All Questions
6,260 questions
4
votes
2
answers
520
views
Can a commutative, associative "multiplication" on an infinite-dimensional vector space be an isomorphism?
Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). ...
- 54.6k
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 2,179
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
2
votes
1
answer
385
views
Does knowing a conjugation of A to A^T determine eigenvalues of A?
Everybody knows that a square matrix $A$ has the same eigenvalues as
$A^T$. And it is clear that if $A^T=BAB^{-1}$ then $B$ maps eigenvectors
of $A$ to those of $A^T$. But I have not found any ...
- 798
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 497
7
votes
1
answer
3k
views
How to resolve a wedge product of vector bundles
Let $X$ be an algebraic variety. Consider an exact sequence
$$0\to A\to B\to C\to 0$$
of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ (...
- 2,444
2
votes
0
answers
328
views
What is this decomposition called?
Let $M$ be a positive semi-definite matrix, symmetric with real entries. Then $M$ can be written as $X X^T$. One way is by a Cholesky decomposition (unique for positive definite but not necessarily ...
- 173
6
votes
3
answers
1k
views
Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 52.4k
4
votes
3
answers
3k
views
Making MATLAB svd robust to transpose operation
I'm playing with MATLAB's svd function to compute the svd of
[ 1 4 7 10
2 5 8 11
3 6 9 12 ]
When I type [U1, ~, ~] = svd(...
- 497
5
votes
2
answers
1k
views
Is the operator norm always attained on a $\{0,1\}$-vector?
Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ ...
- 23k
2
votes
2
answers
369
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 21
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
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 ...
- 44.9k
5
votes
2
answers
612
views
A Boolean function that is not constant on affine subspaces of large enough dimension
I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...
-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 ...
- 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 ...
- 661
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
6
votes
2
answers
2k
views
Computation of a Drazin inverse
I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron value)...
- 20.2k
15
votes
3
answers
1k
views
Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 1,284
21
votes
3
answers
1k
views
Which doubly stochastic matrices can be written as products of pairwise averaging matrices?
A matrix $A$ is called doubly stochastic if its entries are nonnegative, and if all of its rows and columns add up to $1$. A subset of doubly stochastic matrices is the set of pairwise averaging ...
- 415
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 ...
- 25.5k
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 ...
- 3,897
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|, |...
- 73
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 $k \in K$ ...
- 73
4
votes
1
answer
2k
views
Determinant and symmetric power
Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
- 1,510
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.
- 173
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
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 ...
- 3,217
-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?
- 2,513
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?
- 79
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 ...
- 52.4k
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 $\...
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 $Au=0$ ...
- 579
2
votes
1
answer
414
views
Descriptive complexity of Hamel bases of R^ω
(base theory = ZFC)
Are any Hamel bases for the vector space $\mathbb{R}^{\omega}$ in the
1. analytical hierarchy?2. projective hierarchy?
In any of the above cases where the answer is not simply ...
user5810
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/...
- 123
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
$$
\...
- 403
3
votes
0
answers
1k
views
Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix
Hi
From a physics problem, I am trying to evaluate exactly the following kind of determinant:
G = A + M + N.
A is diagonal
M is a product of a column (of 1s) and a row matrix
N is a Hermitian ...
- 31
7
votes
3
answers
3k
views
How many commuting nilpotent matrices are there?
To be precise, fix $n$, fix a field $k$.
What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean ...
- 5,052
4
votes
1
answer
3k
views
SVD complexity for structured sparse matrices
Hello,
For an $n \times n$ real matrix, the SVD (Singular Value Decomposition) algorithm is $O(n^3)$.
I have large matrices (say $10,000 \times 10,000$) that only have elements on few diagonals, i.e. ...
- 2,829
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
13
votes
1
answer
2k
views
Banach-Mazur distance between $\ell^p$-norms
Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$
is an operator norm ...
- 52.4k
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
1
vote
1
answer
2k
views
Principal Minors of Matrix Product
Suppose $A$ is a positive definite matrix and $B$ is a non-symmetric
matrix with all positive principal minors.
Is their product $AB$ a matrix with all positive principal minors?
I believe the ...
- 1,035
1
vote
1
answer
941
views
maximal number of mutually orthogonal vectors
Let $F$ be a field, $n$ be a positive integer. Denote by $h_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, ...
- 109k
0
votes
1
answer
638
views
Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\...
- 10.5k
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
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 $X=G/H$.
- 2,179
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,...
- 54.6k
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 ...
user5810
9
votes
2
answers
1k
views
Other norms for lattice reduction techniques (LLL, PSLQ)?
LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon ...
- 513