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xEnOnn Group Title

How I show that the eigenvalues of matrix A times matrix B is and BA are equal? That's to show eigenvals(AB)=eigenvals(BA).

  • 2 years ago
  • 2 years ago

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  1. hahd Group Title
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    http://en.wikipedia.org/wiki/Characteristic_polynomial think it might be useful

    • 2 years ago
  2. xEnOnn Group Title
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    From this, \[\det(AB-\lambda I)=0\] Since AB and BA are different matrices, I still find it strange that their eigenvalues would turn out the same. How is this so?

    • 2 years ago
  3. Zarkon Group Title
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    Suppose \[ABx=\lambda x\] then multiply by B \[BABx=\lambda Bx\] so \[BA(Bx)=\lambda (Bx)\] thus BA has the same eigenvalues as AB similarly AB has the same eigenvalues as BA.

    • 2 years ago
  4. xEnOnn Group Title
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    wow...this is smart... thanks a lot Zarkon!! :D

    • 2 years ago
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