# Jordan Form

• Dec 4th 2011, 05:23 PM
Borseti
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 ?):

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

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 !!
• Dec 5th 2011, 06:51 AM
dwsmith
Re: Jordan Form
Quote:

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}}]