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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=∑i=1r(λiPi+Ni) 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.
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BibTeX
@misc{bossoney2026direct,
title = {{A Direct Polynomial Approach to Spectral Decomposition}},
author = {Simon Bossoney and Marc Troyanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.20218},
eprint = {2607.20218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20218},
}
RIS
TY - UNPB
TI - A Direct Polynomial Approach to Spectral Decomposition
AU - Simon Bossoney
AU - Marc Troyanov
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2607.20218
UR - https://arxiv.org/abs/2607.20218
AB - 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.
ER -