Linear Algebra Done Right · Problem 8B.21

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · characteristic-polynomial-minimal-polynomial

Problem

Suppose

p, q \in \mathcal{P}(\mathbf{C})

are monic polynomials with the same zeros and

q

is a polynomial multiple of

p

. Prove that there exists

T \in \mathcal{L}(\mathbf{C}^{\deg q})

such that the characteristic polynomial of

T

q

and the minimal polynomial of

T

p

. This exercise implies that every monic polynomial is the characteristic polynomial of some operator.