T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

In this paper,we investigate the tensor similarity and propose the T-Jordan canonical form and its properties.The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed.As a special case,we present properties when two tensors commute based on the tensor T-product.We prove that the Cayley-Hamilton theorem also holds for tensor cases.Then,we focus on the tensor decompositions:T-polar,T-LU,T-QR and T-Schur decompositions of tensors are obtained.When an F-square tensor is not invertible with the T-product,we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases.The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form.The polynomial form of the T-Drazin inverse is also proposed.In the last part,we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

CONTROLLABILITY CANONICAL FORM FOR NONLINEAR CONTROL SYSTEMS

General nonlinear control systems are studied in this paper with the goal to transform them into the so-called controllability canonteal form via state transformation only. The conditions of transformability are given for both single input and multiple input cases. Besides, by an algebraic approach the procedure for constructing the state transformation is established. This paper is formulated in the framework of calculus rather than differential geometry approach.

GH-CONGRUENCE CANONICAL FORM OF QUATERNION MATRIX AND SIMULTANEOUS DIAGONALIZATION OF MATRICES

In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.

Notes on Canonical Forms of Integrable Vector Nonlinear Schr?dinger Systems

We present canonical forms of integrable vector nonlinear Sehr6dinger systems. Mathematically, it is enough to focus on these canonical forms.

ON THE DECOMPOSITION OF COMPLEX VECTOR SPACES AND THE JORDAN CANONICAL FORM OF COMPLEX LINEAR TRANSFORMATIONS

New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.

Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics

In this paper, we study the problem of output feedback stabilization for stochastic nonlin-ear systems. We consider a class of stochastic nonlinear systems in observer canonical form with sta-ble zero-dynamics. We introduce a sequence of state transformations that transform the system into a lower triangular structure that is amenable for integrator backstepping design. Then we design the out-put-feedback controller and prove that the closed-loop system is bounded in probability. Furthermore, when the disturbance vector field vanishes at the origin, the closed-loop system is asymptotically stable in the large. With special care, the controller preserves the equilibrium of the nonlinear system. An ex-ample is included to illustrate the theoretical findings.

Positive-Controllability,Positive-Near-Controllability,and Canonical Forms of Driftless Discrete-Time Bilinear Systems

Controllable canonical forms play important roles in the analysis and design of control systems.In this paper,a fundamental class of discrete-time bilinear systems are considered.Such systems are of interest since,on one hand,they have the most complete controllability theory.On the other hand,they can be nearly-controllable even if controllability fails.Firstly,controllability of the systems with positive control inputs is studied and necessary and sufficient algebraic criteria for positive-controllability and positive-near-controllability are derived.Then,controllable canonical forms and nearly-controllable canonical forms of the systems are presented,respectively,where the corresponding transformation matrices are also explicitly constructed.Examples are given to demonstrate the effectiveness of the derived controllability criteria and controllable canonical forms.

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold.The expressions of Meigenvalues and M-eigenvectors are presented in this paper.As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found.We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold.We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely.We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.

THE PROPERTIES OF A KIND OF RANDOM SYMPLECTIC MATRICES

Several important properties of a kind of random symplectic matrix used by A. Bunse-Gerstner and V. Mehrmann are studied and the following results are obtained: 1) It can be transformed to Jordan canonical form by orthogonal similar transformation; 2) Its condition number is a constant; 3) The condition number of it is about 2.618.

Sensor fault-tolerant observer applied in satellite attitude control

The observing failure and feedback instability might happen when the partial sensors of a satellite attitude control sys- tem （SACS） go wrong. A fault diagnosis and isolation （FDI） method based on a fault observer is introduced to detect and isolate the fault sensor at first. Based on the FDI result, the object system state-space equation is transformed and divided into a correspon- sive triangular canonical form to decouple the normal subsystem from the fault subsystem. And then the KX fault-tolerant observers of the system in different modes are designed and embedded into online monitoring. The outputs of all KX fault-tolerant observers are selected by the control switch process. That can make sense that the SACS is part-observed and in stable when the partial sen- sors break down. Simulation results demonstrate the effectiveness and superiority of the proposed method.

