UNIFIED CANONICAL-FORMS FOR MATRICES OVER A DIFFERENTIAL RING

Linear differential equations with variable coefficients of the vector form (i) x = A(t)x and the scalar form (ii) y(n) + a(n)(t)y(n-1) + ... + a2 (t)y + a1(t)y = 0 can be studied as operators on a differential module over a differential ring. Using this differential algebraic structure and a classical result on differential operator factorization developed by Cauchy and Floquet, a new (variable) eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring. In particular, eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion, Jordan (diagonal), and (generalized) Vandermonde canonical forms, as traditionally used for matrices over a number ring (field). Therefore, these new matrix canonical forms can be viewed as unifications of those classical ones. Two additional canonical matrices that perform order reductions of a Cauchy-Floquet factorization of a linear differential operator are also introduced. General and explicit formulas for obtaining the new canonical forms are derived in terms of the new (variable) eigenvalues. The results obtained here have immediate applications in diverse engineering and scientific fields, such as physics, control systems, communications, and electrical networks, in which linear differential equations (i) and (ii) are used as mathematical models.