# Can you tell how many eigenvectors a matrix has from just the characteristic equation?

If the equation has a repeated root, can you tell without evaluating in the matrix if that repeated root corresponds to more than one eigenvector?

• In general, you only know that the geometric multeplicity does not exceed the algebraic multeplicity of an eigenvalue. – AnyAD Feb 22 at 20:05
• The general topic here is the Jordan normal form, which exists for all square matrices (over $\mathbb C$), unlike diagonalization. – Lee Mosher Feb 22 at 20:30

No. Consider $$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix},$$ and $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$ They have the same characteristic polynomial, but not the same eigenvectors.