= Solution
A <Jordan normal form> of a complex $n\times n$ <matrix> $A$ is a <similarity transformation> $P^{-1}AP=J$ in which $J$ is a direct sum of <Jordan blocks>
$$
J_r(\lambda)=\begin{pmatrix}\lambda&1&&0\\&\lambda&\ddots&\\&&\ddots&1\\0&&&\lambda\end{pmatrix}.
$$
For each <eigenvalue>, the multiset of block sizes is uniquely determined by $A$; only the order of the blocks is arbitrary. The change-of-basis <matrix> $P$ is generally not unique.
For the given <matrix>, expansion of the <characteristic polynomial> gives
$$
\det(tI-A)=(t-1)^2(t-2),\qquad A-I=\begin{pmatrix}-8&3&-5\\7&-2&5\\17&-6&11\end{pmatrix}.
$$
The upper-left $2\times2$ minor of $A-I$ is $-5$, so its <rank> is $2$ and the <eigenspace> for $1$ has dimension $1$. Thus the two occurrences of the <eigenvalue> $1$ belong to a single size-$2$ <Jordan block>. More explicitly, the <Jordan chain> and the remaining <eigenvector> can be chosen as
$$
v_1=(1,1,-1)^T,\quad v_2=(1,3,0)^T,\quad v_3=(0,5,3)^T,\qquad (A-I)v_1=0,\quad(A-I)v_2=v_1,\quad Av_3=2v_3.
$$
Consequently, with $P=(v_1\ v_2\ v_3)$, for which $\det P=1$, the answer is
$$
\boxed{P^{-1}AP=\begin{pmatrix}1&1&0\\0&1&0\\0&0&2\end{pmatrix}.}
$$
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