Similarity Transformation Of Matrices Research Articles

Q-discretization of the Kostant–Toda equation and its asymptotic analysis

Abstract The famous Toda equation is an integrable system related to similarity transformations of tridiagonal matrices. The discrete Toda equation, which is a time-discretization of the Toda equation, is essentially the recursion formula of the quotient-difference (qd) algorithm for computing eigenvalues of tridiagonal matrices. Another time-discretization of the Toda equation is the $q$-discrete Toda equation, which is derived by replacing standard derivatives with the so-called $q$-derivatives that involves a parameter $q$ such that $0<q<1$. In a prior work, we related the $q$-discrete Toda equation to implicit-shift $LR$ transformations (which are similarity transformations) of tridiagonal matrices. Furthermore, we developed the determinantal solution to clarify the convergence as discrete-time goes to infinity. In this paper, we propose an extension of the $q$-discrete Toda equation as a time-discretization of the Kostant–Toda equation and then show the convergence as discrete-time goes to infinity from the perspective of implicit-shift $LR$ transformations of Hessenberg matrices. We also present numerical examples to verify the convergence as discrete-time goes to infinity in the proposed $q$-discrete Kostant–Toda equation.

Синтез параметрически грубых систем управления с регуляторами и наблюдателями состояния на основе грамианных технологий

Automatic control systems based on regulators with state observers are effective means to control linear and linearized objects of a high degree of complexity. One of the key problems in the synthesis of modal systems is the development of parametrically rough (robust) systems that retain their performance, as well as the main quality indicators with possible variations of the parameters of both the control object and the controller itself. The synthesis of a robust system with a state observer is carried out by the modal control method based on the matrix model of the controlled object. To ensure the parametric robustness of the system, the authors have proposed a technique which based on the mathematical apparatus of Gramians of controllability and observability as well as on the similarity transformation of observer matrices. Computer simulation of the system with state controller and observer is carried out by means of the MatLab software. The authors have proposed a method to form an observer state structure that is optimal in the sense of parametric roughness of the system, which is based on the procedure of transforming the similarity of the object model. And the transformation matrix is formed by varying the singular numbers of the controllability and observability gramians. The proposed method allows us to obtain an observer structure with a certain ratio of controllability and observability that ensures the fulfillment of the condition of parametric roughness i.e., the absence of positive feedback in the control device. The results obtained make it possible to clearly demonstrate the high efficiency of using the Gramian method for the synthesis of control systems with state observers with low sensitivity to variations of the controller own parameters.

Matrix regularization for tensor fields

Abstract We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices, and area-preserving diffeomorphisms and local rotations of the orthonormal frame are both realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

The antitriangular factorization of skew-symmetric matrices

In this paper we develop algorithms for orthogonal similarity transformations of skew-symmetric matrices to simpler forms. The first algorithm is similar to the algorithm for the block antitriangular factorization of symmetric matrices, but in the case of skew-symmetric matrices, an antitriangular form is always obtained. Moreover, a simple two-sided permutation of the antitriangular form transforms the matrix into a multi-arrowhead matrix. In addition, we show that the block antitriangular form of the skew-Hermitian matrices has the same structure as the block antitriangular form of the symmetric matrices.

Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases

AbstractConcurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This article proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of the microstructure are calculated via the homogenization method. An efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bidirectional evolutionary structural optimization method is employed to obtain black and white designs for both macrostructure and microstructure. Several two‐dimensional and three‐dimensional numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure.

The Kostant-Toda equation and the hungry integrable systems

The Toda equation, which has applications in many fields, is one of the most important integrable systems. A generalization of the Toda equation is the Kostant-Toda equation, whose Lax representation is expressed using bidiagonal and lower Hessenberg matrices. The discrete Toda equation, which is a time-discretization of the Toda equation, is extended to the so-called discrete hungry Toda (dhToda) equation. The Lax representation for the dhToda equation generates similarity transformations of lower Hessenberg matrices. Although both have been considered, a discrete analogue of the Kostant-Toda equation or a continuous analogue of the dhToda equation has yet to be presented in the literature. In this paper, we present a continuous analogue of the dhToda equation, and then relate this to the Kostant-Toda equation. By using a continuous analogue of the Bäcklund transformation between the dhToda equation and the discrete hungry Lotka-Volterra system, we relate the resulting continuous analogue to the continuous hungry Lotka-Volterra system.

