Kunze Linear Algebra · Problem 7.2.12

Rational and Jordan Forms

Mathematics — Methods · similarity-over-subfields

Problem

Let

F

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

A

and

B

n \times n

matrices over

F

. Prove that if

A

and

B

are similar over the field of complex numbers, then they are similar over

F

. (Hint: Prove that the rational form of

A

is the same whether

A

is viewed as a matrix over

F

or a matrix over

\mathbb{C}

; likewise for

B