A Direct Polynomial Approach to Spectral Decomposition

Abstract

We give a direct construction of the spectral projectors of a complex square matrix, based on explicit interpolation polynomials previously introduced. This yields a spectral resolution $A=\sum_{i=1}^r(λ_iP_i+N_i)$ from which the Primary Decomposition Theorem, the Cayley--Hamilton theorem, criteria for diagonalizability, and the spectral theorem for normal matrices are proved by short formal arguments. A weaker form of the construction, extends to any perfect fields via a Galois invariance argument and produces the Jordan--Chevalley decomposition.