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!