Linear Algebra Done Right · Problem 9C.7

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

T \in \mathcal{L}(V)

is invertible. Let

p

denote the characteristic polynomial of

T

and let

q

denote the characteristic polynomial of

T^{-1}

. Prove that

q(z) = \frac{1}{p(0)} z^{\dim V} p\!\left(\frac{1}{z}\right)

for all nonzero

z \in \mathbf{F}