Generalized Cayley-Hamilton Theorem Research Articles

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem

Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.

Periodic matrix difference equations and companion matrices in blocks: some applications

This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

On the [formula omitted]-dimensional Cayley–Hamilton theorem and its application to an algebraic decision problem inferred from the [formula omitted] norm analysis of delay systems

We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the H2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H2 norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite H2 norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the H2 norm whenever it is finite and by stating an open problem.

Linearized systems and a generalized Cayley–Hamilton theorem

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

Rank in Banach algebras: A generalized Cayley–Hamilton theorem

Let A be a semisimple Banach algebra with non-trivial, and possibly infinite-dimensional socle. Addressing a problem raised in [5, p. 1399], we first define a characteristic polynomial for elements belonging to the socle, and we then show that a Generalized Cayley–Hamilton Theorem holds for the associated polynomial. The key arguments leading to the main result follow from the observation that a purely spectral approach to the theory of the socle carries alongside it an efficient method of dealing with relativistic problems associated with infinite-dimensional socles.

A structured approach to design-for-frequency problems using the Cayley-Hamilton theorem.

An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue design problem using the generalized Cayley-Hamilton theorem is proposed. A method for solving partially described systems is also specified. The Cayley-Hamilton theorem algorithm is shown to be a good design tool for solving inverse eigenvalue problems of mechanical and structural systems.

A generalized Cayley–Hamilton theorem

Let F be a solvable Lie subalgebra of the Lie algebra gln(C) (=Cn×n as a vector space). Let fk(x1,x2,…,xp),(k=1,2,…,r), be polynomials in the commuting variables x1,x2,…,xp with coefficients in C. For n×n matrices M1,M2,…,Mr, let F(x1,x2,…,xp)=∑k=1rMkfk(x1,x2,…,xp) and letδF(x1,x2,…,xp)=detF(x1,x2,…,xp).In this paper, we prove that, for A1,A2,…,Ap,M1,M2,…,Mr∈F, if one value of the matrix-valued function F(A1,A2,…,Ap) (the value depends on the product order of the variables) is nilpotent, then, (a) all values of F(A1,A2,…,Ap) are nilpotent; (b) all values of δF(A1,A2,…,Ap) (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in [7] and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras.

A generalized Cayley–Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if ( AB - BA ) C = C ( AB - BA ) for all A , B , C ∈ F where A , B , C need not be distinct. Let f k ( x 1 , x 2 , … , x p ) , ( k = 1 , 2 , … , r ) , be polynomials in the indeterminates x 1 , x 2 , … , x p with coefficients in the complex field C , and let M 1 , M 2 , … , M r be n × n matrices over C which are not necessarily distinct. Let F ( x 1 , x 2 , … , x p ) = ∑ k = 1 r M k f k ( x 1 , x 2 , … , x p ) and let δ F ( x 1 , x 2 , … , x p ) = det F ( x 1 , x 2 , … , x p ) . In this paper, we prove that, for n × n matrices A 1 , A 2 , … , A p over C , if { A 1 , A 2 , … , A p , M 1 , M 2 , … , M r } is a quasi-commuting family, then F ( A 1 , A 2 , … , A p ) = O implies that δ F ( A 1 , A 2 , … , A p ) = O .

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.

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.

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 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.

Boundary-value descriptor systems: well-posedness, reachability and observability

In this paper we introduce the class of two-point boundary-value descriptor systems (TPBVDS), discrete-time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system 'state' at the two endpoints of the system's interval of definition. By introducing a standard form for regular pencils we obtain a new and simple generalized Cayley-Hamilton theorem that simplifies our investigation of well-posedness, Green's function solutions, and reachability and observability for TPBVDS. There are two distinct notions of reachability and observability that one can define for TPBVDS, associated with processes that propagate inward from and outward toward the boundaries. We investigate each of these in detail, obtaining, among other things, far simpler forms for the reachability and observability results than found previously in the literature. In addition, we describe several methods for the efficient solution of TPBVDS, one involving recursions from each end of t...

On the generalized Cayley-Hamilton theorem

The Cayley-Hamilton theorem is extended to the case where two A and B n \times n matrices are involved. Results similar to the regular case are presented. The results are useful for problems that concern the analysis as well as the synthesis of singular systems, for the definition of a function of two matrices f(A, B) and in various other problems in linear algebra.