Basic Matrix Theory and Linear Algebra

This paper is dedicated to reviewing some of the basic concepts in neutrosophic linear algebra and its generalizations, especially neutrosophic vector spaces, refined neutrosophic, and n-refined neutrosophic vector spaces. Also, this work gives the interested reader a strong background in the study of neutrosophic matrix theory and n-refined neutrosophic matrix theory. We study elementary properties of these concepts such as Kernel, AH-Quotient, and dimension.

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Linear Algebra and Matrix Theory

The Mathematical Gazette ◽

10.2307/3615317 ◽

1972 ◽

Vol 56 (396) ◽

pp. 161

Author(s):

W. G. Kellaway ◽

Evar D. Nering

Keyword(s):

Linear Algebra ◽

Matrix Theory

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1. Background in Matrix Theory and Linear Algebra

Numerical Methods for Large Eigenvalue Problems ◽

10.1137/1.9781611970739.ch1 ◽

2011 ◽

pp. 1-27 ◽

Cited By ~ 6

Keyword(s):

Linear Algebra ◽

Matrix Theory

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Numerical Linear Algebra

10.20850/9781312329850 ◽

2017 ◽

Author(s):

William Layton ◽

Myron Sussman

Keyword(s):

Linear Algebra ◽

Undergraduate Students ◽

Eigenvalue Problems ◽

Matrix Theory ◽

Direct Methods ◽

Numerical Linear Algebra ◽

Good Choice ◽

Computational Mathematics ◽

Diverse Backgrounds ◽

Sparse Systems

This textbook was designed for senior undergraduates in mathematics, engineering and the sciences with diverse backgrounds and goals. It presents modern tools from numerical linear algebra with supporting theory along with examples and exercises, both theoretical and computational with MATLAB. The major topics of numerical linear algebra covered are direct methods for solving linear systems, iterative methods for large and sparse systems (including the conjugate gradient method) and eigenvalue problems. Basic linear algebra (of the type taken in the engineering curriculum) is assumed. Further matrix theory is developed in a self-contained way when needed to expalin why methods work and how they might fail. This book is intended for a one term course for undergraduate students in applied and computational mathematics, the sciences, engineering, computer science, financial mathematics and actuarial science. It is also a good choice for a beginning graduate level course for students who will use the numerical methods to solve problems in their own areas.

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The Number of Spanning Trees in the Composition Graphs

Mathematical Problems in Engineering ◽

10.1155/2014/613685 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Feng Li

Keyword(s):

Linear Algebra ◽

Spanning Trees ◽

Matrix Theory ◽

Structural Parameters ◽

The Other ◽

Arbitrary Graph ◽

Large Graphs ◽

Laplacian Eigenvalues ◽

Number Of Spanning Trees

Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of vertices and eigenvalues of the small graphs only.

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Complexity of Some Generalized Operations on Networks

Complexity ◽

10.1155/2021/9999157 ◽

2021 ◽

Vol 2021 ◽

pp. 1-29

Author(s):

Muhammad Javaid ◽

Hafiz Usman Afzal ◽

Shaohui Wang

Keyword(s):

Linear Algebra ◽

Chebyshev Polynomials ◽

Spanning Trees ◽

Matrix Theory ◽

Connected Components ◽

Number Of Spanning Trees

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of S n ), sum ( S n + W 3 , S n + K 2 , and C n ∘ K 2 + K 1 ), product ( S n ⊠ K 2 and W n ∘ K 2 ), semitotal networks ( Q S n and R S n ), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.

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