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Math Help - Show this matrix is similar to a diagonal matrix

  1. #1
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    Show this matrix is similar to a diagonal matrix

    A square matrix A (of some size n \times n) satisfies the condition A^2 - 8A +15I = 0.

    (a) Show that this matrix is similar to a diagonal matrix.

    (b) Show that for every positive integer k \geq 8 there exists a matrix A satisfying the above condition with tr(A) = k.

    I have little or no idea where to go with this problem. I know that for (a), A^2 - 8A is obviously a diagonal matrix  (-15I), but am not sure if this is of any use. I also am not sure if I can simply say A^2 - 8A + 15I = 0, so (A - 5I) (A - 3I) = 0, in which case, either (i) A = 3I or A = 5I, or (ii) (A - 5I) (A - 3I) = 0 with A - 5I \neq 0 and A -3I \neq 0, and in this case, I'm not sure how to use (ii) to prove anything. Obviously (b) has something to do with the second term being -8A, but apart from this I can't really think of anything.

    Any help would be greatly appreciated.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Show this matrix is similar to a diagonal matrix

    Quote Originally Posted by EuropeanSon View Post
    A square matrix A (of some size n \times n) satisfies the condition A^2 - 8A +15I = 0. (a) Show that this matrix is similar to a diagonal matrix.
    p(\lambda)=\lambda^2-8\lambda+15=(\lambda-3)(\lambda-5) is an annihilator polynomial of A . This implies that the minimum polynomial of A is \mu(\lambda)=\lambda-3 or \mu(\lambda)=\lambda-5 or \mu(\lambda)=(\lambda-3)(\lambda-5) . In all cases, A is diagonalizable. Try (b).
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    Re: Show this matrix is similar to a diagonal matrix

    Quote Originally Posted by FernandoRevilla View Post
    p(\lambda)=\lambda^2-8\lambda+15=(\lambda-3)(\lambda-5) is an annihilator polynomial of A . This implies that the minimum polynomial of A is \mu(\lambda)=\lambda-3 or \mu(\lambda)=\lambda-5 or \mu(\lambda)=(\lambda-3)(\lambda-5) . In all cases, A is diagonalizable. Try (b).
    Thanks. I have no idea, however, what an annihilator polynomial is, and hadn't come across the term before, and so I don't think this was a method which my lecturer intended me to use or would have accepted without proofs (this was an exam question in a paper I had a few months ago).
    Last edited by EuropeanSon; August 2nd 2011 at 07:09 AM.
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    Re: Show this matrix is similar to a diagonal matrix

    I looked over our course contents, and it appears we did have sufficient material covered to solve it in this manner. Thanks, I'll attempt part (b) now.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Show this matrix is similar to a diagonal matrix

    Quote Originally Posted by EuropeanSon View Post
    I looked over our course contents, and it appears we did have sufficient material covered to solve it in this manner. Thanks, I'll attempt part (b) now.
    All right, a little help for (b). If k=3s+5r with s\geq 0,r\geq 0 integers, choose

    A=\textrm{diag}\;(\underbrace{3,\ldots,3}_{\mbox{s times}},\underbrace{5,\ldots,5}_{\mbox{r times}})
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    Re: Show this matrix is similar to a diagonal matrix

    Quote Originally Posted by FernandoRevilla View Post
    All right, a little help for (b). If k=3s+5r with s\geq 0,r\geq 0 integers, choose

    A=\textrm{diag}\;(\underbrace{3,\ldots,3}_{\mbox{s times}},\underbrace{5,\ldots,5}_{\mbox{r times}})
    Ah, yes, in retrospect I'm shocked at not figuring that out. After that it was simple. I was messing around with all sorts of things before, but overlooked letting the s or r variables equal zero for some reason. Thanks!
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  7. #7
    MHF Contributor FernandoRevilla's Avatar
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    Re: Show this matrix is similar to a diagonal matrix

    Don't worry!. Such things usually happen in Mathematics.
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