Abstract
The main focus is on diagonalizable matrices, that is matrices similar to a diagonal one. We completely characterize these matrices and use this to complete the proof of Jordan’s classification theorem for arbitrary matrices with complex entries. Along the way, we prove that diagonalizable matrices with complex entries are dense and use this to give a clean proof of the Cayley–Hamilton theorem.
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This lemma says that if A, B ∈ F[X] are relatively prime polynomials, then we can find polynomials U, V ∈ F[X] such that \(AU + BV = 1\). This easily yields the following more general statement: if \(P_{1},\ldots,P_{k}\) are polynomials whose greatest common divisor is 1, then we can find polynomials \(U_{1},\ldots,U_{k}\) such that \(U_{1}P_{1} +\ldots +U_{k}P_{k} = 1\).
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© 2014 Springer Science+Business Media New York
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Andreescu, T. (2014). Diagonalizability. In: Essential Linear Algebra with Applications. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4636-3_9
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DOI: https://doi.org/10.1007/978-0-8176-4636-3_9
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Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-0-8176-4360-7
Online ISBN: 978-0-8176-4636-3
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