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Math Help - jordan form question

  1. #1
    MHF Contributor
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    jordan form question

    2)A)
    find the definative polinomial and the minimal polinomial of
    A=\left(\begin{array}{ccc}3 & 1 & -1\\2 & 2 & 0\\1 & -1 & 3\end{array}\right)
    find the primary decomposition of R^{3} in this case,and write the diagonal block matrices
    similar to A
    ?
    B)
    find the jordan form of A defined in part A
    and find the transformation matrices to the basis for which it represented in jordan form.
    how i tried:
    regarding A:
    how i tried:
    |tI-A|= \left|\begin{array}{ccc}t-3 & -1 & +1\\-2 & t-2 & 0\\-1 & 1 & t-3\end{array}\right|= 1\left|\begin{array}{cc}-2 & t-2\\-1 & 1\end{array}\right|+0+(t-3)\left|\begin{array}{cc}t-3 & -1\\-2 & t-2\end{array}\right|= -2+t-2+(t-3)[(t-3)(t-2)+2]=t-4+(t-3)[t^{2}-2t-3t+6+2]= t-4+(t-3)[t^{2}-5t+8]
    = t-4+t^{3}-5t^{2}+8t-3t^{2}+15t-24=t^{3}-8t^{2}+24t-28
    there are some roots possible by theory t=1,-1,+28,-28
    t=1: 1-8+24-28
    t=-1: -1-8-24-28
    from the calculator the other dont work too
    ??
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  2. #2
    MHF Contributor

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    Tejas
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    Re: jordan form question

    be very careful with your signs when working out determinants. for |tI - A|, i get:

    (-2)(1) - (t-2)(-1) + (t-3)[(t-3)(t-2) - (-1)(-2)]

    = -2 + t - 2 + (t-3)[t^2 - 5t + 6 - 2]

    = t - 4 + t^3 - 5t^2 + 4t - 3t^2 + 15t - 12 = t^3 - 8t^2 + 20t - 16.

    possible rational roots of this are 1,-1,2,-2,4,-4,8,-8,16,-16. i believe you will find this factors completely.
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