A Direct Polynomial Approach to Spectral Decomposition

2026/07/22 by Simon Bossoney, Marc Troyanov
#math.SP #math.OA

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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=∑_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.

Citations

• A Direct Approach to Hermite Interpolation

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