The Generalized Spectral Decomposition of a Linear Operator
Garret Sobczyk · College Mathematics Journal · 1997
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.