<s>[INST] find the Jordan form and $P$ such that $P^{-1}AP = J$. Consider the matrix $$A = \left(\begin{array}{cccc} -11&0&-9\\32&1&24\\16&0&13 \end{array}\right)$$
I want to find the Jordan form of $A$, with $1$-s at the bottom and the jordan basis, which is $P$ columns such that $P^{-1}AP = J$.
I evaluated the charechteristic polynomial which is $f_A(x) = -(x-1)^3$. So we have one eigenvalue which is $\lambda =1$.
Now, 
$$A-I = \left(\begin{array}{cccc} -12&0&-9\\32&0&24\\16&0&12 \end{array}\right)$$
$$\ker (A-I) = \text{span}\{(0,1,0),(1,0,3/4)\}$$
$$\ker (A-I)^2 =\mathbb{R}^3$$
As I understand I need to find a vector, $v_3$ such that $\ker (A-I) \cup v_3 = \ker (A-I)^2 = \mathbb{R}^3$.
Then, $$v_2 = (A-I)v_3 \\ v_1 = (A-I)v_2$$
And we (suppose to) get $P = (v_1,v_2,v_3)$ such that $P^{-1}AP = J$.
I chose $v_3 = (1,0,0)$, but it doesn't work, and for the matter of fact I'm confused by all this.
I'd be glad if you could explain to me how to correct this.
Thanks.
 [/INST] Saying that 1 is an eigenvalue for matrix $A$ means that there exist non-zero vectors, $v$, such that $Av= v$ or $(A- I)v= 0$.
Here, that means $$\begin{bmatrix}-12 & 0 & -9 \\ 32 & 0 & 24 \\ 16 & 0 & 12\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}\begin{bmatrix}-12x- 9z \\ 32x+ 24z \\ 16x+ 12z \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.$$
You basically have that. Now note that this gives you the three equations $$-12x- 9z= 0,$$ $$32x+ 24z= 0,$$ and $$16x+ 12z= 0.$$  From the first equation, $9z= -12x$ so $z= (-12/9)x= (-4/3)x.$  Setting $z= (-4/3)x$ in the second and third equations gives $$32x+ 24(-4/3)x= 32x- 32x= 0$$ and $$16x+ 12z= 16x+ 12(-4/3)x= 16x- 16x= 0.$$
That is, for any value of $x, z= (-4/3)x$ will satisfy all those equations.  Since $y$ did not enter into this, $y$ can be anything.  That is, the eigenvectors must be of the  form $$\begin{pmatrix}x \\ y \\ z\end{pmatrix}= \begin{pmatrix} x \\ y \\ (-4/3)x \end{pmatrix}= x\begin{pmatrix}1 \\ 0 \\ -4/3\end{pmatrix}+ y\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}.$$  
We can use $\begin{pmatrix}3\\ 0\\ -4\end{pmatrix}=3\cdot \begin{pmatrix}1\\ 0\\ -4/3\end{pmatrix}$ and $\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}$ form a basis for the Eigen space.  since that is a two dimensional subspace, the "Jordan Normal Form" will be of the form $$J= \begin{bmatrix}3 & 1 & 0 \\ 0 & 3 & 0 \\0 & 0 & 3\end{bmatrix}$$ and we can form the matrix $$P^{-1}$$ using the vectors $$\begin{pmatrix}3 \\ 0\\ -4\end{pmatrix},\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}$$ as columns.  
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