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Math Help - Jordan Form

  1. #1
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    Oct 2011
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    Jordan Form

    Hi ALL !!!

    I must find the Jordan form of this operator:

    A(x_1, \ldots, x_6) = (2x_1+x_4+x_6,2x_2+x_5,2x_3,-x_1+3x_4+x_5+x_6,x_1+x_5-x_6,-x_1+x_4+x_5+4x_6)

    I found this matrix (is correct ?):

    \begin{pmatrix}<br />
2 & 0 & 0 & 1 & 0 & 1 \\<br />
0 & 2 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 2 & 0 & 0 & 0 \\<br />
-1 & 0 & 0 & 3 & 1 & 1 \\<br />
1 & 0 & 0 & 0 & 1 &-1 \\<br />
-1 & 0 & 0 & 1 & 1 & 4 <br />
\end{pmatrix}

    Then, I found this characteristic polynomial (is correct ?):

    det(A - x \textit{I}) = (2-x)^3(1-x)(x^2-7x+11)

    But, the Maxima software give me this poli.:

    p_A(x) = (x-2)^4(x-3)^2

    Where is the wrong ????

    Thanks a lot !!
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  2. #2
    MHF Contributor
    Joined
    Mar 2010
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    Florida
    Posts
    3,093
    Thanks
    5

    Re: Jordan Form

    Quote Originally Posted by Borseti View Post
    Hi ALL !!!

    I must find the Jordan form of this operator:

    A(x_1, \ldots, x_6) = (2x_1+x_4+x_6,2x_2+x_5,2x_3,-x_1+3x_4+x_5+x_6,x_1+x_5-x_6,-x_1+x_4+x_5+4x_6)

    I found this matrix (is correct ?):

    \begin{pmatrix}<br />
2 & 0 & 0 & 1 & 0 & 1 \\<br />
0 & 2 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 2 & 0 & 0 & 0 \\<br />
-1 & 0 & 0 & 3 & 1 & 1 \\<br />
1 & 0 & 0 & 0 & 1 &-1 \\<br />
-1 & 0 & 0 & 1 & 1 & 4 <br />
\end{pmatrix}

    Then, I found this characteristic polynomial (is correct ?):

    det(A - x \textit{I}) = (2-x)^3(1-x)(x^2-7x+11)

    But, the Maxima software give me this poli.:

    p_A(x) = (x-2)^4(x-3)^2

    Where is the wrong ????

    Thanks a lot !!
    I don't know where you went wrong since you didn't post your work but the characteristic polynomial Maxima gave you is correct:

    http://www.wolframalpha.com/input/?i=Det[{{2-x%2C0%2C0%2C1%2C0%2C1}%2C{0%2C2-x%2C0%2C0%2C1%2C0}%2C{0%2C0%2C2-x%2C0%2C0%2C0}%2C{-1%2C0%2C0%2C3-x%2C1%2C1}%2C{1%2C0%2C0%2C0%2C1-x%2C-1}%2C{-1%2C0%2C0%2C1%2C1%2C4-x}}]
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