All Questions
6,291 questions
3
votes
1
answer
379
views
Schur product, partial order
Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order.
How to ...
- 478
2
votes
0
answers
124
views
Products of matrices of a certain form
Are $n \times n$ matrices of the form
$$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$
studied anywhere? I am interested in the structure of ...
- 3,081
5
votes
2
answers
2k
views
A basis of the symmetric power consisting of powers
I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a complex vector space ...
- 4,902
3
votes
2
answers
1k
views
Average size of determinants of integer matrices?
I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are ...
- 26.9k
1
vote
2
answers
229
views
Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints
Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants
\begin{align}
w^{H}C_1w>0 \\\
w^{H}C_2w>0 \\\
...~~~~~~~~~~ \\\
....
- 1,421
6
votes
1
answer
1k
views
Efficient computation of Markov chain transition probability matrix
Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...
- 63
6
votes
3
answers
4k
views
Can the sum of two roots of unity be a root of unity?
Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$
Is it true or false that a combination of two (or more, in general) of the ...
- 145
0
votes
0
answers
146
views
Global solution for spectral clustering
I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
- 13
1
vote
0
answers
132
views
Matrices with a common Fischer basis
Let $A$ be a real symmetric $n\times n$ matrix, normalized such that $Tr[A]=1$. Define a 'Fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for ...
- 31
20
votes
4
answers
2k
views
The sum of same powers of all matrices modulo p
The following is a problem from our department algebra competition for
students:
Non-question.
An experimental-math geek was trying to raise all matrices $17\times17$
over the field with 17 ...
- 3,932
0
votes
1
answer
170
views
Tensoring with descending chain of modules
Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
- 11
12
votes
6
answers
7k
views
Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?
I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but ...
- 663
2
votes
1
answer
1k
views
Coercive Symmetric Bilinear form on a Hilbert space
I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
1) Consider a continuous symmetric ...
1
vote
1
answer
188
views
Is there a wedge which operates on multiple vector spaces?
Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...
- 494
1
vote
3
answers
2k
views
$L_p$ space embedding (reference request)
There is a result in the wikipedia article about $L_p$ space embedding:
a. Let $0 ≤ p < q ≤ ∞$. $L_q(S, μ)$ is contained in $L_p(S, μ)$ iff $S$ does not contain sets of arbitrarily large measure;...
user14319
6
votes
1
answer
987
views
Bounding the second derivative of the log-determinant
I'm trying to use the log-determinant to regularize an optimization problem. To make the argument work, I need to bound the second derivative of the log-determinant.
I need to prove that $\text{Tr}\...
- 777
1
vote
1
answer
113
views
Expected rank - computable approximations
I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general).
Computing $\mathbb{E} \ \mathrm{...
- 3,323
0
votes
1
answer
312
views
Deriving the fundamental equation (with regards to computer vision)
I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
4
votes
1
answer
2k
views
lipschitz constant of a multivariate function
I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
- 41
2
votes
1
answer
6k
views
Inversion of complex matrix
Hello all,
Assume matrix of complex numbers described as a sum of real matrices $A$ which is diagonal and $B$ which is symmetric (and block symmetric if the term is correct):
$A+ Bi$
I want to ...
5
votes
1
answer
670
views
The minimal norm of a shifted stochastic matrix
Given a row-stochastic matrix $M$ with singular values $\sigma_{1} \geq \cdots \geq \sigma_{n}$, I am looking for an upper bound on the expression
$$\min_{\alpha} \left\| M- \frac{\alpha}{n}J_{n} \...
- 225
4
votes
1
answer
255
views
On the divisibility of the special linear group of degree $n$ over an algebraically closed field
Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But ...
- 10.5k
0
votes
1
answer
912
views
Maximal isotropic subspaces of $V\oplus V^*$
Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and denote the dual of $V$ by $V^*$. Then for $E$ a subspace of $V$ and $\epsilon\in \Lambda^2E^*$ clearly the space $$L=L(E,\epsilon):=\{...
- 673
2
votes
0
answers
51
views
Conditions under which a set of points have a low weight representation under some basis
Let $S \subset R^d$ be a convex polytope in $d$ dimensional real space. Say that $S$ has weight $w$ if there exists some basis $x_1,\ldots,x_d$ such that for every point $v \in S$, we can write $v = \...
- 794
1
vote
1
answer
387
views
All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k
The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \...
- 10.5k
2
votes
1
answer
568
views
integral basis of orthogonal complement
Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$.
My goal is to find an ...
- 145
2
votes
2
answers
841
views
When can a matrix with negative entries have a completely non-negative dominant eigenvector?
