scholarly journals A Study on the Determinants of some Hessenberg Toeplitz Bohemians

2020 ◽  
Author(s):  
Jishe Feng ◽  
Hongtao Fan
Keyword(s):  
Linear Algebra ◽  
Toeplitz Matrices ◽  
Electronic Journal ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Hessenberg Matrices

In this paper, we deduce explicit formulas to evaluate the determinants of nonsymmetrical structure Toeplitz Bohemians by two determinants of specific Hessenberg Toeplitz matrices, which are linear combinations in terms of determinants of specific Hessenberg Toeplitz matrices. We get some new results very di¤erent from [Massimiliano Fasi, Gian Maria Negri Porzio, Determinants of normalized upper Hessenberg matrices, Electronic Journal of Linear Algebra, Volume 36, pp. 352-366, June 2020].

scholarly journals The Explicit Formulae of the Determinants of Specific Pentadiagonal Toeplitz Hessenberg Matrices

2021 ◽  
Vol 2068 (1) ◽  
pp. 012007
Author(s):  
Jishe Feng ◽  
Hongtao Fan
Keyword(s):  
Linear Algebra ◽  
Toeplitz Matrix ◽  
Recurrence Relations ◽  
Sparse Matrix ◽  
Toeplitz Matrices ◽  
Special Kind ◽  
Computational Mathematics ◽  
Hessenberg Matrices ◽  
Explicit Formulae

Abstract The pentadiagonal Toeplitz matrix is a special kind of sparse matrix widely used in linear algebra, combinatorics, computational mathematics, and has been attracted much attention. We use the determinants of two specific Hessenberg matrices to represent the recurrence relations to prove two explicit formulae to evaluate the determinants of specific pentadiagonal Toeplitz matrices proposed in a recent paper [3]. Further, four new results are established.

scholarly journals Computation of several Hessenberg determinants

2020 ◽  
Vol 70 (6) ◽  
pp. 1521-1537
Author(s):  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Linear Algebra ◽  
Simple Formula ◽  
Binomial Coefficients ◽  
Explicit Formulas ◽  
Methods And Techniques

AbstractIn the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

Preface: Special volume of Electronic Journal of Linear Algebra dedicated to Professor Ravindra B. Bapat

2015 ◽  
Vol 29 ◽  
pp. 1-2
Author(s):  
Rajendra Bhatia ◽  
Steve Kirkland ◽  
K. Prasad ◽  
Simo Puntanen
Keyword(s):  
Linear Algebra ◽  
Matrix Theory ◽  
60Th Birthday ◽  
Electronic Journal ◽  
Special Volume ◽  
Matrix Methods ◽  
International Conference ◽  
Combinatorial Matrix Theory

This special volume of the Electronic Journal of Linear Algebra is dedicated to Professor Ravindra B. Bapat on the occasion of his 60th birthday. The volume contains papers related to the International Conference on Linear Algebra & its Applications, which was held December 18--20, 2014 at the Department of Statistics of Manipal University in Manipal, India. The theme of conference focused on (i) Matrix Methods in Statistics, (ii) Combinatorial Matrix Theory and (iii) Classical Matrix Theory covering different aspects of Linear Algebra.

Special functions, infinite divisibility and transcendental equations

1979 ◽  
Vol 85 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
C. Ping May
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Infinite Divisibility ◽  
Probability Density Functions ◽  
Integral Representations ◽  
Density Functions ◽  
Modified Bessel Functions ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Transcendental Equations

AbstractWe establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

scholarly journals Computation of several Hessenberg determinants

2020 ◽  
Author(s):  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Linear Algebra ◽  
Simple Formula ◽  
Binomial Coefficients ◽  
Explicit Formulas ◽  
Methods And Techniques

In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

Matrix Properties, Fundamental Spaces, Orthogonality

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Linear Algebra ◽  
Null Space ◽  
Vector Spaces ◽  
Linear Combinations ◽  
Matrix Rank ◽  
Column Space ◽  
Matrix Properties ◽  
The Matrix ◽  
Schmidt Orthogonalization ◽  
Row Space

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an M × N matrix is associated with four fundamental spaces. The column space is the set of all M-vectors that are linear combinations of the columns. If the matrix has M independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left null space, the set of all M-vectors that the matrix’s s transpose maps to the zero N-vector. The row space is the set of all N-vectors that are linear combinations of the rows. If the matrix has N independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the null space, the set of all N-vectors that the matrix maps to the zero M-vector.

scholarly journals The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2147
Author(s):  
Marina Tošić ◽  
Eugen Ljajko ◽  
Nataša Kontrec ◽  
Vladica Stojanović
Keyword(s):  
Linear Combination ◽  
Linear Algebra ◽  
Linear Combinations ◽  
Explicit Formulae

Baksalary et al. (Linear Algebra Appl., doi:10.1016/j.laa.2004.02.025, 2004) investigated the invertibility of a linear combination of idempotent matrices. This result was improved by Koliha et al. (Linear Algebra Appl., doi:10.1016/j.laa.2006.01.011, 2006) by showing that the rank of a linear combination of two idempotents is constant. In this paper, we consider similar problems for k-potent matrices. We study the rank and the nullity of a linear combination of two commuting k-potent matrices. Furthermore, the problem of the nonsingularity of linear combinations of two or three k-potent matrices is considered under some conditions. In these situations, we derive explicit formulae of their inverses.

A Linear Algebra Approach to the Analysis of Rigid Body Displacement From Initial and Final Position Data

1982 ◽  
Vol 49 (1) ◽  
pp. 213-216 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Rigid Body ◽  
Linear Algebra ◽  
Initial Position ◽  
The Body ◽  
Final Position ◽  
Explicit Formulas ◽  
Algebra Approach ◽  
Other Information ◽  
Position Data ◽  
Rigid Body Displacement

The location and orientation of a rigid body in space can be defined in terms of three noncollinear points in the body. As the rigid body is moved through space, the motion may be described by a series of rotations and translations. The sequence of displacements may be conveniently represented in matrix form by a series of displacement matrices that describe the motion of the body between successive positions. If the rotations and translations (and hence the displacement matrix) are known then succeeding positions of a rigid body may be easily calculated in terms of the initial position. Conversely, if successive positions of three points in the rigid body are known, it is possible to calculate the parameters of the corresponding rotation and translation. In this paper, a new solution is presented which provides explicit formulas for the rotation and translation of a rigid body in terms of the initial and final positions of three points fixed in the rigid body. The rotation matrix is determined directly whereupon appropriate rotation angles and other information can subsequently be calculated if desired.

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