanczos). Suppose further that I know a ...
CommunityBot
- 1
4
votes
2
answers
1k
views
Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?
Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
1
vote
1
answer
201
views
How can I characterize the type of solution vector that comes out of a matrix?
Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will ...
- 156k
0
votes
1
answer
1k
views
For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 9,756
3
votes
2
answers
3k
views
distributed incremental SVD
Hello all,
I need some theoretical pointers (formulas, articles, online links) on how to merge Singular Value Decompositions (SVD) of two matrices (two different sets of observations over the same ...
- 153
13
votes
2
answers
1k
views
Combinatorial proof of (a special case of) the spectral theorem?
The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...
- 85.6k
34
votes
2
answers
4k
views
Symmetric powers and duals of vector bundles in char p
Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals:
...
CommunityBot
- 1
9
votes
2
answers
2k
views
A generalization of Boolean matrix multiplication for order-3 tensors
The Boolean matrix product of two 0-1 $n \times n$ matrices $A$ and $B$ is the matrix $C$ defined as
$$C[i,j] = \vee_{k=1}^n (A[i,k] \wedge B[k,j]).$$ If $A = B$ and the matrix is an adjacency matrix ...
- 2,071
5
votes
2
answers
457
views
fast merging of orthogonal bases
Given two matrices $U_1, U_2$, we can use QR factorization to find orthogonal basis for the subspace spanned by (columns of) $\begin{bmatrix}U_1,U_2\end{bmatrix}$.
Now this generally makes no use of ...
- 93