Jordan Form

**Borseti** · December 4th 2011, 04:23 PM

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 ?):

$\begin{pmatrix} 2 & 0 & 0 & 1 & 0 & 1 \\ 0 & 2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ -1 & 0 & 0 & 3 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 &-1 \\ -1 & 0 & 0 & 1 & 1 & 4 \end{pmatrix}$

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 !!

**dwsmith** · December 5th 2011, 05:51 AM

Originally Posted by Borseti

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 ?):

$\begin{pmatrix} 2 & 0 & 0 & 1 & 0 & 1 \\ 0 & 2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ -1 & 0 & 0 & 3 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 &-1 \\ -1 & 0 & 0 & 1 & 1 & 4 \end{pmatrix}$

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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