Kunze Linear Algebra · Problem 7.3.13

Rational and Jordan Forms

Mathematics — Methods · jordan-form-transpose

Problem

N

is a

k \times k

elementary nilpotent matrix, i.e.,

N^k = 0

but

N^{k-1} \neq 0

, show that

N^t

is similar to

N

. Now use the Jordan form to prove that every complex

n \times n

matrix is similar to its transpose.