The Effect of Perturbation of an Off-diagonal Entry Pair on the Geometric Multiplicity of an Eigenvalue

2020 ◽  
Author(s):  
Kenji Toyonaga ◽  
Charles R. Johnson
Keyword(s):  
Diagonal Entry ◽  
Geometric Multiplicity ◽  
Multiplicity Of An Eigenvalue

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 An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 745
Author(s):  
Weiqian Zhang ◽  
Chaoqian Li
Keyword(s):  
Toeplitz Matrices ◽  
Diagonal Entry ◽  
Main Diagonal ◽  
Multilinear Algebra ◽  
Inclusion Set

A set to locate all eigenvalues for matrices with a constant main diagonal entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant main diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.

VECTOR RANDOM FIELDS WITH LONG-RANGE DEPENDENCE

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 249-258 ◽  
Author(s):  
CHUNSHENG MA
Keyword(s):  
Long Range ◽  
Random Fields ◽  
Random Function ◽  
Short Range ◽  
Diagonal Entry ◽  
Random Functions ◽  
Cross Covariance ◽  
Direct Covariance ◽  
Crucial Ingredient ◽  
Special Case

It is well-known that the crucial ingredient for a vector Gaussian random function is its covariance matrix, where a diagonal entry termed a direct covariance is simply the covariance function of a component but it seems no simple interpretation for an off-diagonal entry termed a cross covariance, which often make it hard to specify. In this paper we employ three approaches to derive vector random functions in space and/or time, which are not homogeneous (stationary) in general but contain the stationary case as a special case, and have long-range or short-range dependence.

scholarly journals On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs

2016 ◽  
Vol 31 ◽  
pp. 187-199
Author(s):  
Keivan Hassani Monfared ◽  
Paul Horn ◽  
Franklin Kenter ◽  
Kathleen Nowak ◽  
John Sinkovic ◽  
...  
Keyword(s):  
Binary Sequence ◽  
Diagonal Entry ◽  
Symmetric Matrices ◽  
Nonnegative Matrices ◽  
Characteristic Sequence ◽  
The Family ◽  
Characteristic Sequences

The principal permanent rank characteristic sequence is a binary sequence r_0 r_1 · · · r_n, where r_k = 1 if there exists a principal square submatrix of size k with nonzero permanent and r_k = 0 otherwise, and r_0 = 1 if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.

Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point

2021 ◽  
Vol 32 (5) ◽  
pp. 847-864
Author(s):  
A. Budylin
Keyword(s):  
Stationary Point ◽  
Hilbert Problem ◽  
Conjugacy Problem ◽  
Diagonal Entry ◽  
Content Type ◽  
Quadratic Phase ◽  
Shock Region ◽  
The Cauchy Problem ◽  
Leading Term ◽  
Power Orders

The ( 2 × 2 ) (2\times 2) matrix conjugacy problem (the Riemann–Hilbert problem) with rapidly oscillating off-diagonal entries and quadratic phase function is considered, specifically, the case when one of the diagonal entries vanishes at a stationary point. For solutions of this problem, the leading term of the asymptotics is found. However, the method allows us to construct complete expansions in power orders. These asymptotics can be used, for example, to construct the asymptotics of solutions of the Cauchy problem for the nonlinear Schrödinger equation for large times in the case of the so-called collisionless shock region.

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