# Eigenvalue with ranks

• Apr 26th 2008, 01:54 PM
Let T be a linear operator on a finite dimensional vector space V. Let $\lambda$ be an eigenvalue of T. Prove that if $rank((T - \lambda I )^m = rank ((T - \lambda I ) ^{m+1} )$ for some integer m, then $K_{ \lambda } = N ((N- \lambda I)^m$