scholarly journals One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 539
Author(s):  
Seda Yamaç Akbiyik ◽  
Mücahit Akbiyik ◽  
Fatih Yilmaz
Keyword(s):  
Symmetric Matrix ◽  
Harmonic Numbers ◽  
Pell Numbers ◽  
Algebraic Properties ◽  
The Matrix ◽  
Matrix Norms ◽  
Exponential Matrix

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

Pell Numbers, Pell–Lucas Numbers and Modular Group

2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat
Keyword(s):  
Modular Group ◽  
Symmetric Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Pell Numbers ◽  
The Matrix

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

scholarly journals A Convenient Method to Obtain Stellar Eigenfrequencies

1983 ◽  
Vol 66 ◽  
pp. 331-341
Author(s):  
M. Knölker ◽  
M. Stix
Keyword(s):  
Eigenvalue Problem ◽  
Model Comparison ◽  
Symmetric Matrix ◽  
Outer Boundary ◽  
Solar Model ◽  
Pressure Perturbation ◽  
Stellar Oscillations ◽  
The Matrix ◽  
First Results ◽  
Algebraic Eigenvalue Problem

AbstractThe differential equations describing stellar oscillations are transformed into an algebraic eigenvalue problem. Frequencies of adiabatic oscillations are obtained as the eigenvalues of a banded real symmetric matrix. We employ the Cowling-approximation, i.e. neglect the Eulerian perturbation of the gravitational potential, and, in order to preserve selfadjointness, require that the Eulerian pressure perturbation vanishes at the outer boundary. For a solar model, comparison of first results with results obtained from a Henyey method shows that the matrix method is convenient, accurate, and fast.

A Trust Evaluation of Networks Utilizing Matrix Operations Based on t-norms and t-conorms

2012 ◽  
Vol 16 (1) ◽  
pp. 154-161
Author(s):  
Masaya Nohmi ◽  
◽  
Aoi Honda ◽  
Yoshiaki Okazaki
Keyword(s):  
Trust Evaluation ◽  
Matrix Operations ◽  
Fuzzy Graphs ◽  
Algebraic Properties ◽  
The Matrix ◽  
Membership Value

A new scheme for numerical trust evaluation of networks is proposed. Matrix operations based on t-norms and t-conorms are used for the evaluation. The algebraic properties of the matrix operations are studied. Fuzzy graphs, in which nodes are linked with some membership value, are proposed, using the matrices as adjacent matrices. Furthermore, the fuzzinesses of the trustability distribution are calculated.

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.

scholarly journals Optimized Cramer’s Rule in WZ Factorization and Applications

2020 ◽  
Vol 13 (4) ◽  
pp. 1035-1054
Author(s):  
Olayiwola Babarinsa ◽  
Azfi Zaidi Mohammad Sofi ◽  
Mohd Asrul Hery Ibrahim ◽  
Hailiz Kamarulhaili
Keyword(s):  
Sparse Matrices ◽  
Schur Complement ◽  
Matrix Group ◽  
Lu Factorization ◽  
Triangular Matrices ◽  
Performance Time ◽  
The Matrix ◽  
Cramer’S Rule ◽  
And Performance ◽  
Matrix Norms

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.

Complexity issues for the symmetric interval eigenvalue problem

2014 ◽  
Vol 13 (1) ◽  
Author(s):  
Milan Hladík
Keyword(s):  
Computational Complexity ◽  
Eigenvalue Problem ◽  
Relative Error ◽  
Symmetric Matrix ◽  
Approximation Error ◽  
Np Hard ◽  
Approximation Errors ◽  
The Matrix ◽  
Polynomially Solvable ◽  
Relative Approximation

AbstractWe study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.

scholarly journals Newton's method for the eigenvalue problem of a symmetric matrix

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 75-82
Author(s):  
V. V. Verbitskyi ◽  
A. G. Huk
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Symmetric Matrix ◽  
Simple Eigenvalue ◽  
Normalization Condition ◽  
Nonlinear System Of Equations ◽  
Computational Costs ◽  
The Matrix ◽  
Eigenvalue And Eigenvector

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

scholarly journals The enhanced principal rank characteristic sequence over a field of characteristic 2

2017 ◽  
Vol 32 ◽  
pp. 273-290 ◽  
Author(s):  
Xavier Martínez-Rivera
Keyword(s):  
Symmetric Matrix ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Characteristic Sequence ◽  
The Matrix ◽  
Characteristic 2

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over $\Z_2$ possess very special structures. For more general fields of characteristic $2$, some restrictions on attainable epr-sequences are obtained.

scholarly journals Bounds for the Completely Positive Rank of a Symmetric Matrix over a Tropical Semiring

2018 ◽  
Vol 34 ◽  
pp. 152-162
Author(s):  
David Dolžan ◽  
Polona Oblak
Keyword(s):  
Upper Bound ◽  
Symmetric Matrix ◽  
Completely Positive ◽  
Tropical Semiring ◽  
The Matrix ◽  
Rank Of A Matrix ◽  
Positive Rank

In this paper, an upper bound for the CP-rank of a matrix over a tropical semiring is obtained, according to the vertex clique cover of the graph prescribed by the positions of zero entries in the matrix. The graphs that beget the matrices with the lowest possible CP-ranks are studied, and it is proved that any such graph must have its diameter equal to $2$.

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