
jordan form question
2)A)
find the definative polinomial and the minimal polinomial of
$\displaystyle A=\left(\begin{array}{ccc}3 & 1 & 1\\2 & 2 & 0\\1 & 1 & 3\end{array}\right)$
find the primary decomposition of $\displaystyle R^{3}$ 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:
tIA=$\displaystyle \left\begin{array}{ccc}t3 & 1 & +1\\2 & t2 & 0\\1 & 1 & t3\end{array}\right$=$\displaystyle 1\left\begin{array}{cc}2 & t2\\1 & 1\end{array}\right+0+(t3)\left\begin{array}{cc}t3 & 1\\2 & t2\end{array}\right$=$\displaystyle 2+t2+(t3)[(t3)(t2)+2]=t4+(t3)[t^{2}2t3t+6+2]$=$\displaystyle t4+(t3)[t^{2}5t+8]$
=$\displaystyle t4+t^{3}5t^{2}+8t3t^{2}+15t24=t^{3}8t^{2}+24t28$
there are some roots possible by theory t=1,1,+28,28
t=1: 18+2428
t=1: 182428
from the calculator the other dont work too
??

Re: jordan form question
be very careful with your signs when working out determinants. for tI  A, i get:
$\displaystyle (2)(1)  (t2)(1) + (t3)[(t3)(t2)  (1)(2)]$
$\displaystyle = 2 + t  2 + (t3)[t^2  5t + 6  2]$
$\displaystyle = t  4 + t^3  5t^2 + 4t  3t^2 + 15t  12 = t^3  8t^2 + 20t  16$.
possible rational roots of this are 1,1,2,2,4,4,8,8,16,16. i believe you will find this factors completely.