Stabilization of a class of switched nonlinear systems

The stabilization of a class of switched nonlinear systems is investigated in the paper. The systems concerned are of （generalized） switched Byrnes-Isidori canonical form, which has all switched models in （generalized） Byrnes- Isidori canonical form. First, a stability result of switched systems is obtained. Then it is used to solve the stabilization problem of the switched nonlinear control systems. In addition, necessary and sufficient conditions are obtained for a switched affine nonlinear system to be feedback equivalent to （generalized） switched Byrnes-Isidori canonical systems are presented. Finally, as an application the stability of switched lorenz systems is investigated.

ASYMPTOTIC CANONICAL FORMS AND ITERATED LOGARITHM RATE RESULTS OF LEAST SQUARES ESTIMATES FOR UNSTABLE ARMA MODELS

Herein we give the asymptotic canonical forms of the design mains Pn where is an unstable ARMA process B denotes the backshift operator such that B, and p is the order of the polynomial having all roots outside or on the unit circle. These asymptotic canonical forms for Pn, which behave a.s. approximately diagonally, are then used to obtain the itersted logarithm rates of almost sure convergence of the least-squares estimates to the unknown true parameter for an unstable time series.

Max-plus-linear model-based predictive control for constrained hybrid systems: linear programming solution

In this paper, a linear programming method is proposed to solve model predictive control for a class of hybrid systems. Firstly, using the （max, ＋） algebra, a typical subclass of hybrid systems called max-plus-linear （MPL） systems is obtained. And then, model predictive control （MPC） framework is extended to MPL systems. In general, the nonlinear optimization approach or extended linear complementarity problem （ELCP） were applied to solve the MPL-MPC optimization problem. A new optimization method based on canonical forms for max-min-plus-scaling （MMPS） functions （using the operations maximization, minimization, addition and scalar multiplication） with linear constraints on the inputs is presented. The proposed approach consists in solving several linear programming problems and is more efficient than nonlinear optimization. The validity of the algorithm is illustrated by an example.

COMPUTING THE EIGENVECTORS OF A MATRIX WITH MULTIPLEX EIGENVALUES BY SVD METHOD

Every matrix is similar to a matrix in Jordan canonical form,which has very important sense in the theory of linear algebra and its engineering application.For a matrix with multiplex eigenvalues,an algorithm based on the singular value decomposition(SVD) for computing its eigenvectors and Jordan canonical form was proposed.Numerical simulation shows that this algorithm has good effect in computing the eigenvectors and its Jordan canonical form of a matrix with multiplex eigenvalues.It is superior to MATLAB and MATHEMATICA.

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.

A New Approach for Inverting the Pascal Matrix Plus a Scalar

In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn+aIn for arbitrary real number a≠1.

T_(M)-Eigenvalues of Odd-Order Tensors

In this paper,we propose a definition for eigenvalues of odd-order tensors based on some operators.Also,we define the Schur form and the Jordan canonical form of such tensors,and discuss commuting families of tensors.Furthermore,we prove some eigenvalue ine-qualities for Hermitian tensors.Finally,we introduce characteristic polynomials of odd-order tensors.

Some Results on the Differential Geometry of Spacelike Curves in De-Sitter Space

The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of spherecal motion. In this work, we give some results for differential geometry of spacelike curves in 3-dimensional de-Sitter space. Also, we study the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions to analyze and characterize the shape of the curves in 3-dimensional de-Sitter space.

Einstein’s Elevator in Cosmology

The metrics of gravitational and cosmological models are brought into canonical form in comoving coordinates. The FWR curvature parameter k is read from this and it is shown that k=0 does not correlate to a flat model, but for a spatially positively curved geometry in which reference systems which are in free fall exist. This also corresponds to Einstein’s elevator principle. Moreover, we will show that our subluminal cosmos is associated with the Rh=ct model of Melia, assuming that k=0 is related to a free-falling system in the sense described above.

THE EIGENVALUE PERTURBATION BOUND FOR ARBITRARY MATRICES

In this paper we present some new absolute and relative perturbation bounds for the eigenvalue for arbitrary matrices, which improves some recent results. The eigenvalue inclusion region is also discussed.