Depth of an eigen value (generalized eigenspaces)

[Doom of Doom]

Let A be a matrix and \mu be an eigenvalue of that matrix. Suppose that for some k, \tex{ker}\left(A-\mu I\right)^k=\tex{ker}\left(A-\mu I\right)^{k+1}. Then show that \tex{ker}\left(A-\mu I\right)^{k+r}=\tex{ker}\left(A-\mu I\right)^{k+r+1} for all r\geq0.

 

[Mark44]

What have you tried? Do you know what the kernel of a matrix is?

 

[HallsofIvy]

Important point: for any matrix A, A0= 0!