William A. Adkins

The Cayley-Hamilton and Frobenius theorems via the Laplace transform

The Cayley–Hamilton theorem on the characteristic polynomial of a matrix A and Frobenius’ theorem on minimal polynomial of A are deduced from the familiar Laplace transform formula L(eAt)=(sI−A)−1. This formula is extended to a formal power series ring over an algebraically closed field of characteristic 0, so that the argument applies in the more general setting of matrices over a field of characteristic 0.

Matrices over differential fields which commute with their derivative

Abstract A theorem is proved concerning the diagonalizability of a matrix over a differential field by means of a similarity transformation from the field of constants of the differential field. This result contains, as a special case, known results concerning the diagonalizability over the complex numbers of a Hermitian matrix of analytic functions under the hypothesis that the matrix commutes with its derivative.

Normal matrices over hermitian discrete valuation rings

Abstract A generalization of the classical spectral theorem for normal complex matrices is proved for matrices with entries from a class of discrete valuation rings which we have called hermitian. The hermitian discrete valuation rings include the convergent complex power series in one variable and the complex formal power series in one variable. Thus one obtains an algebraic proof of Rellichs theorem on diagnolizability of hermitian analytic matrices which is simultaneously valid for both the rings of convergent and formal power series.

Simultaneous diagonalization of matrices parametrized by a projective algebraic curve

Abstract If K is an algebraic function field in one variable over an algebraically closed field k, then conditions are presented to insure that a matrix A ∈ Mn(K) is diagonalizable by means of a similarity transformation T ∈ GL(n, k). This result generalizes results of Friedland [1] and Motzkin-Taussky [4].

Linear Systems of Differential Equations

In previous chapters, we have discussed ordinary differential equations in a single unknown function, y(t). These are adequate to model real-world systems as they evolve in time, provided that only one state, that is, the number y(t), is needed to describe the system. For instance, we might be interested in the temperature of an object, the concentration of a pollutant in a lake, or the displacement of a weight attached to a spring. In each of these cases, the system we wish to describe is adequately represented by a single function of time. However, a single ordinary differential equation is inadequate for describing the evolution over time of a system with interdependent subsystems, each with its own state.

Putzer's Algorithm for e At via the Laplace Transform

Summary A method due to E. J. Putzer computes the matrix exponential eAt for an n × n matrix A without transforming A to Jordan canonical form. A variation of Putzers algorithm is presented. This approach is based on an algorithmically produced formula for the resolvent matrix (sI - A) -1 that is combined with simple Laplace transform formulas to give a formula, similar to Putzers, for eAt.

The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.

The pointwise-local-global principle for solutions of generic linear equations

Abstract The problem studied in this paper is to determine some conditions on a matrix A over a ring R which will insure that the matrix equation Au = f is solvable over R if it is solvable over the residue field ( ) for every ∈ Spec R . If R is a regular local ring (containing a field), a polynomial ring over an algebraically close field, or the ring of holomorphic functions on a Stein manifold, then a sufficient condition on A for pointwise solvability to imply global solvability is that A be generic, a concept which is defined in the paper. For the rings of functions, pointwise solvability will mean solvability over R / M for a certain set of maximal ideals. The relationship between this notion of pointwise solvability and solvability over ( ) for all prime ideals is studied by introducing various types of closure operations on submodules. Mather has previously proved a theorem similar to the main result of this paper for the case of rings of smooth real valued functions on open subsets of Euclidean space.

Power Series Methods

Thus far in our study of linear differential equations, we have imposed severe restrictions on the coefficient functions in order to find solution methods. Two special classes of note are the constant coefficient and Cauchy–Euler differential equations. The Laplace transform method was also useful in solving some differential equations where the coefficients were linear. Outside of special cases such as these, linear second order differential equations with variable coefficients can be very difficult to solve.

Second Order Constant Coefficient Linear Differential Equations

This chapter begins our study of second order linear differential equations, which are equations of the form