On Matrix Representations of Geometric (Clifford) Algebras

Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its matrix representation of order $n$. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.

On the Use of Lie Group Homomorphisms for Treating Similarity Transformations in Nonadiabatic Photochemistry

A formulation based on Lie group homomorphisms is presented for simplifying the treatment of unitary similarity transformations of Hamiltonian matrices in nonadiabatic photochemistry. A general derivation is provided whereby it is shown that a similarity transformation acting on a traceless, Hermitian matrix through a unitary matrix ofSU(n)is equivalent to the product of a single matrix ofOn2-1by a real vector. We recall how Pauli matrices are the adequate tool whenn=2and show how the same is achieved forn=3with Gell-Mann matrices.

Positive stable realizations with system Metzler matrices

Positive stable realizations with system Metzler matricesConditions for the existence of positive stable realizations with system Metzler matrices for linear continuous-time systems are established. A procedure for finding a positive stable realization with system Metzler matrix based on similarity transformation of proper transfer matrices is proposed and demonstrated on numerical examples. It is shown that if the poles of stable transfer matrix are real then the classical Gilbert method can be used to find the positive stable realization.

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systemsThe notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.

ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics

It is shown that every ray transfer matrix, often called the ABCD matrix, can be written as a similarity transformation of one of the Wigner matrices that dictate the internal space-time symmetries of relativistic particles, while the transformation matrix is a rotation preceded by a squeeze. The implementation of this mathematical procedure is described, and how it facilitates the calculations for scattering processes in periodic systems is explained. Multilayer optics and resonators such as laser cavities are discussed in detail. For both cases, the one-cycle transfer matrix is written as a similarity transformation of one of the Wigner matrices, rendering the computation of the ABCD matrix for an arbitrary number of cycles tractable.

An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices.

Similarity transformations of matrices in the stationary equations of gas-dynamics

Similarity transformations of matrices in the stationary equations of gas-dynamics

Some basic algebraic relations in the order‐reduction of large‐scale systems by aggregation

AbstractIn this paper, we prove conditions for the existence of a full‐rank matrix K which satisfies the aggregation relations (4)–(6). These conditions are convenient for checking and offer some rules for selecting K. Theorem 3 shows the properties of the matrix F in the aggregated model. It is actually a generalization of the important algebraic theorem on the similarity transformation of matrices.

Algebraic solution of matrix linear equations in control theory

Matrix linear equations ATX+XB=P and ΦTXψ+P=X arise in control theory, and the efficient computer solution of these equations is an important problem. This paper develops an algebraic method of solution of the equations, via a similarity transformation of system matrices to companion form. Solution evaluation by this method is three to five times faster than by standard methods. The method also shows the formal equivalence of state-space and transform expressions for the output covariance of single-input/single-output systems, and provides an improved algorithm for evaluating such terms.

Note on the Practical Computation of Proper Values

It has been suggested by Householder [1] and by Householder and Bauer [2] that orthogonal similarity transformations of matrices are particularly stable with respect to the practical computation of proper values. It is the purpose of this note to examine this question and to demonstrate in terms of a “condition number” to be defined below a sense in which this conjecture is true. Broadly speaking, any problem may be termed “ill-conditioned” for which the solution is acutely sensitive to slight variations in the parameters of the problem. Examples of ill-conditioning occur in many contexts. Computers are familiar with the phenomenon as it manifests itself, for example, in the study of matrix inversion. Since our purpose is to study the conditioning of matrices specifically for the proper value problem, it is convenient to have a nomenclature and notation which will avoid confusion with conditioning as it is used in other senses.