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Math Help - Diagonalizable matrix used in polynomial form

  1. #1
    LHS
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    Diagonalizable matrix used in polynomial form

    I'm rather unsure what is going on with the last part of question c any help would be greatly appreciated!

    Last edited by LHS; March 30th 2011 at 12:12 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by LHS View Post
    I'm rather unsure what is going on with the last part of question c any help would be greatly appreciated!



    If you don`t rotate the problem, we'll suffer from neck ache.
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  3. #3
    LHS
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    sorry.. was suffering from some technical difficulties!
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  4. #4
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    Hi there,

    Your matrix is diagonalizable. In particular,

    <br />
$B=PDP^{-1}$, where <br />
$D=\begin{pmatrix}<br />
6 & 0 & 0 \\<br />
0 & 6 & 0 \\<br />
0 & 0 & 3\\<br />
\end{pmatrix}\\<br />
    Hence,

    <br />
B^n = (PDP^{-1})^n = (PDP^{-1})(PDP^{-1}) ... (PDP^{-1})\\<br />
= PD^nP^{-1}\\<br />

    Substituting this into your equation yields,

    <br /> <br />
a_nPD^nP^{-1} + ... + a_1PDP^{-1} + a_0& = C\\<br /> <br />

    and pre and post multiplying both sides by P inv. and P (respectively) yields:


    <br /> <br />
a_nD^n + ... + a_0& = P^{-1}CP\\<br /> <br />


    where C is that diagonal matrix you are given on the right-hand side of the equation (with entries 1, 2, 3).

    However, you will notice that when you multiply P^{-1}CP out, you will get a matrix that is not diagonal on the right hand side of the last equation.

    But a power of a diagonal matrix is again a diagonal matrix, a scalar multiple of a diagonal matrix is also diagonal, and the sum of diagonal matrices is again diagonal. Hence, you will never get the RHS, so it is impossible.
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  5. #5
    LHS
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    Can't really thank you enough.. looks worryingly simple now! You know, one of those questions you have a mental block on, that was bothering the life out of me
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  6. #6
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    Don't worry about it. It's something I might not have figured out some other day.

    When I am facing problems like this, and a potential way of solving it doesn't occur to me within a few minutes, I consciously remind myself of "what I have in my toolbox." This involves not only summoning up what I know about diagonalizable matrices (and perhaps even making lists and drawing arrows to indicate potential connections between ideas), but also reminding myself of the basic methods of proof: direct proof, contrapositive, contradiction, etc. When in doubt -- proof by induction!
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  7. #7
    LHS
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    That's good advice, i'll bear it in mind cheers mate!
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