scholarly journals Lorentz transformation from an elementary point of view

2016 ◽  
Vol 31 ◽  
pp. 794-833 ◽  
Author(s):  
Arkadiusz Jadczyk ◽  
Jerzy Szulga
Keyword(s):  
Lorentz Transformation ◽  
Symmetric Matrix ◽  
Point Of View ◽  
Singular Case ◽  
Elementary Methods ◽  
Skew Symmetric Matrix

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

scholarly journals Overlapping Pfaffians

10.37236/1263 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Donald E. Knuth
Keyword(s):  
Symmetric Matrix ◽  
History Of ◽  
Combinatorial Construction ◽  
Skew Symmetric Matrix

A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.

DECOMPOSITION OF STOCHASTIC FLOWS AND ROTATION MATRIX

2002 ◽  
Vol 02 (01) ◽  
pp. 93-107 ◽  
Author(s):  
PAULO R. C. RUFFINO
Keyword(s):  
Asymptotic Behaviour ◽  
Constant Curvature ◽  
Symmetric Matrix ◽  
Rotation Matrix ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Affine Transformations ◽  
Simply Connected ◽  
Skew Symmetric Matrix

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φt = ξt°Ψt° Θt, where ξt and Ψt have the same properties of Liao's decomposition and (ξt°Ψt) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξt via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.

A Further Extension of Cayley's Parameterization

1959 ◽  
Vol 11 ◽  
pp. 48-50 ◽  
Author(s):  
Martin Pearl
Keyword(s):  
Symmetric Matrix ◽  
Arbitrary Field ◽  
Complex Field ◽  
Result Theorem ◽  
Characteristic 2 ◽  
Image Position ◽  
Row Space ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In a recent paper (3)* the following theorem was proved for real matrices.Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then1is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.

scholarly journals On Turnbull identity for skew-symmetric matrices

2000 ◽  
Vol 43 (2) ◽  
pp. 379-393 ◽  
Author(s):  
Tôru Umeda ◽  
Takeshi Hirai
Keyword(s):  
Differential Operators ◽  
Symmetric Matrix ◽  
Point Of View ◽  
Symmetric Matrices ◽  
Invariant Differential Operators ◽  
Type Identity ◽  
Central Elements ◽  
Theoretic Point ◽  
Invariant Differential ◽  
Matrix Spaces

AbstractIn the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements inU(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

Cayley formula in Minkowski space-time

2015 ◽  
Vol 12 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Time ◽  
Rotation Matrix ◽  
Symmetric Matrices ◽  
Rotation Matrices ◽  
Minkowski Space Time ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in [Formula: see text]. For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi-skew-symmetric matrices A1 and A2 satisfying the properties [Formula: see text] and [Formula: see text] Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay (A).

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