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Norms of Dual Complex Vectors and Dual Complex Matrices
Abstract
In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Hölder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.
... It can be readily demonstrated that the dual-valued absolute function and the dual-valued power functions are expressed as |a + bϵ| = |a| + D b |a|ϵ = |a| + sign(a)bϵ, a ̸ = 0, |b|ϵ, a = 0. (3.5a) (a + bϵ) p = bϵ, p = 1, a = 0, b ∈ R, 0, p > 1, a = 0, b ∈ R, a p + pa p−1 bϵ, p ≥ 1, a > 0, b ∈ R. Indeed, (3.5a) is consistent with the conclusion proposed by Qi, Ling, and Yan in [22]. Besides, (3.5b) and (3.5c) extend the results developed by Miao and Huang in [17]. ...
... A useful class of vector norms is the vector p-norm defined by ∥y∥ p = (|y 1 | p + · · · + |y n | p ) 1 p for 1 ≤ p < ∞ and ∥y∥ ∞ = max 1≤i≤n |y i |, where y = [y 1 , · · · , y n ] ⊤ ∈ R n . This section derives the dual-valued vector p-norm (1 ≤ p ≤ ∞) and discusses the consistency between the dual-valued vector p-norm and the result proposed by Miao and Huang in [17]. ...
... Additionally, the following theorem claims that our proposed dual-valued vector p-norm (1 ≤ p ≤ ∞) is accordant with the element-wise method, not only limited to positive integer p put forth in [17]. ...
Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the G\^ateaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector p-norm and the unitarily invariant dual-valued Ky Fan p-k-norm . The equivalence between the dual-valued Ky Fan p-k-norm and the dual-valued vector p-norm of the first k singular values of the dual matrix is then demonstrated. Practically, we define the dual transitional probability matrix (DTPM), as well as its dual-valued effective information (). Additionally, we elucidate the correlation between the , the dual-valued Schatten p-norm, and the dynamical reversibility of a DTPM. Through numerical experiments on a dumbbell Markov chain, our findings indicate that the value of k, corresponding to the maximum value of the infinitesimal part of the dual-valued Ky Fan p-k-norm by adjusting p in the interval [1,2), characterizes the optimal classification number of the system for the occurrence of the causal emergence.
... In [23,52,53], the definition of absolute value of dual real numbers is proposed. Afterwards, the norms of dual quaternion vectors and dual quaternion matrices are given. ...
... Therefore, we still have U x 2 = ∥ x∥ 2 in this case. Definition 1. ( [52]) For a given A = A + εA 0 ∈ DR n×n , the spectral norm of a dual real matrix can be defined as ...
Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A^, (which is usually represented in the form A^=A+εA0, A and A0 are, respectively, the standard part and the infinitesimal part of A^) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0.
We introduce a total order and an absolute value function for dual numbers. The absolute value function of dual numbers takes dual number values, and has properties similar to those of the absolute value function of real numbers. We define the magnitude of a dual quaternion, as a dual number. Based upon these, we extend 1-norm, ∞-norm, and 2-norm to dual quaternion vectors.
The paper provides an efficient method for obtaining powers and roots of dual complex matrices based on a far reaching generalization of De Moivre’s formula. We also resolve the case of normal and matrices using polar decomposition and the direct sum structure of . The compact explicit expressions derived for rational powers formally extend (with loss of periodicity) to real, complex or even dual ones, which allows for defining some classes of transcendent functions of matrices in those cases without referring to infinite series or alternatively, obtain the sum of those series (explicit examples may be found in the text). Moreover, we suggest a factorization procedure for , based on polar decomposition and generalized Euler type procedures recently proposed by the author in the real case. Our approach uses dual biquaternions and their projective version referred to in the Euclidean setting as Rodrigues’ vectors. Restrictions to certain subalgebras yield interesting applications in various fields, such as screw geometry extensively used in classical mechanics and robotics, complex representations of the Lorentz group in relativity and electrodynamics, conformal mappings in computer vision, the physics of scattering processes and probably many others. Here we only provide brief comments on these subjects with several explicit examples to illustrate the method.
In this study, we generalize the well-known formulae of De-Moivre and Euler of complex numbers to dual-complex numbers. Furthermore, we investigate the roots and powers of a dual-complex number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.
Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. With the
purpose of exploiting new applications, in this paper are presented the dual version of some classical linear algebra algorithms.
These algorithms have been tested for the position analysis of the RCCC mechanism and computational improvements over existing
methods obtained.
