Page 200 - Advanced Linear Algebra

184 Advanced Linear Algebra

)
b What do the matrices

(~ > ? and ) ~ > ?

have to do with this issue?
c Show that even without the assumption on invertibility the matrices
)
() and )( have the same characteristic polynomial. Hint : Write
(~ 70 8
Á
7
8
0
where and are invertible and Á is an d matrix that has the
d identity in the upper left-hand corner and 's elsewhere. Write
Z
)~ 8)7. Compute () and )( and find their characteristic
polynomials.
14. Let be a linear operator on - with minimal polynomial
2

²%³ ~ ²%b ³²%c ³. Find the rational canonical form for if
1

-~ r -~ s, or -~ d .
B

15. Suppose that the minimal polynomial of ²= ³ is irreducible. What can
you say about the dimension of ?
=
16. Let B ²= ³ where = is finite-dimensional. Suppose that ²%³ is an

irreducible factor of the minimal polynomial ²%³ of . Suppose further
that "Á #  = have the property that ²"³ ~ ²#³ ~ ²%³ . Prove that
" ~ ² ³# for some polyjomial ²%³ if and only if # ~ ² ³" for some

polynomial ²%³ .