Perron-Forbenius obviously answers this question for positive and for certain non-negative matrices. I want to know whether these conditions can be weakened at all. In other words, what, if anything, ...
- 31
3
votes
0
answers
176
views
Extending a Hilbert space isometrically
Let $H$ be a Hilbert space, and let $X$ be a topological vector space.
Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?
...
- 8,512
4
votes
1
answer
316
views
standard practice for large dense truncated svd computations?
What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices.
I assume for k small that you use a ...
- 922
4
votes
0
answers
806
views
(Co)limit computations for diagrams of Vector Spaces
Fix a field $K$ and consider a finite directed graph $\Gamma$ where multiple edges between a pair of vertices are allowed so long as the total number of edges is finite. Associate to each vertex $v$ a ...
- 15.5k
0
votes
1
answer
173
views
Avoiding epsilon in mixed integer linear and quadratically constrained programs
I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$.
$(x = 0 \wedge y = 1) \vee (x \neq 0 \...
- 1
1
vote
0
answers
1k
views
Algebraic Independence of Polynomials in n Variables with Real Coefficients
I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
- 11
8
votes
1
answer
2k
views
Symplectic block-diagonalization of a real symmetric Hamiltonian matrix
Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...
- 315
7
votes
2
answers
5k
views
Relationship between the derivative of a matrix and its eigenvalues
Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing?
...
- 71
5
votes
3
answers
5k
views
Eigenvalues of principal minors Vs. eigenvalues of the matrix
Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x?
(Here "semidefinite" can not be taken to ...
- 51
2
votes
1
answer
493
views
How to find the nilpotent submatrices of a symmetric, real matrix?
Given a symmetric, real $n \times n$-matrix $M$, is there a way to find all $m \times m$-submatrices ($1 < m < n$) that are nilpotent?
By the Cauchy interlacing theorem, I know that $M$ must ...
- 5,565
4
votes
0
answers
287
views
Eigenvalues of "modified" Johnson scheme via the representation theory of the symmetric group
I am interested in eigenvalues of the following association scheme, which somewhat resembles the Johnson scheme.
Let $n$ and $k\leq n$ be positive integers.
The $n!/(n-k)!$ vertices of the scheme ...
2
votes
0
answers
132
views
Characterizing the singular values of a matrix with structure
Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...
- 21
3
votes
1
answer
531
views
A little question on certain parallel-lines-preserving maps
Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties:
1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine $\mathbb{...
- 23.4k
1
vote
1
answer
3k
views
Eigenvalues of a second difference matrix
I'm trying to efficiently calculate the eigenvalues for this matrix. It looks like this:
(source)
It's diagonalized by the DST matrix in a similar way that a circulant matrix ...
- 11
0
votes
1
answer
1k
views
Applications of the natural bilinear forms on the direct sum between a vector space and its dual
As is known, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew-symmetric bilinear forms
$$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 (\xi(Y) \pm \eta(X)).$$
I am interested in ...
- 4,312
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
21
votes
7
answers
2k
views
Modern developments in finite-dimensional linear algebra
Are there any major fundamental results in finite-dimensional linear algebra discovered after early XX century? Fundamental in the sense of non-numerical (numerical results, of course, are still ...
- 263
0
votes
1
answer
172
views
orthogonality in a lattice
Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3.
...
- 107
3
votes
0
answers
2k
views
Relation between the eigenspace of a covariance matrix and eigenspace of correlation matrix
I was discussing applying Principal Component Analysis to a covariance matrix versus applying PCA to the corresponding correlation matrix with a collegue. This led me to think about the following ...
- 131
1
vote
1
answer
246
views
orthogonal base in unimodular lattice
Let $\Lambda$ be an unimodular lattice with a quadratic form $(-,-)$ of signature $(m,n)$ , $m,n>0$.
I know that, fixed a base $e_1,\cdots,e_{m+n}$ for $\Lambda$, the matrix which has entries $a_{...
- 107
2
votes
1
answer
665
views
Covering the cone of positive semidefinite matrices by intervals
Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the ...
- 6,962
20
votes
3
answers
3k
views
Small-index subgroups of SL(3,Z)
I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$.
The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$,
along the bottom row. I could not figure out ...
- 743
7
votes
0
answers
294
views
Largest entry of the inverse matrix?
I wonder if there is a "qualitative way" of predicting from the structure ix of the matrix $A$ which entry of $A^{-1}$ will be the largest. I am specially interested in the case
that $A$ is a ...
- 6,962
8
votes
2
answers
5k
views
When is spectral norm of AB equal to that of BA?
I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$
The argument I was using to show this is that for any two square ...
- 922