I'm trying to prove in two cases where two similar matrices are diagonalizable or non-diagonalizable? If they are diagonalizable, obviously the statement holds. But what if they are not? Can non-diagonalizable matrices still be similar? If yes, show me an example please. Thank you.
[Math] Prove that similar matrices have the same geometric multiplicity
diagonalizationlinear algebramatrices
Best Answer
Let $A,B\in M_n(\Bbb K)$ so that $\exists P\in GL_n(\Bbb K), A = P B P^{-1}$
$\forall \lambda \in \Bbb K, \gamma_A(\lambda)=\dim \operatorname{Ker} \left( A - \lambda I_n\right) = \dim \left\{X \in \Bbb K ^n, AX=\lambda X\right\}=\dim \left\{X \in \Bbb K ^n, P B P^{-1}X=\lambda X\right\} =\dim \left\{X \in \Bbb K ^n, B (P^{-1}X)=\lambda (P^{-1}X)\right\}=\dim \left\{X \in \Bbb K ^n, B X=\lambda X\right\}=\dim \operatorname{Ker} \left( B - \lambda I_n\right)=\gamma_B(\lambda)$