Kunze Linear Algebra · Problem 7.3.12

Rational and Jordan Forms

Mathematics — Methods · jordan-form-similarity

Problem

Use the result of Exercise 11 and the Jordan form to prove the following: Let

A

and

B

n \times n

matrices over the field

F

which have the same characteristic polynomial

f = (x - c_1)^{d_1} \cdots (x - c_k)^{d_k}

and the same minimal polynomial. Suppose also that for each

i

the solution spaces of

(A - c_i I)

and

(B - c_i I)

have the same dimension. If no

d_i

is greater than 6, then

A

and

B

are similar.