Displaying similar documents to “Investigating generalized quaternions with dual-generalized complex numbers”

G-matrices, J -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A - T = D 1 A D 2 , where A - T denotes the transpose of the inverse of A . Denote by J = diag ( ± 1 ) a diagonal (signature) matrix, each of whose diagonal entries is + 1 or - 1 . A nonsingular real matrix Q is called J -orthogonal if Q T J Q = J . Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let A , B be n × m matrices. The concept of matrix majorization means the j th column of A is majorized by the j th column of B and this is done for all j by a doubly stochastic matrix D . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

On the combinatorial structure of 0 / 1 -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A 0 / 1 -simplex is the convex hull of n + 1 affinely independent vertices of the unit n -cube I n . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0 / 1 -simplices in I n can be represented by 0 / 1 -matrices P of size n × n whose Gramians G = P P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse P - of P is doubly stochastic and has the...

( m , r ) -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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For integers m > r 0 , Brietzke (2008) defined the ( m , r ) -central coefficients of an infinite lower triangular matrix G = ( d , h ) = ( d n , k ) n , k as d m n + r , ( m - 1 ) n + r , with n = 0 , 1 , 2 , , and the ( m , r ) -central coefficient triangle of G as G ( m , r ) = ( d m n + r , ( m - 1 ) n + k + r ) n , k . It is known that the ( m , r ) -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = ( d , h ) with h ( 0 ) = 0 and d ( 0 ) , h ' ( 0 ) 0 , we obtain the generating function of its ( m , r ) -central coefficients and give an explicit representation for the ( m , r ) -central Riordan array G ( m , r ) in terms of the Riordan array G ....

Computing the greatest 𝐗 -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector x is said to be an eigenvector of a square max-min matrix A if A x = x . An eigenvector x of A is called the greatest 𝐗 -eigenvector of A if x 𝐗 = { x ; x ̲ x x ¯ } and y x for each eigenvector y 𝐗 . A max-min matrix A is called strongly 𝐗 -robust if the orbit x , A x , A 2 x , reaches the greatest 𝐗 -eigenvector with any starting vector of 𝐗 . We suggest an O ( n 3 ) algorithm for computing the greatest 𝐗 -eigenvector of A and study the strong 𝐗 -robustness. The necessary and sufficient conditions for strong 𝐗 -robustness are introduced...

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

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Let be a commutative ring, 𝒢 be a generalized matrix algebra over with weakly loyal bimodule and 𝒵 ( 𝒢 ) be the center of 𝒢 . Suppose that 𝔮 : 𝒢 × 𝒢 𝒢 is an -bilinear mapping and that 𝔗 𝔮 : 𝒢 𝒢 is a trace of 𝔮 . The aim of this article is to describe the form of 𝔗 𝔮 satisfying the centralizing condition [ 𝔗 𝔮 ( x ) , x ] 𝒵 ( 𝒢 ) (and commuting condition [ 𝔗 𝔮 ( x ) , x ] = 0 ) for all x 𝒢 . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) 𝔗 𝔮 has the so-called proper form from a new perspective. Using the aforementioned...

Possible isolation number of a matrix over nonnegative integers

LeRoy B. Beasley, Young Bae Jun, Seok-Zun Song (2018)

Czechoslovak Mathematical Journal

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Let + be the semiring of all nonnegative integers and A an m × n matrix over + . The rank of A is the smallest k such that A can be factored as an m × k matrix times a k × n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A . For A with isolation number k , we investigate the possible values of the...

Row Hadamard majorization on 𝐌 m , n

Abbas Askarizadeh, Ali Armandnejad (2021)

Czechoslovak Mathematical Journal

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An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let 𝐌 m , n be the set of all m × n real matrices. For A , B 𝐌 m , n , we say that A is row Hadamard majorized by B (denoted by A R H B ) if there exists an m × n row stochastic matrix R such that A = R B , where X Y is the Hadamard product (entrywise product) of matrices X , Y 𝐌 m , n . In this paper, we consider the concept of row Hadamard majorization as a relation on 𝐌 m , n and characterize the structure of all linear operators T : 𝐌 m , n 𝐌 m , n preserving...

The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos Martins da Fonseca (2016)

Czechoslovak Mathematical Journal

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Suppose that A is a real symmetric matrix of order n . Denote by m A ( 0 ) the nullity of A . For a nonempty subset α of { 1 , 2 , ... , n } , let A ( α ) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α . When m A ( α ) ( 0 ) = m A ( 0 ) + | α | , we call α a P-set of A . It is known that every P-set of A contains at most n / 2 elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

( 0 , 1 ) -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let m and n be positive integers, and let R = ( r 1 , ... , r m ) and S = ( s 1 , ... , s n ) be nonnegative integral vectors. Let A ( R , S ) be the set of all m × n ( 0 , 1 ) -matrices with row sum vector R and column vector S . Let R and S be nonincreasing, and let F ( R ) be the m × n ( 0 , 1 ) -matrix, where for each i , the i th row of F ( R , S ) consists of r i 1’s followed by ( n - r i ) 0’s. Let A A ( R , S ) . The discrepancy of A, disc ( A ) , is the number of positions in which F ( R ) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over...

Distance matrices perturbed by Laplacians

Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel (2020)

Applications of Mathematics

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Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s . Let D i j denote the sum of all the weights lying in the path connecting the vertices i and j of T . We now say that D i j is the distance between i and j . Define D : = [ D i j ] , where D i i is the s × s null matrix and for i j , D i j is the distance between i and j . Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of order...

L p , q spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the L p , q spaces................................. 114. Integral representation of bounded linear functionals on L p , q ( B ) ........ 235. Examples in L p , q theory...................................................................................

Ground states of supersymmetric matrix models

Gian Michele Graf (1998-1999)

Séminaire Équations aux dérivées partielles

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We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the d = 9 model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in d = 9 . Moreover, it would be unique. Other values of d , where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation....

Certain additive decompositions in a noncommutative ring

Huanyin Chen, Marjan Sheibani, Rahman Bahmani (2022)

Czechoslovak Mathematical Journal

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We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a 2 × 2 matrix A over a projective-free ring R is strongly J -clean if and only if A J ( M 2 ( R ) ) , or I 2 - A J ( M 2 ( R ) ) , or A is similar to 0 λ 1 μ , where λ J ( R ) , μ 1 + J ( R ) , and the equation x 2 - x μ - λ = 0 has a root in J ( R ) and a root in 1 + J ( R ) . We further prove that f ( x ) R [ [ x ] ] is strongly J -clean if f ( 0 ) R be optimally J -clean.

On linear preservers of two-sided gut-majorization on 𝐌 n , m

Asma Ilkhanizadeh Manesh, Ahmad Mohammadhasani (2018)

Czechoslovak Mathematical Journal

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For X , Y 𝐌 n , m it is said that X is gut-majorized by Y , and we write X gut Y , if there exists an n -by- n upper triangular g-row stochastic matrix R such that X = R Y . Define the relation gut as follows. X gut Y if X is gut-majorized by Y and Y is gut-majorized by X . The (strong) linear preservers of gut on n and strong linear preservers of this relation on 𝐌 n , m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of gut on n and 𝐌 n , m .