Cayley-Hamilton Research Articles

Exact analytical polynomial formulation of the exponential of fully matrices

An analytical polynomial formulation of the exponential of fully symmetric matrices is proposed. This polynomial formula is derived using the Cayley - Hamilton theorem and the polynomial Euclidean division. The application field of these developments is time-dependent wavepacket calculations involving multiple potential energy surface crossing. The use of this polynomial formulation in time-dependent calculations is directly connected to the choice of the split operator, proposed by Feit and Fleck (1982 J. Comput. Phys. 47 412) as a numerical propagation scheme. We show in this paper that this analytical polynomial formula is numerically more efficient than the scheme proposed by Almeida and Metiu (1988 Chem. Phys. Lett. 146 47) where the exponential of the kinetic operator is evaluated in the momentum and diabatic representation and the exponential of the potential operator is evaluated in the coordinate and adiabatic representation. The use of an optical potential ( negative imaginary potential) to absorb the wavepacket near the edges of the grid point is quite simple to implement as it appears as a multiplicative factor in front of the polynomial expression. Our scheme is exemplified on the photodissociation of a diatomic molecule involving three potential energy curves.

Determinants, integrality and Noether’s theorem for quantum commutative algebras

The aim of this paper is to generalize Noether’s theorem for finite groups acting on commutative algebras, to finite-dimensional triangular Hopf algebras acting on quantum commutative algebras. In the process we construct a non-commutative determinant function which yields an analogue of the Cayley-Hamilton theorem for certain endomorphisms.

The Matrix Exponential

There are many different methods to calculate the exponential of a matrix: series methods, differential equations methods, polynomial methods, matrix decomposition methods, and splitting methods, none of which is entirely satisfactory from either a theoretical or a computational point of view. How then should the matrix exponential be introduced in an elementary differential equations course, for engineering students for example, with a minimum of mathematical prerequisites? In this note, a method is given that uses the students’ knowledge of homogeneous linear differential equations with constant coefficients and the Cayley–Hamilton theorem. The method is not new; what is new is the approach.

Observation and simulation of a large signal mechanical vibrating system

The equation of motion of n-degrees of freedom non-conservative mechanical system is represented by a Vector-Matrix differential equation. The matrix method is usually used where the natural frequencies and natural modes of vibrations are obtained, to solve this equation and a similarity transformation is always required to obtain the steady state solution. The purpose of this paper is to propose a solution, based on Cayley-Hamilton theorem, to the equation of motion of n-degrees of freedom discrete non-conservative mechanical system. The similarity transformation is not required in the proposed method, and contrary to the classical matrix method, the computations to be carried out are moderate, thus enabling use of personal computers. Simulated examples are presented to illustrate the results given in this paper.

On an alternative supermatrix reduction

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even reduction) which arises in connection with the possible extension of manifold structure group reductions. The study was initiated by consideration of generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated together. They can be unified into some kind of ‘sandwich’ semigroups. We also define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.

Matrix description of multivariable polynomials

A new multiway description of multivariable polynomials is presented. Applications to linear transformations of polynomial variables, the Cayley-Hamilton theorem, and polynomial factorizability are given.

The identities of the algebraic invariants of the four-dimensional Riemann tensor

It is well known that there are fourteen independent algebraic invariants of the four-dimensional Riemann tensor. Several authors have written down a set of fourteen independent invariants, but it is still not known how other invariants can be expressed in terms of these sets. This paper investigates this problem by looking for relationships between invariants of the Riemann tensor. Essentially the problem turns out to be analogous to finding relationships between the invariants of two 3×3 matrices, one of which is symmetric and trace-free and the other Hermitian. A number of identities between the invariants can be obtained simply by using a generalization of the Cayley–Hamilton theorem but others, which depend on the symmetry of the matrices, are considerably more complex.

Generalized Cayley-Hamilton theorem for polynomial matrices with arbitrary degree

We give an alternative, more convenient expression of the well known Cayley-Hamilton theorem when polynomial matrices of arbitrary degree are involved. Based on the results of a recently developed algorithm (Fragulis et al. 1991) for the computation of the inverse of a polynomial matrix, certain relationships among the coefficient matrices of the given polynomial matrix are obtained. We also propose two ways of finding the powers of a polynomial matrix: one in terms of its coefficient matrices and the other making use of the generalized Cayley-Hamilton theorem. These methods are of closed form and are easily implementable in a digital computer.

A Topological Proof of the Cayley-Hamilton Theorem

A Topological Proof of the Cayley-Hamilton Theorem

The exponential map for the conformal group O(2,4)

We present a general method to obtain a closed, finite formula for the exponential map from the Lie algebra to the Lie group, for the defining representation of the orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group, and is also independent of the metric. We present an explicit formula for the exponential of generators of the $SO_+(p,q)$ groups, with $p+q = 6$, in particular we are dealing with the conformal group $SO_+(2,4)$, which is homomorphic to the $SU(2,2)$ group. This result is needed in the generalization of U(1) gauge transformations to spin gauge transformations, where the exponential plays an essential role. We also present some new expressions for the coefficients of the secular equation of a matrix.

The Cayley-Hamilton theorem for supermatrices

Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned determinant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are considered.

