All Questions
5,883 questions
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Number of Symmetric matrices
Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{...
- 179
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184
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Oja's rule gives unit eigenvectors
Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
- 101
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0
answers
98
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Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
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283
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
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93
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Changing Couplings of Discrete Random Variables
Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
- 329
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100
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
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290
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Need any information about an affine lattice
Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ ...
- 18.4k
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40
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Question about a systematic row reduction algorithm for compressive sensing
Suppose a ''brute-force'' algorithm is designed to systematically select from the first $n$ columns of an $m \times (n+1),$ $m<n$ augmented matrix $G$ representing a consistent underdetermined ...
- 259
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89
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Show that a certain ratio of diagonal entries dominates a certain ratio of singular values
Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
- 723
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263
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Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
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answers
84
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Number of solutions to non-linear equations
As part of our project, we are required to determine the total number of distinct solutions to the following equations.
There are $n$ variables of one type, say $\{p_i\}_{i=1}^n$, $m$ variables of ...
- 109
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votes
1
answer
81
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Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 3
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68
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A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
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1
answer
250
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Find the minimum distance of some bad binary code
Let $M$ be a $n \times n$ matrix over the finite field of two elements that satisfies the following property$\colon$ the total number of 1's in each row coincides with one in each column. In other ...
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69
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Quasi-stationary measure on a finite graph equals stationary measure?
Assume the simple random walk $X$ on the graph $G(V,E)$, s.t. $G$ is simple, undirected, finite, connected and let $B \subset V$, s.t. $V\setminus B$ is connected. Let $\sigma_B$ be the quasi-...
- 71
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336
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Pfaffian minors of skew symmetric matrix under perturbation
Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers.
My ...
- 493
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0
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203
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Conical combination of rank 1 matrices with nonnegative entries
Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in the nonnegative real numbers $\mathbb{R}_+$. Suppose that, for each $i = 1,\ldots, n$, the sum $b_i := a_{i1} + \cdots + a_{in}$ of the ...
user94803
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182
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The minimal angle between a vector and a subspace, given another minimal angle
Let $X$ be a set of vectors in $\mathbb{R}^d$. Denote $\theta$ to be the minimal angle between any two vectors in $X$.
Denote $\alpha$ to be an angle between (1) some vector $y\in X$ and (2) some ...
- 968
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166
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Endomorphism ring as ind-pro object
Let $Vect_f$ be the category of finite-dimensional vector spaces. This category comes with a very well-behaved duality functor. Now the ind-completion of this category (if I understand correctly) is ...
- 7,952
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55
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Continuous Functions On Grassmannans under containment restrictions
Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...
- 689
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698
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Singular Values of Linearly Combined Matrices
I have a question related with singular values of matrix sums.
Let's assume I have matrices $A$, $B$, and $D$ (positive, semi-definite) where $D = A + B$. For singular values of $D$, I know that
$$
...
- 101
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183
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How to understand the change of basis in a certain differential equation?
Let $w:\mathbb{R} \times \mathbb{R}^n \rightarrow Mat(n,\mathbb{R})$ be a smooth function, $R_{ij}$ be a fixed skew-symmetric $n\times n$ real matrix, and $A\in\mathbb{R}$. Consider the equation $$\...
- 1,903
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185
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Sum of unit vectors always has a binary span after constrained permutations
Conjecture:
Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, ...
- 121
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1
answer
307
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Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators
Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...
- 195
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111
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Sandwich rule for Lie algebras
On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
- 1,959
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252
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Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
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838
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Simplifying product of matrix exponential?
Is there a known generalization for n-term matrix exponential multiplication?
I am aware that the Baker–Campbell–Hausdorff formula could be used, e.g.:
...
- 111
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0
answers
114
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A linear combination problem
Given $0/1$ $n\times n$ matrix $M$.
Suppose we have $2$ vectors $\lambda,\mu\in\Bbb R^{1\times n}$ such that both
$$\lambda M\in\{0,1\}^{1\times n}$$
$$M\mu'\in\{0,1\}^{n\times 1}$$
holds with $'$ ...
user76479
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322
views
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
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212
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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
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84
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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
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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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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
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votes
1
answer
355
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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
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votes
2
answers
321
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
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1
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130
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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
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0
answers
89
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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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614
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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
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0
answers
89
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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
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0
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83
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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
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161
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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
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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
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182
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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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0
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227
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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
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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
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0
answers
163
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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
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0
answers
81
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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
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votes
0
answers
695
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