LIE ALGEBRAS AND LINEAR DIFFERENTIAL EQUATIONS
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Publisher Summary This chapter discusses certain symmetry properties possessed by the solutions of linear differential equations. The two results are established on the Lie algebra generated by a pair ofn by nmatrices. In order to avoid undue repetition, one agrees to call a matrix of rational functionsG(s) regular if it is square and approaches zero as|s| approached infinity. It is possible to associate a Lie algebra with each regular matrix of rational functions in a natural way. The symplectic matrices form a group and the Eigen values of symplectic matrices occur in reciprocal pairs. That is to say, if λ is an eigenvalue of a symplectic matrix. then so is 1/λ. This observation together with the basic ideas of Floquet theory enables one to show that for 0 ≤ t