Page 186 - Advanced Linear Algebra

170 Advanced Linear Algebra

The Characteristic Polynomial
To continue our translation project, we need a definition. Recall that in the
characterization of cyclic modules in Theorem 6.17, we made reference to the
product of the elementary divisors, one from each associate class. Now that we
have singled out a special representative from each associate class, we can make
a useful definition.

Definition Let B ²= ³ . The characteristic polynomial ²%³ of is the

product of all of the elementary divisors of :

²%³ ~ ²%³
Á

Á
Hence,
deg² ²%³³ ~ dim²= ³

Similarly, the characteristic polynomial ²%³ of a matrix 4 is the product of
4
the elementary divisors of 4 .
The following theorem describes the relationship between the minimal and
characteristic polynomials.

B
Theorem 7.8 Let ²= ³ .
1 )(The Cayley–Hamilton theorem ) The minimal polynomial of divides the

characteristic polynomial of :

²%³ “ ²%³

Equivalently, satisfies its own characteristic polynomial, that is,

² ³ ~
2 The minimal polynomial
)
Á Á
²%³ ~ ²%³Ä ²%³

and characteristic polynomial
Á
²%³ ~ ²%³

Á

of have the same set of prime factors ²%³ and hence the same set of

)
(
roots not counting multiplicity .
We have seen that the multiset of elementary divisors forms a complete
invariant for similarity. The reader should construct an example to show that the
pair ² ²%³Á ²%³³ is not a complete invariant for similarity, that is, this pair of