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Math Help - Determining a matrix according to its charateristic polynomial

  1. #1
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    Determining a matrix according to its charateristic polynomial

    I am wondering whether it is possible to determine a matrix if its characteristic polynomial is known. Of course, there is more than one matrix for a fixed characteristic polynomial but there must also be something special that other matrixes dont have. For instance, if the c.p. is (-1)^n \left(t^n-1\right) with n the order, are there some simple conditions that can be applied to its matrix?
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  2. #2
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    The eigenvalues (and so the characteristic polynomial) determine a matrix up to "similarity".

    Matrices having the same eigenvalues (characteristic polynomial) must be "similar"- that is, P= QDQ^{-1} for some invertible matrix Q.
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