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The characteristic equation

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The Theory of Matrices

Part of the book series: Ergebnisse der Mathematik und Ihrer Grenƶgebiete ((MATHE1,volume 5))

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Abstract

The minimum equation. If A is a matrix of order n over a field p, the matrices I, A, A 2,..., A n 2 constitute n 2 + 1 sets of n 2 numbers each, and hence are linearly dependent in p. Thus A satisfies some equation

$$m{\text{} }(\lambda){\text{} } = {\text{} }\lambda ^u {\text{} } + {\text{} }m_1 \lambda ^{u - 1} {\text{} } + {\text{} }...{\text{} } + {\text{} }m_u {\text{} } = {\text{} }0$$

with coefficients in p of minimum degree μ. We shall call μ the index of A. The index of a scalar matrix is 1. Every matrix except 0 has an index.

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Authors
  1. C. C. Mac Duffee
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© 1933 Springer-Verlag Berlin Heidelberg

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Mac Duffee, C.C. (1933). The characteristic equation. In: The Theory of Matrices. Ergebnisse der Mathematik und Ihrer Grenƶgebiete, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-99234-6_2

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  • DOI: https://doi.org/10.1007/978-3-642-99234-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-98421-1

  • Online ISBN: 978-3-642-99234-6

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