Page 187 - Advanced Linear Algebra

The Structure of a Linear Operator 171

polynomials does not uniquely determine the multiset of elementary divisors of
the operator .

In general, the minimal polynomial of a linear operator is hard to find. One of
the virtues of the characteristic polynomial is that it is comparatively easy to
find and we will discuss this in detail a bit later in the chapter.

Note that since ²%³ “ ²%³ and both polynomials are monic, it follows that

deg
²%³ ~ ²%³ ¯ ² ²%³³ ~ deg ² ²%³³

Definition A linear operator B ²= ³ is nonderogatory if its minimal
polynomial is equal to its characteristic polynomial:
²%³ ~ ²%³

or equivalently, if
deg² ²%³³ ~ deg² ²%³³

or if
deg² ²%³³ ~ dim²= ³

Similar statements hold for matrices.
Cyclic and Indecomposable Modules
We have seen (Theorem 6.17) that cyclic submodules can be characterized by
their elementary divisors. Let us translate this theorem into the language of =
(and add one more equivalence related to the characteristic polynomial).

B
Theorem 7.9 Let ²= ³ have minimal polynomial

²%³ ~ ²%³Ä ²%³

where ²%³ are distinct monic primes. The following are equivalent:

) is cyclic.
1 =
) is the direct sum
2 =
= ~ ºº# »» l Ä l ºº# »»

of -cyclic submodules ºº# »» of order ²%³ .

3 The elementary divisors of are
)

ElemDiv² ³ ~ ¸ ²%³ÁÃÁ ²%³¹

)
4 is nonderogatory, that is,
²%³ ~ ²%³