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Math Help - Are these matrices similar?

  1. #1
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    Are these matrices similar?

    Surprise.. stuck on yet another hw problem Are these two matrices similar?

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  2. #2
    Member mohammadfawaz's Avatar
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    Re: Are these matrices similar?

    Well, the simplest check is usually the determinant. In this case, it doesn't help since both matrices have determinant equal to 16.
    On the other hand, you can look at the characteristic equation of each matrix (The equation with eigenvalues as roots). The first matrix gives you (\lambda-2)(\lambda-2)(\lambda-4) while the second gives you (\lambda-1)(\lambda-4)(\lambda-4). Since they are not the same, these two matrices are not similar!
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  3. #3
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    Re: Are these matrices similar?

    similar matrices have the same characteristic polynomial (although not necessarily vice versa). if the characteristic polynomials are different, they cannot be similar. this will give you an answer, in this case.
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    Re: Are these matrices similar?

    So in other words I can also say that if the eiganvalues of matrix A are not all the same as the eiganvalues of matrix B, they aren't similar?
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  5. #5
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    Re: Are these matrices similar?

    yes, but...

    what you said is true, however, two matrices with the same eigenvalues may not be similar. tests (in increasing order of strength):

    determinants < eigenvalues < characteristic polynomials < Jordan forms

    if any of these things are different for 2 matrices, we can conclude the matrices are NOT similar. but for any of these things except the jordan form, it might be the case that they have the same (determinant, eigenvalues, char. polyn.) but are not similar.
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