Useful Relations for Partial Expansion of Proper Rational Functions and Transition Matrices

It is shown that the sum of the residues of a proper rational function X(s) = K(N(s)/D(s)) at all its poles is given by Kδm,n—1. N(s) and D(s) are monic polynomials of degree m and n, respectively, with m ≤ n—1. δm,n—1 is the Kronecker symbol. This result simplifies calculations encountered in the partial fraction inversion of proper rational Laplace transforms with repeated poles. A similar result is obtained for partial fraction expansion of the transition matrix (sI-A)—1 which arises in Laplace transform solution of the vector— matrix equation ẋ = Ax + Bu: the sum of all the residue matrices associated with the eigenvalues of A is equal to the unit matrix. Each residue matrix associated with a simple eigenvalue is a dyadic, and is, therefore, completely determined by its first row and column.