We attach the degenerate signature (n,0,1) to the projectivized dual
Grassmann algebra over R(n+1). We explore the use of the resulting Clifford
algebra as a model for euclidean geometry. We avoid problems with the
degenerate metric by constructing an algebra isomorphism between this Grassmann
algebra and its dual, that yields non-metric meet and join operators. We review
the Cayley-Klein construction of the projective (homogeneous) model for
euclidean geometry leading to the choice of the signature (n,0,1). We focus on
the cases of n=2 and n=3 in detail, enumerating the geometric products between
simple k- and m-vectors. We establish that versor (sandwich) operators provide
all euclidean isometries, both direct and indirect. We locate the spin group, a
double cover of the direct euclidean group, inside the even subalgebra of the
Clifford algebra, and provide a simple algorithm for calculating the logarithm
of such elements. We conclude with an elementary account of euclidean rigid
body motion within this framework.
Complex dual numbers w̌1=x1+iy1+εu1+iεv1 which form a commutative ring are for the first time introduced in this paper. Arithmetic operations and functions of complex dual numbers are defined. Complex dual numbers are used to solve dual polynomial equations. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism. Like the dual number, the complex dual number is a useful mathematical tool for analytical and numerical treatment of spatial mechanisms.
This paper presents a new parameterization approach for the graph-based SLAM problem and reveals the differences of two popular over-parameterized ways in the optimization procedure. In the SALM problem, constraints or relative transformations between any two poses are generally separated into translations plus 3D rotations, which are then described in a homogeneous transformation matrix (HTM) to simplify computational operations. This however introduces added complexities in frequent conversions between the HTM and state variables, due to their different representations. This new approach, unit dual quaternion (UDQ), describes a spatial transformation as a screw with only elements. We show that state variables can be directly represented by UDQs, and how their relative transformations can be written with the UDQ product, without the trivial computations of HTM. Then, we explore the performances of the unit quaternion and the axis–angle representations in the graph-based SLAM problem, which have been successfully applied to over parameterize perturbations under the assumption of small errors. Based on public synthetic and real-world datasets in 2D and 3D environments, experimental results show that the proposed approach reduces greatly the computational complexity while obtaining the same optimization accuracies as the HTM-based algorithm, and the axis–angle representation is superior to be the quaternion in the case of poor initial estimations.
We introduce a new presentation of the two dimensional rigid transformation
which is more concise and efficient than the standard matrix presentation.
By modifying the ordinary dual number construction for the complex numbers,
we define the ring of anti-commutative dual complex numbers,
which parametrizes two dimensional rotation and translation all together.
With this presentation, one can easily interpolate or blend
two or more rigid transformations at a low computational cost.
We developed a library for C++ with the MIT-licensed source code.
In SCARA robots, which are often used in industrial applications, all joint axes are parallel, covering three degrees of freedom in translation and one degree of freedom in rotation. Therefore, conventional approaches for the hand-eye calibration of articulated robots cannot be used for SCARA robots. In this paper, we present a new linear method that is based on dual quaternions and extends the work of Daniilid is 1999 (IJRR) for SCARA robots. To improve the accuracy, a subsequent nonlinear optimization is proposed. We address several practical implementation issues and show the effectiveness of the method by evaluating it on synthetic and real data.
This paper focuses on finding a dual quaternion solution to attitude and position control for multiple rigid body coordination. Representing rigid bodies in 3-D space by unit dual quaternion kinematics, a distributed control strategy, together with a specified rooted-tree structure, are proposed to control the attitude and position of networked rigid bodies simultaneously with notion concision and nonsingularity. A property called pairwise asymptotic stability of the overall system is then analyzed and validated by an example of seven quad-rotor formation in the Urban Search And Rescue Simulation (USARSim) platform. As a separate but related issue, a maximum depth condition of the rooted tree is found with respect to error accumulation along each path using dual quaternion algebra, such that a given safety bound on attitude and position errors can be satisfied.
To relate measurements made by a sensor mounted on a mechanical link to the robot’s coordinate frame, we must first estimate the transformation between these two frames. Many algorithms have been proposed for this so-called hand-eye calibration, but they do not treat the relative position and orientation in a unified way. In this paper, we introduce the use of dual quaternions, which are the algebraic counterpart of screws. Then we show how a line transformation can be written with the dual-quaternion product. We algebraically prove that if we consider the camera and motor transformations as screws, then only the line coefficients of the screw axes are relevant regarding the hand-eye calibration. The dual-quaternion parameterization facilitates a new simultaneous solution for the hand-eye rotation and translation using the singular value decomposition. Real-world performance is assessed directly in the application of hand-eye information for stereo reconstruction, as well as in the positioning of cameras. Both real and synthetic experiments show the superiority of the approach over two other proposed methods.
Preliminary sketch of bi-quaternions
- W K Clifford
On the homogeneous model of Euclidean geomery
- C Gunn
- L Dorst
- J Lasenby