Compounds of Skew-Symmetric Matrices

1964 ◽  
Vol 16 ◽  
pp. 473-478 ◽  
Author(s):  
Marvin Marcus ◽  
Adil Yaqub
Keyword(s):  
Symmetric Matrices ◽  
The Real ◽  
Real Solutions ◽  
Interesting Paper ◽  
The Matrix ◽  
Image Position ◽  
Square Matrix

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

The Real Solutions of Equation of in Control Theory

2010 ◽  
Vol 121-122 ◽  
pp. 911-915
Author(s):  
Xue Ting Liu
Keyword(s):  
Control Theory ◽  
Chemical Engineering ◽  
Research Field ◽  
Matrix Equations ◽  
The Real ◽  
Real Solutions ◽  
Nonsingular Solution ◽  
The Matrix ◽  
Active Research ◽  
Solutions Of Equations

. The research of matrix equations is an active research field, matrix equations have applied in many physical applications in recent years. As one of them, the equation is applied more and more extensively, such as control theory, chemistry and chemical engineering and so on. In this paper, motivated by [1], we give two discriminations about the real solutions of equation . The matrix is proved that it is a nonsingular solution of equation whenever are nonsingular solutions of equations at last.

On the Schur product ofH-matrices and non-negative matrices, and related inequalities

1964 ◽  
Vol 60 (3) ◽  
pp. 425-431 ◽  
Author(s):  
M. S. Lynn
Keyword(s):  
Real Line ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Schur Product ◽  
Symmetric Positive Definite ◽  
The Matrix ◽  
Image Position ◽  
Symmetric Positive Definite Matrices

1.Introduction. Let ℛndenote the set of alln×nmatrices with real elements, and letdenote the subset of ℛnconsisting of all real,n×n, symmetric positive-definite matrices. We shall use the notationto denote that minor of the matrixA= (aij) ∈ ℛnwhich is the determinant of the matrixTheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned bywhereA= (aij),B= (bij),C= (cij) andLet ϕ be the mapping of ℛninto the real line defined byfor allA∈ ℛn, where, as in the sequel,.

A Geometrical Approach to the Second-Order Linear Differential Equation

1962 ◽  
Vol 14 ◽  
pp. 349-358 ◽  
Author(s):  
C. M. Petty ◽  
J. E. Barry
Keyword(s):  
Differential Equation ◽  
Minkowski Plane ◽  
Geometrical Approach ◽  
Order Linear Differential Equation ◽  
The Real ◽  
Additional Tool ◽  
Real Solutions ◽  
Convex Curves ◽  
Linear Differential ◽  
Image Position

In this paper various concepts intrinsically defined by the differential equation1.1are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).Throughout this paper, R(t) is a real-valued, continuous function of t on the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.

Summability Methods on Matrix Spaces

1961 ◽  
Vol 13 ◽  
pp. 63-77 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Symmetric Matrix ◽  
Complex Numbers ◽  
Symmetric Matrices ◽  
Bounded Symmetric Domains ◽  
Summability Methods ◽  
Distinguished Boundary ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Image Position ◽  
Matrix Spaces

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by(1)where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying(2)where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by(3)(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

The real solutions of a certain non-linear system of equations

1961 ◽  
Vol 57 (3) ◽  
pp. 503-506
Author(s):  
N. R. Lebovitz
Keyword(s):  
Linear System ◽  
System Of Equations ◽  
Inhomogeneous System ◽  
Complete Proof ◽  
The Real ◽  
Linear System Of Equations ◽  
Real Solutions ◽  
Non Linear ◽  
Image Position ◽  
Non Linear System

In a recent paper on the behaviour of a system of disk dynamos(1), a problem of a purely algebraic character arose. The problem is to find all the real solutions of the non-linear, inhomogeneous system of 2n equationswhere x0 = xn and the parameter ρ is real and not zero, but otherwise arbitrary. The real solutions were correctly given in (1), but the proof that they are the only real solutions was incomplete. A different, and complete, proof is given here.

Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

1974 ◽  
Vol 26 (1) ◽  
pp. 78-90 ◽  
Author(s):  
Philip G. Buckhiester
Keyword(s):  
Finite Field ◽  
Matrix Equation ◽  
Symmetric Matrices ◽  
Number Of Solutions ◽  
The Matrix ◽  
Image Position

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation1.1where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

scholarly journals On the Canonical Form of a Rational Integral Function of a Matrix

1932 ◽  
Vol 3 (2) ◽  
pp. 135-143 ◽  
Author(s):  
D. E. Rutherford
Keyword(s):  
Canonical Form ◽  
Integral Function ◽  
The Other ◽  
Singular Matrix ◽  
The Matrix ◽  
Image Position ◽  
Rational Integral ◽  
Square Matrix

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

scholarly journals A Matrix Form of Taylor's Theorem

1930 ◽  
Vol 2 (1) ◽  
pp. 33-54 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Matrix Form ◽  
Mathematical Society ◽  
Principal Diagonal ◽  
Independent Variables ◽  
Fundamental Operator ◽  
The Matrix ◽  
Image Position ◽  
Square Matrix

The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated thatwhere s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.

scholarly journals Inverse Displacement Analysis of a General 6R Manipulator Based on the Hyper-Chaotic Least Square Method

10.5772/50909 ◽  
2012 ◽  
Vol 9 (1) ◽  
pp. 7 ◽  
Author(s):  
Youxin Luo ◽  
Wei Yi ◽  
Qiyuan Liu
Keyword(s):  
Nonlinear Equations ◽  
Discrete System ◽  
Least Square Method ◽  
Least Square ◽  
Displacement Analysis ◽  
The Real ◽  
Real Solutions ◽  
Chaotic Sequences ◽  
The Matrix ◽  
Constrained Equations

The hyper-chaotic least square method for finding all real solutions of nonlinear equations was proposed and the inverse displacement analysis of a general 6R manipulator was completed. Applying the D-H method, a 4 × 4 matrix transform was obtained and the first type twelve constrained equations were established. Analysing the characteristics of the matrix, the second type twelve constrained equations were established by adding variables and restriction. Combining the least square method with hyper-chaotic sequences, the hyper-chaotic least square method based on utilizing a hyper-chaotic discrete system to obtain and locate initial points to find all the real solutions of the nonlinear questions was proposed. The numerical example was given for two type constrained equations. The results show that all the real solutions have been obtained, and it proves the correctness and validity of the proposed method.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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