数式処理による固有値問題の解法

記号行列の有理式演算に基づくJordan分解

抄録

A method is presented to deal with the eigenvalue problem for symbolic matrices on the basis of rational operations. From difficulties in solving higher order algebraic equations with symbolic coefficients, it is impossible to obtain the Jordan normal form of a higher order symbolic matrix in a usual sense. Jordan decomposition on the field of rational expressions is considered, which can be performed by symbolic manipulation. The semisimple and the nilpotent components are obtained by substitution of the matrix into polynomials with coefficients of rational expressions. The polynomials are calculated from the characteristic polynomial by rational operations such as irreducible factor decomposition, partial fractional decomposition and multiple reduction. The method for operations of linear subspaces by symbolic manipulation proposed by the authors are applicable to derivations of the generalized eigenspaces and the Jordan normal forms for the eigenvalues in rational expressions.