Linear Algebra Done Right · Problem 7

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · minimal-polynomial-jordan-basis

Problem

Suppose

T \in \mathcal{L}(V)

is nilpotent and

v_1, \ldots, v_n

is a Jordan basis for

T

. Prove that the minimal polynomial of

T

z^{m+1}

, where

m

is the length of the longest consecutive string of 1's that appears on the line directly above the diagonal in the matrix of

T

with respect to

v_1, \ldots, v_n