An Algebraic Approach for Solving Boundary Value Matrix Problems: Existence, Uniqueness, and Closed Form Solutions
In this paper we show that in an analogous way to the scalar case, the general
solution of a non homogeneous second order matrix differential equation may
be expressed in terms of the exponential functions of certain matrices related
to the corresponding characteristic algebraic matrix equation. We introduce the
concept of co-solution of an algebraic equation of the type
,
that allows us to obtain a method of the variation of the parameters for the
matrix case and further to find existence, uniqueness conditions for solutions of
boundary value problems. These conditions are of algebraic type, involving the
Penrose-Moore pseudoinverse of a matrix related to the problem. A computable
closed form for solutions of the problem is given.
1980 Mathematics Subject Classification (1985 revision): 15A24, 34B05, 34B10, 15A09.