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This chapter is devoted to matrix decompositions. We study the Schur decomposition, spectral decomposition, singular value decomposition (SVD), polar decomposition, and the Jordan decomposition. We begin with commuting matrices, study the polynomials that annihilate matrices, especially the minimal and characteristic polynomials, and discuss the similarity of the matrices associated with a given matrix.

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Matrix Polynomials and Canonical Forms

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  • 发表日期：

    2026年

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  • Fuzhen Zhang

摘要

This chapter is devoted to matrix decompositions. We study the Schur decomposition, spectral decomposition, singular value decomposition (SVD), polar decomposition, and the Jordan decomposition. We begin with commuting matrices, study the polynomials that annihilate matrices, especially the minimal and characteristic polynomials, and discuss the similarity of the matrices associated with a given matrix.