scholarly journals Pairs of matrices with a non-zero commutator

1955 ◽  
Vol 51 (4) ◽  
pp. 551-553 ◽  
Author(s):  
L. S. Goddard ◽  
H. Schneider
Keyword(s):  
Characteristic Root ◽  
Column Vector ◽  
Closed Field ◽  
Complex Matrix ◽  
Algebraically Closed Field ◽  
Characteristic Roots ◽  
Linearly Independent ◽  
Special Cases ◽  
The Matrix ◽  
Square Complex

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.

scholarly journals Approximating the Matrix Sign Function Using a Novel Iterative Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
F. Soleymani ◽  
P. S. Stanimirović ◽  
S. Shateyi ◽  
F. Khaksar Haghani
Keyword(s):  
Iterative Method ◽  
Asymptotical Stability ◽  
Complex Matrix ◽  
Initial Matrix ◽  
Matrix Sign Function ◽  
Sign Function ◽  
The Matrix ◽  
Square Complex

This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

Triangular representations of linear algebras

1953 ◽  
Vol 49 (4) ◽  
pp. 595-600 ◽  
Author(s):  
M. P. Drazin
Keyword(s):  
Commutative Algebra ◽  
Similarity Transformation ◽  
Closed Field ◽  
Triangular Form ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Common Eigenvector ◽  
General Systems ◽  
The Matrix ◽  
Matrix Case

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).

Pseudo-Tournament Matrices and Their Eigenvalues

2014 ◽  
Vol 4 (3) ◽  
pp. 205-221
Author(s):  
Chuanlong Wang ◽  
Xuerong Yong
Keyword(s):  
Directed Graph ◽  
Round Robin ◽  
Column Vector ◽  
Complex Matrix ◽  
The Matrix ◽  
Tournament Matrices ◽  
Class Of Matrices

AbstractA tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An n × n complex matrix A is called h-pseudo-tournament if there exists a complex or real nonzero column vector h such that A + A* = hh* − I. This class of matrices is a generalisation of well-studied tournament-like matrices such as h-hypertournament matrices, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of an h-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.

scholarly journals The Matrix Equation XA- AX=Xαg(X) over Fields or Rings

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Gerald Bourgeois
Keyword(s):  
Commutative Ring ◽  
Matrix Equation ◽  
Zero Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Reduced Ring ◽  
The Matrix ◽  
Generic Matrix

Let n,α∈N≥2 and let K be an algebraically closed field with characteristic 0 or greater than n. We show that if f∈K[X] and A,B∈Mn(K) satisfy [A,B]=f(A), then A,B are simultaneously triangularizable. Let R be a reduced ring such that n! is not a zero divisor and let A be a generic matrix over R; we show that X=0 is the sole solution of AX-XA=Xα. Let R be a commutative ring with unity; let A be similar to diag(λ1In1,…,λrInr) such that, for every i≠j,λi-λj is not a zero divisor. If X is a nilpotent solution of XA-AX=Xαg(X) where g∈R[X], then AX=XA.

On bad groups, bad fields, and pseudoplanes

1991 ◽  
Vol 56 (3) ◽  
pp. 915-931 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Projective Planes ◽  
Closed Field ◽  
Morley Rank ◽  
Algebraically Closed Field ◽  
Desarguesian Projective Planes ◽  
Natural Geometry ◽  
Special Cases ◽  
Finite Morley Rank ◽  
Rank 2 ◽  
Open Question

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and1. K = A − A,2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.If K is finite this is equivalent to 1 and2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.Finite sharp-fields are special cases of difference sets [De]

Thermal Stresses in a Generally Anisotropic Body With an Elliptic Inclusion Subject to Uniform Heat Flow

1998 ◽  
Vol 65 (1) ◽  
pp. 51-58 ◽  
Author(s):  
C. K. Chao ◽  
M. H. Shen
Keyword(s):  
Heat Flow ◽  
Thermal Stresses ◽  
Anisotropic Body ◽  
Analytical Continuation ◽  
Complex Matrix ◽  
Elliptic Inclusion ◽  
Uniform Heat ◽  
Special Cases ◽  
The Matrix ◽  
Stress Functions

A general analytical solution for the elliptical anisotropic inclusion embedded in an infinite anisotropic matrix subjected to uniform heat flow is provided in this paper. Based upon the method of Lekhnitskii formulation, the technique of conformal mapping, the method of analytical continuation, and the concept of superposition, both the solutions of the temperature and stress, functions either in the matrix or in the inclusion are expressed in complex matrix notation. Numerical results are carried out and provided in graphic form to elucidate the effect of material and geometric parameters on the interfacial stresses. Since the general solutions have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our solutions presented here are exact and general.

Codimension growth of central polynomials of Lie algebras

2020 ◽  
Vol 32 (1) ◽  
pp. 201-206
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Adjoint Representation ◽  
Closed Field ◽  
Polynomial Identities ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Linearly Independent

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like {(\dim L)^{n}}.

On the Anisotropic Elastic Inclusions in Plane Elastostatics

1993 ◽  
Vol 60 (3) ◽  
pp. 626-632 ◽  
Author(s):  
Chyanbin Hwu ◽  
Wen J. Yen
Keyword(s):  
Major Axis ◽  
Analytical Continuation ◽  
Complex Matrix ◽  
Arbitrary Loading ◽  
Special Cases ◽  
The Matrix ◽  
Point Forces ◽  
Elastic Inclusions ◽  
Anisotropic Elastic ◽  
Plane Elastostatics

By combining the method of Stroh’s formalism, the concept of perturbation, the technique of conformal mapping and the method of analytical continuation, a general analytical solution for the elliptical anisotropic elastic inclusions embedded in an infinite anisotropic matrix subjected to an arbitrary loading has been obtained in this paper. The inclusion as well as the matrix are of general anisotropic elastic materials which do not imply any material symmetry. The special cases when the inclusion is rigid or a hole are also studied. The arbitrary loadings include in-plane and antiplane loadings. The shapes of ellipses cover the lines or circles when the minor axis is taken to be zero or equal to the major axis. The solutions of the stresses and deformations in the entire domain are expressed in complex matrix notation. Simplified results are provided for the interfacial stresses along the inclusion boundary. Some interesting examples are solved explicitly, such as point forces or dislocations in the matrix and uniform loadings at infinity. Since the general solutions have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our results are exact and universal.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

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