I have the following question:
For the following matrix $A$, find an invertible matrix $P$ over $\mathbb{C}$ such that $P^{-1}AP$ is upper triangular:
\begin{equation}
A= \pmatrix{
1 & 1 & 1 \\
1 & 1 & 1 \\
0 & 0 & 0}
\end{equation}
I have calculated the characteristic polynomial: $ \lambda^2(2-\lambda) $. From that I deduced that we have two eigenvalues:
- $\lambda=0$ with algebraic multiplicity $2$.
- $\lambda=2$ with algebraic multiplicity $1$.
For $\lambda=0$ the eigenvectors are of the form:
\begin{equation}
\alpha \pmatrix{-1\\0\\1} + \beta \pmatrix{-1\\1\\0},
\end{equation}
where $\alpha$ and $\beta$ constants.
Hence, the eigenspace is two-dimensional, i.e the geometric multiplicity is $2$.
Similarly, for $\lambda=2$ we get: \begin{equation}
\gamma \pmatrix{1\\1\\0}
\end{equation}
where $\gamma$ a constant.
Hence, the geometric multiplicity of $\lambda=2$ is $1$.
Now the JCF form of $A$ is: $J_1(0) \oplus J_1(0) \oplus J_1(2)$.
This is the part I get confused. The JCF gives me the matrix:$\pmatrix{0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 2}$, which I have checked is what I get if I use $P= \pmatrix{-1 & -1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0}$ in $P^{-1}AP$. This is the diagonal form not the upper triangular form.
I know for a fact that using for e.g. $P= \pmatrix{1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1}$ will result in indeed an upper triangular matrix, namely $\pmatrix{2 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0}$.
It is to my understanding that that if we take the eigenvectors as columns of $P$ then computing the $P^{-1}AP$ will result in an upper triangular matrix.
What exactly is the process I should follow to produce an upper triangular matrix? Is the process I am doing above not used for triangularisation? How are Jordan matrices involved in triangularisation?
I have also tried to find the generalised eigenvectors using $(A−\lambda I)^2 v=0$ , $(A−\lambda I)^3 v=0$ etc. but due to the nature of $A$ I am not getting any solutions. I am a bit lost on how to deduce the matrix $P$.
(A note: I have not done Schur decomposition and is not in my course's syllabus.)
Thank you in advance for any solutions/suggestions.