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Thread: Diagonalizable

  1. #1
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    Diagonalizable

    I am trying to prove that the charactertic polynomial of a diagonalizable matrix will "split".

    I know intuitively this has to be true because a diagonalizable matrix would be similar to a diagonal one, which would have the same charactristic polynomial... and obviously that polynomial will split since its determinant will just be determined by the diagonal entries.. but I am stuck on how to formally prove it.

    Thank you in advance for any tips!!!
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  2. #2
    Senior Member jakncoke's Avatar
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    What do you mean by "split"?
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  3. #3
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    Split

    split meaning factors... so the polynomial f(t) splits over the field if there are scalars c, a_1, a_2, ..., a_n in F such that f(t)=c(t-a_1)(t-a_2)...(t-a_n)
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  4. #4
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    Quote Originally Posted by MatthewD View Post
    split meaning factors... so the polynomial f(t) splits over the field if there are scalars c, a_1, a_2, ..., a_n in F such that f(t)=c(t-a_1)(t-a_2)...(t-a_n)
    So the characteristic polynomial
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