All Questions
6,260 questions
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322
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Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
0
votes
0
answers
212
views
Can we drop commutativity assumption?
Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
- 25
0
votes
0
answers
84
views
Can we increase spectral norms of All maximum size square submatrices by orthogonal perturbation?
Let the matrix $A$ consist of $k$ columns from some $n \times n$ orthogonal (unitary) matrix. It is obvious that there is no perturbation of $A$ which
leaves its columns orthonormal,
increases ...
- 1
0
votes
0
answers
53
views
Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
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votes
0
answers
369
views
Bounds on the smallest eigenvalue of a Hankel matrix
Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue.
What bounds are known on $\lambda_n$ in terms of the entries on $H$.
I can see some results ...
- 265
0
votes
1
answer
355
views
Lower bound on Spectral Gap of Rank one + Diagonal
For some $x\in\mathbb{R}^n, \|x\|_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix
$$
X_\alpha := xx^T + \alpha\sum_{k=1}^nx_k^2e_ke_k^T.
$$
Suppose for simplicity that the ...
- 205
0
votes
2
answers
322
views
Solving sparse linear least squares or a positive definite 5-band matrix system fast
I want to quickly solve the following linear least-squares problem
$$\min_{x \in \mathbb{R}^n} \left\| A x - b \right\|_2^2$$
with a special sparse structure where each row in $A$ has only up to $4$ ...
- 101
0
votes
1
answer
130
views
Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
- 209
0
votes
0
answers
89
views
Degree of permutation of hypercube
Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
- 13.9k
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votes
0
answers
614
views
Hadamard / matrix product adjoint
First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself).
I apologize if this question seems dumb; I'm a new ...
- 1
0
votes
0
answers
89
views
Decomposing matrices to lower ranks
Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...
- 13.9k
0
votes
0
answers
83
views
Bits of precision matrix reconstruction
We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
- 13.9k
0
votes
0
answers
161
views
Way to parameterise sparse multi diagonal matrix
I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$
where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is
$$
\Lambda = \begin{bmatrix}
x &...
- 1
0
votes
0
answers
345
views
Shift functor and the origin of the linear decalage isomorphism
Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of $\mathbb{Z}$-...
- 2,074
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0
answers
182
views
Question about majorization of eigenvalues after conjugation
Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...
- 2,339
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votes
0
answers
227
views
Canonical forms of symmetric/skewsymmetric quaternionic matrix
$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim ...
- 293
0
votes
0
answers
68
views
Estimate bounds on Minkowski distance from point to a segment in Lp space
Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
- 101
0
votes
0
answers
163
views
Applying a linear operator to a basis set following SVD orthonormalization
Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases.
My intention is to:
decompose $\Phi$ via SVD:
$U \Lambda V^T = \Phi$
to create it's ...
- 1
0
votes
0
answers
80
views
Finding gradient of an optimization
I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...
- 161
0
votes
0
answers
81
views
A Optimization problem using co-ordinates of joint numerical range.
Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as
\begin{align}
\mathcal{S}=\{\left(\mathbf{u}^H\...
- 1,421
0
votes
0
answers
696
views
Heat equation with graph laplacian
I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...
- 161
0
votes
0
answers
103
views
Perturbed linear system, particular form
We have a linear system $Ax=b$ where $A$ is real and symmetric, all elements of its main diagonal are strictly positive and all off-diagonal elements are $\leq 0$. Further, $A_{ii} > -A_{ij} \; \...
- 311
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
0
votes
0
answers
91
views
Complexity of turning a d-degree polynomial to 2-degree polynomial
For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
- 187
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0
answers
251
views
Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components
I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
- 11
0
votes
0
answers
561
views
What are the properties of this linear operator?
Suppose $f(x)$ is a function which satisfies the following condition:
$$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$
Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
- 10.1k
0
votes
0
answers
131
views
The largest size of a boolean subgraph (a hypercube) of a given graph
Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
- 313
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votes
0
answers
49
views
Forming orthogonal bases in different orders
Let $\alpha_1, \dots, \alpha_n$ be unit vectors in some vector space $V = R^d$. For any permutation $\pi: [n] \rightarrow [n]$, we can form the Gram-Schmidt orthogonal bases $\beta_{\pi,1}, \dots, \...
- 3,475
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votes
0
answers
104
views
Linear system with many solutions from a finite set
Basically I am looking for a linear system with
many solutions from a finite set.
Choose a finite set of rationals $S$ and fix
positive integer $k$.
Let $A$ be a linear system with $n$ variables $...
- 25.4k
0
votes
0
answers
320
views
Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
- 801
0
votes
2
answers
153
views
Union of linear inequalities cover whole space?
We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$.
There are $k$ sets of linear inequality constraints on the $a_i$.
I need to check that any choice of $a_i$ satisfies at least ...
- 109
0
votes
0
answers
117
views
"Almost orthogonalizing" matrices using a signature matrix
Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ||A||_{op}||...
- 949
0
votes
0
answers
704
views
expected matrix inverse of circulant plus diagonal matrix with chi-square variables
Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
- 96
0
votes
0
answers
115
views
a very elementary question on the conjugated matrices
Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero .
We suppose that they have the same ...
- 3,472
0
votes
0
answers
266
views
Finding the effective maximum number of subspaces in a finite dimensional vector space
Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search.
For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
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0
answers
262
views
Lattice basis reductions and finding minimal values
While reading several articles about lattice basis reduction I am left with a few questions.
For one, I came across this piece of text
Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and $...
- 1
0
votes
1
answer
311
views
Subspace generated by positive vectors
Hi everyone, first of all i must admit i'm very familiar with quadratic forms and positive subspaces, so i'm sorry if my question is too trivial. So, here's my problem:
Let $L$ be a real vector space ...
0
votes
1
answer
775
views
Positive subspaces of quadratic forms
here's my question:
Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$.
We have $W\subset V$ a positive (with respect to the ...
0
votes
0
answers
957
views
Diagonal of the inverse of a 6x6 symmetric partitioned matrix
Let
$$M = \begin{bmatrix}
A & B \\
B & C
\end{bmatrix}$$
in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...
- 1
0
votes
0
answers
52
views
Dense Matrix Estimation
I have a matrix $X \in \mathbb{R}^{m\times n}$ and I want to estimate it with a dense matrix $Y^{m\times n}$ such that $Y$ is still close to $X$ in some distance measure. Is this doable in a ...
- 137
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
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 ...
0
votes
0
answers
270
views
Solution Existence of a System of Complex Quadratic Equations
Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following
quadratic set of ...
- 342
0
votes
0
answers
257
views
What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?
Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties:
$X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the $\...
0
votes
0
answers
155
views
Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
- 155
0
votes
0
answers
151
views
Ratio of Eigen values and Mutual Independence
Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix.
If $X$ contains $M$ observations ...
- 1
0
votes
1
answer
590
views
Strictly diagonally dominant hermitian matrices eigenvalues sign
Let $A\in \mathcal{M}_{n\times n}(\mathbb{C})$ be a strictly diagonally dominant hermitian matrix.
My main goal is to tell how many positive eingenvalues $A$ has in terms of its leading diagonal ...
- 3
0
votes
0
answers
166
views
Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
- 87
0
votes
0
answers
440
views
Foliations in positive characteristic
Ekedahl wrote about foliations in positive characteristic, over the field $\mathbb{Z}/p\mathbb{Z}$ as a subsheaf of the tangent sheaf, that are closed with respect to involution and $p$-power.
My ...
- 1
0
votes
0
answers
237
views
Geometric Mean of Positive Matrices
Hello all,
My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by:
$M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-...
- 155