I have the matrix
$$A=\begin{pmatrix} 5 & 1 & 0\\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$
and I should determine generalised eigenvectors, if they exist.
I found one eigenvalue with algebraic multiplicity $3$.
$$\lambda=5$$
I calculated two eigenvectors:
$$\vec{v_{1}} =\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad{}
\vec{v_{2}} =\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$
Also, I know this formula for generalized vector
$$\left(A-\lambda I\right)\vec{x} =\vec{v}$$
Finally, my question is:
How do I know how many generalised eigenvectors I should calculate?
For every eigenvector one generalised eigenvector or?
My university book is really confusing, and I saw there that they calculated generalised eigenvector only for some eigenvectors, and for some not. But I don't understand how to know that.
Best Answer
Your matrix is in Jordan normal form. You can read on the matrix that $e_1$ and $e_3$ are eigenvectors for the eigenvalue $2$, and $e_2$ is a generalised eigenvector.