scholarly journals Eventually DSDD Matrices and Eigenvalue Localization

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 448 ◽  
Author(s):  
Caili Sang ◽  
Jianxing Zhao
Keyword(s):  
Sufficient Condition ◽  
Infinity Norm ◽  
Eigenvalue Localization ◽  
Inclusion Set ◽  
Diagonally Dominant

Firstly, the relationships among strictly diagonally dominant ( S D D ) matrices, doubly strictly diagonally dominant ( D S D D ) matrices, eventually S D D matrices and eventually D S D D matrices are considered. Secondly, by excluding some proper subsets of an existing eigenvalue inclusion set for matrices, which do not contain any eigenvalues of matrices, a tighter eigenvalue inclusion set of matrices is derived. As its application, a sufficient condition of determining non-singularity of matrices is obtained. Finally, the infinity norm estimation of the inverse of eventually D S D D matrices is derived.

scholarly journals New Sufficient Condition for the Positive Definiteness of Fourth Order Tensors

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 303 ◽  
Author(s):  
Jun He ◽  
Yanmin Liu ◽  
Junkang Tian ◽  
Zhuanzhou Zhang
Keyword(s):  
Spectral Radius ◽  
Fourth Order ◽  
Upper Bound ◽  
Positive Definiteness ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Weakly Symmetric ◽  
Localization Set

In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

scholarly journals Note on error bounds for linear complementarity problems involving $ B^S $-matrices

2021 ◽  
Vol 7 (2) ◽  
pp. 1896-1906
Author(s):  
Deshu Sun ◽  
Keyword(s):  
Error Bounds ◽  
Complementarity Problems ◽  
Matrix Theory ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Upper Bounds ◽  
Theory Analysis ◽  
Infinity Norm ◽  
S Matrix ◽  
Diagonally Dominant

<abstract><p>Using the range for the infinity norm of inverse matrix of a strictly diagonally dominant $ M $-matrix, some new error bounds for the linear complementarity problem are obtained when the involved matrix is a $ B^S $-matrix. Theory analysis and numerical examples show that these upper bounds are more accurate than some existing results.</p></abstract>

New Criteria for Nonsingular H-Matrices

2014 ◽  
Vol 1006-1007 ◽  
pp. 1043-1046
Author(s):  
Gui Chun Han
Keyword(s):  
Sufficient Condition ◽  
Equivalent Representation ◽  
Numerical Example ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
New Criteria

The concept of quasi-double diagonally dominant matrices is introduced, and it’s equivalent representation is given. As its application, one sufficient condition for nonsingular H-matrices is obtained. The effectiveness of the proposed criterion for nonsingular H-matrices is showed by numerical example.

scholarly journals Two new eigenvalue localization sets for tensors and theirs applications

2017 ◽  
Vol 15 (1) ◽  
pp. 1267-1276 ◽  
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Linear Algebra ◽  
Symmetric Tensor ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Even Order ◽  
Weakly Symmetric ◽  
Localization Set ◽  
Theoretical Results ◽  
Localization Sets

Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

scholarly journals The Diagonally Dominant Degree and Disc Separation for the Schur Complement of Ostrowski Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Jianxing Zhao ◽  
Feng Wang ◽  
Yaotang Li
Keyword(s):  
Upper Bound ◽  
Schur Complement ◽  
Numerical Examples ◽  
Infinity Norm ◽  
Diagonally Dominant ◽  
Inequality Techniques

By applying the properties of Schur complement and some inequality techniques, some new estimates of diagonally and doubly diagonally dominant degree of the Schur complement of Ostrowski matrix are obtained, which improve the main results of Liu and Zhang (2005) and Liu et al. (2012). As an application, we present new inclusion regions for eigenvalues of the Schur complement of Ostrowski matrix. In addition, a new upper bound for the infinity norm on the inverse of the Schur complement of Ostrowski matrix is given. Finally, we give numerical examples to illustrate the theory results.

scholarly journals Schur Complement-Based Infinity Norm Bounds for the Inverse of GDSDD Matrices

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 186
Author(s):  
Yating Li ◽  
Yaqiang Wang
Keyword(s):  
Lower Bound ◽  
Schur Complement ◽  
Upper Bounds ◽  
Singular Value ◽  
Numerical Examples ◽  
Infinity Norm ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Norm Bound ◽  
Smallest Singular Value

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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