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

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Benjamin Lasorne
Keyword(s):  
Lie Group ◽  
Hermitian Matrix ◽  
Similarity Transformation ◽  
Unitary Matrix ◽  
Unitary Similarity ◽  
Similarity Transformations ◽  
Pauli Matrices ◽  
General Derivation ◽  
Single Matrix ◽  
Unitary Similarity Transformations

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.

Diagonalizability with respect to perplectic and pseudo-unitary similarity transformations

2020 ◽  
Vol 591 ◽  
pp. 61-71 ◽  
Author(s):  
Edgar Aivan Afable ◽  
Ralph John de la Cruz ◽  
Agnes T. Paras ◽  
Mary Elizabeth Segui
Keyword(s):  
Unitary Similarity ◽  
Similarity Transformations ◽  
Unitary Similarity Transformations

scholarly journals Some Theorems On Matrices With Real Quaternion Elements

1955 ◽  
Vol 7 ◽  
pp. 191-201 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Similarity Transformation ◽  
Elementary Divisor ◽  
Quaternion Matrix ◽  
Unitary Similarity ◽  
Triangular Form ◽  
Quaternion Matrices ◽  
Divisor Theory ◽  
Real Quaternion ◽  
Unitary Similarity Transformation

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS

2005 ◽  
Vol 02 (01) ◽  
pp. 111-125 ◽  
Author(s):  
PAOLO ANIELLO
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Similarity Transformation ◽  
Perturbative Expansion ◽  
Analytic Perturbation ◽  
The Matrix ◽  
Matrix Formula

We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

scholarly journals State feedback linearization using block companion similarity transformation

2021 ◽  
Vol 10 (2) ◽  
pp. 658-667
Author(s):  
Kessal Farida ◽  
Hariche Kamel ◽  
Bentarzi Hamid ◽  
Boushaki Razika
Keyword(s):  
Feedback Linearization ◽  
State Feedback ◽  
Similarity Transformation ◽  
Research Work ◽  
Multivariable System ◽  
Similarity Transformations ◽  
State Dependent ◽  
Linearized System ◽  
Determine System ◽  
The Given

In this research work, a new method is proposed for linearizing a class of nonlinear multivariable system; where the number of inputs divides exactly the number of states. The idea of proposed method consists in representing the original nonlinear system into a state-dependent coefficient form and applying block similarity transformations that allow getting the linearized system in block companion form. Because the linearized system’s eigenstructure can determine system performance and robustness far more directly and explicitly than other indicators, the given class multivariable system is chosen. Examples are used to illustrate the application and show the effectiveness of the given approach.

scholarly journals Spectral theory for self-adjoint quadratic eigenvalue problems - a review

2021 ◽  
Vol 37 ◽  
pp. 211-246
Author(s):  
Peter Lancaster ◽  
Ion Zaballa
Keyword(s):  
Eigenvalue Problems ◽  
Direct Reduction ◽  
Unitary Similarity ◽  
Matrix Polynomials ◽  
Similarity Transformations ◽  
Physical Problems ◽  
Matrix Coefficients ◽  
Expository Paper ◽  
Eigenvalue Multiplicities ◽  
Quadratic Eigenvalue

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

scholarly journals CONSTRAINTS ON MASS MATRICES DUE TO MEASURED PROPERTY OF THE MIXING MATRIX

2006 ◽  
Vol 21 (11) ◽  
pp. 907-910 ◽  
Author(s):  
S. CHATURVEDI ◽  
VIRENDRA GUPTA
Keyword(s):  
Hermitian Matrix ◽  
Main Diagonal ◽  
Unitary Matrix ◽  
Matrix Elements ◽  
Mass Matrices ◽  
The Matrix ◽  
Measured Property ◽  
Mixing Matrix ◽  
The Asymmetry

It is shown that two specific properties of the unitary matrix V can be expressed directly in terms of the matrix elements and eigenvalues of the hermitian matrix M which is diagonalized by V. These are the asymmetry Δ(V) = |V12|2-|V21|2, of V with respect to the main diagonal and the Jarlskog invariant [Formula: see text]. These expressions for Δ(V) and J(V) provide constraints on possible mass matrices from the available data on V.

scholarly journals MAPS AND TWISTS RELATING U(sl(2)) AND THE NONSTANDARD Uh(sl(2)): UNIFIED CONSTRUCTION

1999 ◽  
Vol 14 (12) ◽  
pp. 765-777 ◽  
Author(s):  
B. ABDESSELAM ◽  
A. CHAKRABARTI ◽  
R. CHAKRABARTI ◽  
J. SEGAR
Keyword(s):  
Series Expansion ◽  
Similarity Transformation ◽  
General Construction ◽  
Twist Map ◽  
Similarity Transformations ◽  
Twist Operator ◽  
Twist Operators

A general construction is given for a class of invertible maps between the classical U ( sl (2)) and the Jordanian U h( sl (2)) algebras. Here the role of the maps is studied in the context of construction of twist operators relating the cocommutative and non-cocommutative coproducts of the U(sl(2)) and U h( sl (2)) algebras respectively. It is shown that a particular map called the "minimal twist map" implements the simplest twist given directly by the factorized form of the ℛh matrix of Ballesteros–Herranz. For a "non-minimal" map the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. Our general prescription may be used to evaluate the series expansion in powers of h of the twist operator corresponding to an arbitrary "non-minimal" map. The classical and the Jordanian antipode maps may also be interrelated by suitable similarity transformations.

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