All Questions
2,026 questions with no upvoted or accepted answers
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Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
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67
views
Integration of matrix form of Vasicek variance (Python/Matlab)
$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
- 1
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254
views
The coevaluation map for a projective module and its dual
$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
- 145
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133
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A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"
Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...
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116
views
Inequality involving $\ell_p$-sum of matrices
Let $A_1, \cdots, A_n, B_1, \cdots, B_n$ be square matrices of same dimension, and suppose that $\left|A_i\right| \leq \left|B_i\right|$ for every $1 \leq i \leq n$.
Is it true that $\Big(\sum\limits_{...
- 565
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105
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Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces
Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
- 31
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46
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What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?
What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$?
I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
- 1
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159
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Double summation of matrices as constraints in convex optimization in CVX
I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53:
\begin{align}
\text{minimize} &\qquad s\\
\text{subject to} & \...
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answers
71
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Generalization of Levi-Civita type construction towards divergent integrals and corresponding questions
A known generalization of Levi-Civita field is a field of Hahn power series of $\varepsilon$ of the form
$\mathbb{R}[[\varepsilon^{\mathbb{Q}}]]$. Assuming $\varepsilon=1/\omega$, we can naturally ...
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248
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Is there a relationship between infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I have found a paper that used the following ...
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253
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Determinant of chain complexes
Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
user30211
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145
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Is it correct to use extensionality axiom in an algebraic theory? Is "extensionality theory" appropriate name for the identity theory plus this axiom?
There is something about extensionality axiom which makes debatable its use in any theory, not only in an algebraic one -- this law looks more like a definition than a statement when written like this:...
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199
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 111
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71
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Improving a basis
Given any triple of $n \times n$ matrices $(A, B, J)$ and invertible $2n \times 2n$ matrix $U$ for which $U(J \oplus -J)U^{-1} = \begin{pmatrix}0 & A \\ B & 0\end{pmatrix}$, can we find a pair ...
- 4,943
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229
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Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201
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240
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Fundamental theorem of algebra for sedenions
The Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.²
Does ...
- 1,590
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54
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Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
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60
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Hamming distance globally and Euclidean distance locally to a cycle
Given a permutation matrix 'the question is to decide if there is a permutation matrix representing a cycle within Hamming distance $d$ from given matrix'. Is there an efficient algorithm for it?
...
- 13.9k
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227
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Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
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46
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Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
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193
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Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
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173
views
Can the notion of algebraic closedness be generalized to the rings with zero divisors?
Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero?
For instance, it seems that any polynomial of non-zero-divisor-...
- 10.1k
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150
views
How to classify rings by combinatorial structures?
There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
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293
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Quotient of monoids and monoid algebras
Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
- 327
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249
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What is the computational complexity of solving a highly underdetermined system?
Let $F$ be a finite field with $q$ elements. Consider an underdetermined system of linear equations with $m$ equations and $n$ variables where $n\gg m$. What is the complexity of solving such a highly ...
- 131
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72
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Explicit isomorphism between two Jordan algebras
I have seen written that the space of $m\times m$-complex matrices $M_m(\mathbb C)$ endowed with the usual Jordan product is isomorphic, as a Jordan algebra to the complexification of the space $Herm(...
- 1,463
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86
views
Idempotent Matrix non diagonal terms as a function of diagonal terms
Assuming the construction of an idempotent symmetric matrix ($𝐴=𝐴^𝑇$ and $𝐴=𝐴^2$). Is there any way to write the terms of this matrix as a function of its diagonal terms ?
For a 3x3 matrix, the ...
- 1
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145
views
Square root of a circulant matrix block
I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...
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299
views
Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$.
$B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$.
$T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
- 13.9k
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90
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
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425
views
Linear independence of vectors in Graph Theory
I have poste this question on StackExchange but there were no takers - would I be luckier on this site?
Most of this is well known, so let me just restate the corresponding Math:
Given a connected, ...
- 419
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113
views
Reference for matrix Lyapunov function / matrix dynamic system / stability
We usually consider $\dot{x} = f(x)$, where $x$ is a vector.
Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...
- 101
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110
views
Decomposition an $A$-module to irreducible ones
Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra.
Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...
- 4,058
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48
views
Is it possible that in certain rings the power series representing special functions are expressable via series representing elementary functions?
Let's consider the Taylor power series of a function on real numbers.
Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be ...
- 10.1k
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250
views
Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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62
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Criterion for the existence of roots of a system of polynomial equations on a torus
Given a system of polynomial equations in $n$ indeterminates $z_{1}%
,\ldots,z_{n}$:
\begin{align*}
p_{1}\left( z_{1},\ldots,z_{n}\right) & =0\\
& \vdots\\
p_{m}\left( z_{1},\ldots,z_{n}\...
- 61
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74
views
Estimate of the nilpotency class from the subgroup
Let $G$ be a nilpotent group and $H \vartriangleleft G$ a normal subgroup such that $[G:H] \le m$.
Assume $H$ has the nilpotency class $ \le n$. Can we show the nilpotency class of $G$ is bounded by a ...
- 2,535
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149
views
L_q matrix inequality
The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
- 101
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202
views
Ratio of maximum to minimum value
Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
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166
views
Minimize a vector from a matrix operation
I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem.
Lets say we have
$$
(L+A)*s = v
$$
L is the ...
- 101
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113
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Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
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76
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Isomorphism problem for enveloping algebras
Let A and B be finite-dimensional algebras over a field k. We denote the enveloping algebra of A by A^e, which is the tensor product (over k) of the algebra A and its opposite algebra. Suppose A^e and ...
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108
views
Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
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96
views
Gelfand–Kirillov dimension of the first Weyl algebra by using the definition
$\DeclareMathOperator\GKdim{GKdim}$Here I am trying to find the Gelfand–Kirillov dimension of the first Weyl algebra just by using the definition of the Gelfand–Kirillov dimension.
Let $A$ be an ...
- 131
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140
views
Field of algebraic functions
We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...
- 1
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79
views
Eigendecomposition of $A=I+BDB^H$
Suppose that we have $$A = I_m + BDB^H$$ where matrix $A$ is $m \times m$, matrix $B$ is $m \times k$, $BB^H \neq I_m$ and $D$ is a $k \times k$ diagonal matrix. Can we obtain the eigendecomposition ...
- 13
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61
views
Combining quadratic and linear matrix terms into a quadratic term
Given
$$ C = AFA^T + A\bar{F} $$
where $A = [A_1 A_2]$, $F = \begin{bmatrix}
F_1 & F_2 \\
F_2^T & F_3
\end{bmatrix}$, $\bar{F} = 2 \begin{bmatrix}
\bar{F}_1 \\
\bar{F}_2
\end{bmatrix}$ such ...
- 1
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0
answers
236
views
Eigenvectors of a matrix
Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have
$$\xi_{i}=(\lambda_1, 0,...
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54
views
Is there a method to find a vector that optimizes a Rayleigh quotient over a subspace?
Let $M\in\mathbb{C}^{n\times n}$ be an arbitary Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
Is there a method to find vectors $y,z\in E$ such that
$$\dfrac{y^*My}{y^*y}=\sup_{x\in E\\...
- 596
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votes
0
answers
141
views
Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
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