I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 & 6 & 0\end{bmatrix}$$ I have already found the eigenvalue $\lambda = 2$ which has algebraic multiplicity of 3 and geometric multiplicity of 2. The corresponding eigenvectors to this eigenvalue are $$v_1=\begin{bmatrix} -3\\1\\0 \end{bmatrix} \quad\quad \text{and} \quad\quad v_2=\begin{bmatrix} 1\\0\\1 \end{bmatrix}$$ I know that I need to solve either $(A-2I)w = v_1$ or $(A-2I)w = v_2$ to find a generalized eigenvector $w$. However, both system of equations have no solutions. I need some guidance on how to find the generalized eigenvector, or if there is some other way for me to decompose my matrix.
2026-02-23 20:39:15.1771879155
Finding generalized eigenvectors of a matrix
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