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$% L. Jódar, An algebraic approach for solving boundary value matrix problems: existence, uniqueness, and closed form solutions, Rev. Mat. Univ. Complut. Madrid 1 (1988), no. 1, 2, 3, 145–155.%$

An Algebraic Approach for Solving Boundary Value Matrix Problems: Existence, Uniqueness, and Closed Form Solutions

Department of Applied Mathematics
Polytechnical University of Valencia
P.O. 22012 Valencia Spain

Received: May 18, 1988
Revised: September 13, 1988

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  2
X + A1X + A0 =0  , 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.