Linear Algebra Done Right · Problem 5C.7

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

V

is finite-dimensional,

T \in \mathcal{L}(V)

, and

v \in V

. (a) Prove that there exists a unique monic polynomial

p_v

of smallest degree such that

p_v(T)v = 0

. (b) Prove that the minimal polynomial of

T

is a polynomial multiple of

p_v