Algebraic Properties of Self-Adjoint Systems

Dunham Jackson
1916 Transactions of the American Mathematical Society  
The general definition of adjoint systems of boundary conditions associated with ordinary linear differential equations was given by Birkhoff.t In a paper of Bôcher,î in which the idea is further developed, there is obtained a condition that a system of thé second order be self-adjoint. It is proposed here to extend the discussion of this problem to the case of systems of any order. § A condition for self-adjointness of the boundary conditions is simply expressed in matrix form, without any
more » ... rm, without any requirement that a corresponding property be possessed by the differential equations; the condition gains in symmetry if it is assumed that the differential expression which forms the lefthand member of the given differential equation is itself identical with its adjoint or with the negative of its adjoint. As a preliminary, it will be well to recall a well-known rule for the combination of matrices, which will be found particularly convenient. Let «i, a2, a3, and «4 be M-rowed square matrices. The square matrix of order 2m which is made up of the elements of these four, with the elements of «i, arranged in order, in its upper left-hand corner, and the elements of the other matrices correspondingly disposed, may be indicated by the notation a=(ai a2V \ «3 OH/
doi:10.2307/1988829 fatcat:7solzvjdqzh6zdbq7tyo64gisa