A Direct Polynomial Approach to Spectral Decomposition

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 $A={\sum }_{i=1}^r({\lambda }_i{P}_i+N_i)$ 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.