I have a matrix $A$ and I want to put it into Jordan canonical form. Let $A$ be a $3x3$ matrix and have one eigenvalue, $\lambda$ with algebraic multiplicity 3.
To create a matrix $P$ that is a basis for $\mathbb R^3$, I find the eigenvector $v_1$ for $\lambda$, and then I need to find two more generalized eigen vectors for $\lambda$ as well correct?
But from working problems it seems that I can not just use any two other generalized eigenvectors to form $P$, correct? They have to be two specific ones, that when I carry out:
I will have $A$ in its Jordan form. I thought it could be any two generalized eigen vectors?