Page 199 - Advanced Linear Algebra

The Structure of a Linear Operator 183

where ²%³ are distinct monic primes. Prove that the following are

equivalent:
a) = is -cyclic.

b)deg² ²%³³ ~ dim²= . ³

)
c The elementary divisors of are the prime power factors ²%³ and so

= ~ ºº# »» l Ä l ºº# »»

is a direct sum of -cyclic submodules ºº# »» of order ²%³ .

3. Prove that a matrix ( C ²-³ is nonderogatory if and only if it is similar
to a companion matrix.
4. Show that if and are block diagonal matrices with the same blocks, but
)
(
in possibly different order, then and are similar.
)
(
5. Let ( C ²-³ . Justify the statement that the entries of any invariant
factor version of a rational canonical form for are “rational” expressions
(
in the coefficients of , hence the origin of the term rational canonical
(
form. Is the same true for the elementary divisor version?
=
B

6. Let ²= ³ where is finite-dimensional. If ²%³  -´%µ is irreducible
and if ² ³ is not one-to-one, prove that ²%³ divides the minimal

polynomial of .

7. Prove that the minimal polynomial of B ²= ³ is the least common
multiple of its elementary divisors.
8. Let B ²= ³ where is finite-dimensional. Describe conditions on the
=
minimal polynomial of that are equivalent to the fact that the elementary

divisor version of the rational canonical form of is diagonal. What can

you say about the elementary divisors?
(
9. Verify the statement that the multiset of elementary divisors or invariant
factors is a complete invariant for similarity of matrices.
)
10. Prove that given any multiset of monic prime power polynomials
Á Á
4 ~ ¸ ²%³Á ÃÁ ²%³ÁÃÁÃÁ Á ²%³Á ÃÁ Á ²%³¹

and given any vector space of dimension equal to the sum of the degrees
=
of these polynomials, there is an operator B ²= ³ whose multiset of
elementary divisors is 4 .
11. Find all rational canonical forms up to the order of the blocks on the
²
)
diagonal for a linear operator on s 6 having minimal polynomial

1
²% c ³ ²% b ³ .
1

)
12. How many possible rational canonical forms up to order of blocks are
(
1 ?
there for linear operators on s 6 with minimal polynomial ²% c 1³²% b ³
)
(
)
13. a Show that if and are d matrices, at least one of which is
invertible, then () and )( are similar.