scholarly journals New Graphs of Finite Mutation Type

10.37236/863 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Harm Derksen ◽  
Theodore Owen
Keyword(s):  
Directed Graph ◽  
Symmetric Matrix ◽  
Cluster Algebras ◽  
Symmetric Matrices ◽  
Finite Graphs ◽  
Isomorphism Classes ◽  
Geometric Type ◽  
Symmetrizable Matrices ◽  
Marked Points ◽  
Mutation Type

To a directed graph without loops or $2$-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph $\Gamma$ is the set of all isomorphism classes of graphs that can be obtained from $\Gamma$ by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs "of geometric type". Besides graphs with $2$ vertices, and graphs of geometric type, there are only 9 other "exceptional" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes.

scholarly journals Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

10.37236/6230 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
John W Lawson
Keyword(s):  
Mapping Class Groups ◽  
Cluster Algebras ◽  
Mapping Class ◽  
Symmetric Matrices ◽  
Class Groups ◽  
Exchange Graph ◽  
Symmetrizable Matrices

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

scholarly journals On Turnbull identity for skew-symmetric matrices

2000 ◽  
Vol 43 (2) ◽  
pp. 379-393 ◽  
Author(s):  
Tôru Umeda ◽  
Takeshi Hirai
Keyword(s):  
Differential Operators ◽  
Symmetric Matrix ◽  
Point Of View ◽  
Symmetric Matrices ◽  
Invariant Differential Operators ◽  
Type Identity ◽  
Central Elements ◽  
Theoretic Point ◽  
Invariant Differential ◽  
Matrix Spaces

AbstractIn the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements inU(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

Recovery of Low Rank Symmetric Matrices via Schatten p Norm Minimization

2016 ◽  
Vol 33 (01) ◽  
pp. 1650003
Author(s):  
Li Cui ◽  
Lu Liu ◽  
Di-Rong Chen ◽  
Jian-Feng Xie
Keyword(s):  
Symmetric Matrix ◽  
Linear Measurement ◽  
Low Rank ◽  
Symmetric Matrices ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Matrix Formula ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Recovery Problem

In this paper, we give an application of the perturbation inequality to the low rank matrix recovery problem and provide a condition on the linear map of underdetermined linear system that every minimal rank symmetric matrix [Formula: see text] can be exactly recovered from the linear measurement [Formula: see text] via some Schatten [Formula: see text] norm minimization. Moreover it is shown that the explicit bound on exponent [Formula: see text] in the Schatten [Formula: see text] norm minimization can be exactly extracted.

LU-FACTORIZATIONS OF SYMMETRIC MATRICES WITH APPLICATIONS

2010 ◽  
Vol 03 (01) ◽  
pp. 133-143 ◽  
Author(s):  
Guoyou Qian ◽  
Jingya Lu
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Lu Factorization

In this paper, we describe explicitly the LU -factorization of a symmetric matrix of order n with n ≤ 7 when each of its ordered principal minors is nonzero. By using this result and some other related results on non-singularity previously given by Smith, Beslin, Hong, Lee and Ligh in the literature, we establish several theorems concerning LU -factorizations of power GCD matrices, power LCM matrices and reciprocal power GCD matrices and reciprocal power LCM matrices.

scholarly journals A special property of the matrix Riccati equation

1974 ◽  
Vol 10 (2) ◽  
pp. 245-253 ◽  
Author(s):  
A.N. Stokes
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Symmetric Matrix ◽  
Special Property ◽  
Positive Definiteness ◽  
Symmetric Matrices ◽  
Unusual Property ◽  
Riccati Differential Equation ◽  
The Matrix ◽  
Independent Variable

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

Cayley formula in Minkowski space-time

2015 ◽  
Vol 12 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Time ◽  
Rotation Matrix ◽  
Symmetric Matrices ◽  
Rotation Matrices ◽  
Minkowski Space Time ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in [Formula: see text]. For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi-skew-symmetric matrices A1 and A2 satisfying the properties [Formula: see text] and [Formula: see text] Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay (A).

scholarly journals A class of constrained inverse eigenvalue problem and associated approximation problem for symmetrizable matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1903-1909
Author(s):  
Xiangyang Peng ◽  
Wei Liu ◽  
Jinrong Shen
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Original Matrix ◽  
Approximation Solution ◽  
Solvability Conditions ◽  
Inverse Eigenvalue ◽  
Symmetrizable Matrices

The real symmetric matrix is widely applied in various fields, transforming non-symmetric matrix to symmetric matrix becomes very important for solving the problems associated with the original matrix. In this paper, we consider the constrained inverse eigenvalue problem for symmetrizable matrices, and obtain the solvability conditions and the general expression of the solutions. Moreover, we consider the corresponding optimal approximation problem, obtain the explicit expressions of the optimal approximation solution and the minimum norm solution, and give the algorithm and corresponding computational example.

scholarly journals On the smallest singular value of symmetric random matrices

2021 ◽  
pp. 1-22
Author(s):  
Vishesh Jain ◽  
Ashwin Sah ◽  
Mehtaab Sawhney
Keyword(s):  
Random Matrices ◽  
Symmetric Matrix ◽  
Geometric Approach ◽  
Random Variable ◽  
Symmetric Matrices ◽  
Common Denominator ◽  
Geometric Framework ◽  
Small Constant ◽  
Special Case ◽  
Smallest Singular Value

Abstract We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

scholarly journals On the Generalized Cluster Algebras of Geometric Type

2020 ◽  
Author(s):  
Liqian Bai ◽  
◽  
Xueqing Chen ◽  
Ming Ding ◽  
Fan Xu ◽  
...  
Keyword(s):  
Cluster Algebras ◽  
Geometric Type
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