The Generalized Spectral Decomposition of a Linear Operator

The generalized spectral decomposition expressing a linear operator as a sum of idempotent and nilpotent operators is derived from the partial fraction decomposition of the reciprocal of its minimal polynomial, and makes possible a functional calculus of analytic functions of the operator. The classical spectral and polar decompositions for matrices are shown to be easy consequences. The idempotents and associated nilpotents provide a basis for the commutative algebra generated by an operator that is far more convenient for computations than the usual basis of powers of the operator. (The College Mathematics Journal, January 1997) ∗I gratefully acknowledge the support given by INIP of the Universidad de las Americas, and CONACYT (The Mexican National Council for Science and Technology) grant 3803E.