Determinant theory for D01-lattice matrices

We study determinant theory for D 01-lattice matrices and consider the Cayley-Hamilton theorem for these matrices (a distributive lattice L is said to be D 01-lattice, if 0, 1 ϵ L, and 0 ⩽ x ⩽ 1, for all x ϵ L). We prove that det( AB) ⩾ det( A)det( B), det( Aadj( A)) = det( A) = det(adj( A) A), where adj( A) denotes the adjoint matrix of a D 01-lattice matrice A.

The Square Roots of 2 × 2 Matrices

Introduction In a recent article MacKinnon [1] describes four methods that may be used to find square roots of 2 X 2 matrices. The first of these methods requires that the matrix for which the square roots are sought be diagonalizable and, subsequently, this method was used by Scott [2] to determine all the square roots of 2 X 2 matrices. A surprising conclusion is that scalar 2 X 2 matrices possess double-infinities of square roots whereas nonscalar 2 X 2 matrices have only a finite number of square roots. The purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of 2 X 2 matrices. These formulae indicate exactly when a 2 X 2 matrix has square roots, and the number of such roots. By definition, the square roots of a 2 X 2 matrix, A, are those 2 X 2 matrices, X, for which

Invariant factors of an endomorphism and finite free resolutions

Polynomial invariants are associated to an endomorphism u of a module M that has a finite free resolution. The first invariant χ( u, X) (characteristic polynomial of u) is a monic polynomial of degree the Euler characteristic of M. Its construction is based on the MacRae invariant. When M is a finite free module, χ( u, X) is the classical characteristic polynomial of u. With additional assumptions there is constructed a finite sequence of monic polynomials { d i ( u, X)} i ⩾ 0 such that their product is χ( u, X) and d i ( u, X) divides d i + 1 ( u, X). When R is a field, these polynomials are the invariant factors of u. A generalized Cayley-Hamilton theorem is given. The generic behavior of the polynomials χ( u, X) and { d i ( u, X)} i ⩾ 0 in Spec R is proved. Finally it is shown, under certain assumptions that there exists a free submodule F of M, invariant with respect to u, such that the restriction of u to F is similar to the endomorphism of F defined by the diagonal block matrix where the ith block is the companion matrix of d i ( u, X).

The Cayley-Hamilton theorem for supermatrices

Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are discussed.

A simplified version of the Cayley-Hamilton theorem and exponential forms of the 2 × 2 and 3 × 3 matrices

In this note we present a simple proof of a theorem allowing us to castf(Â), where  is a non-singular matrix andf a function admitting a McLaurin expansion, as a finite sum. We also discuss the complementary version of the theorem and, limiting ourselves to 2 × 2 and 3×3 matrices, we show how they can be cast in an exponential form. Such form greatly simplifies the task of finding the π-th power (with π being any real or complex number) of a given matrix. The applications to physical problems like the optical-resonator stability and the Cabibbo-Kobayashi-Maskawa matrix are also discussed.

Exact solutions of buckling for simply supported thick laminates

The three-dimensional state equations for the initial buckling of orthotropic thick plates are established in the Cartesian co-ordinate system in this paper without imposing any assumptions about the displacement models or stress distribution across the thickness of the cross-section. By using the continuity conditions of the displacement and stresses on each interface between any two adjacent layers, the state equations for the laminates, which consist of orthotropic layers, were obtained. A unified exact solution for the buckling of simply supported rectangular laminates with any given number of orthotropic layers is presented in the paper by means of the well-known Cayley-Hamilton theorem. All the equations of elasticity can be satisfied exactly and the nine elasticity constants of orthotropy for each layer are fully taken into account. Regardless of the number of the layers considered, only a set of simultaneous linear algebraic equations of 3rd order needs to be solved in the final stage of the solution. The numerical results were calculated and compared with those of thin plate theory, Mindlin analysis and those due to Srinivas et al. who also solved the problems with the three-dimensional theory of elasticity.

The generalized Cayley-Hamilton theorem for standard pencils

The Cayley-Hamilton theorem is extended from the case of one matrix to that of two matrices. The developed theorem is suitable for the standard form pencil sE − A. Let Δ(s) := det(sE − A) = Σ r=0 na rs r . Then we show that Σ r=0 na rA rE n−r = 0 . Such a neat result, which is very similar to the original theorem also, is due to some nice properties of the standard pair {E, A}. For example, E and A commute and share the same eigenspaces.

Generalized Gantmacher formulas through functions of matrices

Gantmacher formulas are generalized by including the effect of the centripetal and angular accelerations in addition to the usual Coriolis term. The integration is accomplished by using a method based on functions of matrices, which permits the summation of an infinite series in closed form through the use of the Cayley–Hamilton theorem. The problem is further extended by allowing arbitrary matrix coefficients in the vector equation of motion, provided they obey a certain commutativity condition. In addition to standard examples, the problem of a particle moving on a plane rotating with variable angular velocity is considered. This example differs from the ones considered in the literature because neither the centripetal nor the angular acceleration terms can be neglected. Solutions of various orders are developed by truncating the exact solution when appropriate.

The minimal realization of discrete-event systems

In this paper, it is proved that an impulse response sequence is periodical if it is realizable and the dimension of its minimal realization is not smaller than its order d. On the basis of G. J. Olsder, minimal realizations of some specific p x q impulse response matrix sequences, which are analogous to that of conventional control theory, are constructed. Sufficient conditions to simplify some dependent relationships of a impulse response sequence are provided. It is shown that Cayley-Hamilton theorem of “max” algebra take an important role in minimal realization problem of discrete-event systems.