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Diagonalization of matrices over rings
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2013 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 376, 123-138 p.Article in journal (Refereed) Published
Abstract [en]

We propose a method for diagonalizing matrices with entries, in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory.

Place, publisher, year, edition, pages
2013. Vol. 376, 123-138 p.
Keyword [en]
Matrices, Determinant, Diagonalization, Eigenvector, Eigenvalue, Symmetric function, Splitting algebra, Universal root, Regularity, Spectral mapping
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-118590DOI: 10.1016/j.jalgebra.2012.10.029ISI: 000314134300009Scopus ID: 2-s2.0-84875059023OAI: oai:DiVA.org:kth-118590DiVA: diva2:607190
Note

QC 20130222

Available from: 2013-02-22 Created: 2013-02-21 Last updated: 2017-12-06Bibliographically approved

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  • apa
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  • de-DE
  • en-GB
  • en-US
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  • nn-NO
  • nn-NB
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  • Other locale
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