16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.2.15

The Rational and Jordan Forms

Mathematics — Linear Algebra · minimal-polynomial-over-subfield

Problem

Let

F

be a subfield of the field of complex numbers, and let

A

be an

n \times n

matrix over

F

. Let

p

be the minimal polynomial for

A

. If we regard

A

as a matrix over

\mathbb{C}

, then

A

has a minimal polynomial

f

as an

n \times n

matrix over

\mathbb{C}

. Use a theorem on linear equations to prove

p = f

. Can you also see how this follows from the cyclic decomposition theorem?