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Not every characteristic polynomial splits. However, we can still consider factors of the characteristic polynomials to analyze the structure of these linear transformations.
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One canonical form is the rational canonical form for the linear operator
on . This involves an ordered basis for such that Where each
is the companion matrix of some polynomial where is a monic irreducible divisor of the characteristic polynomial of and A companion matrix for
Is the matrix
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Notation: We order every rational canonical basis so that all
-cyclic bases associated with the same irreducible monic divisor of the characteristic polynomial are grouped together. Within each grouping, the
-cyclic bases are in order of decreasing size. -
Let
be the number of dots in the -th row of the corresponding dot diagram and the number of dots in the -th column -
For matrices, the definition is similar. The rational canonical form of
is the rational canonical form of the linear operator . The accompanying basis is the rational canonical basis of -
The rational canonical basis is not unique.
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Let
be a linear operator on a finite dimensional vector space with characteristic polynomial Where the
’s are distinct irreducible monic polynomials, each . The generalized eigenspace corresponding to
is defined -
(Friedberg Lem.7.15) Let
be a linear operator on a finite dimensional vector space and let be a nonzero vector in and suppose that the -annihilator of is of the form for some irreducible monic polynomial . Then divides the minimal polynomial of and so does . -
(Friedberg 7.16) Let
be a linear operator on a finite dimensional vector space and an ordered basis for . Then is a rational canonical basis of if and only if is the disjoint union of -cyclic bass where each for some irreducible monic divisor of the characteristic polynomial of . -
(Friedberg 7.17) Let
be a linear operator on a finite dimensional vector space and let Be the minimal polynomial of
where the ’s are distinct irreducible monic factors of and the ’s are positive integers. Then - For each
, is a nontrivial -invariant subspace of . - For
, - For
, is invariant under and the restriction of to is one-to-one. - For each
- For each
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(Friedberg Lem.7.18) Let
be a linear operator on a finite dimensional vector space and let be the distinct irreducible monic divisors of the minimal polynomial of . Let such that Then
for all . -
(Friedberg 7.18) Suppose that
is a linear operator on a finite dimensional vector space . Let be distinct irreducible monic divisors of the minimal polynomial of and for each , let be a linearly independent subset of . Then for and Is a linearly independent subset of
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(Friedberg 7.19) Let
be distinct vectors in such that is linearly independent. For each
, let be a vector in such that . Then is also linearly independent.
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(Friedberg Lem.7.20) Let
be a -invariant subspace of and a basis for . Then if then is linearly independent. - For some
, can be extended to a linearly independent set such that
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(Friedberg 7.20) If the minimal polynomial of
is of the form then has a rational canonical basis. -
(Friedberg 7.20.1)
has a basis consisting of a union of -cyclic basis - Intuition: Apply (Friedberg 7.20) to the restriction of
to
- Intuition: Apply (Friedberg 7.20) to the restriction of
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(Friedberg 7.21) Every linear operator on a finite dimensional vector space has a rational canonical basis, and hence a rational canonical form.
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(Friedberg 7.22) Let
be a linear operator on an -dimensional vector space with characteristic polynomial where the
’s are distinct irreducible monic polynomials and the ’s are positive integers. Then - For each
, divides the minimal polynomial of - For each
, - If
is a rational canonical basis for then for each , is a basis for - If
is a basis for then is a basis for . In particular, if each is a disjoint union of -cyclic bases, then is a rational canonical basis for .
- For each
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(Friedberg 7.23) Let
be a linear operator on a finite dimensional vector space . Let be an irreducible monic divisor of the characteristic polynomial of of degree . Then -
(Friedberg Lem.7.23.1) Let
be the total number of dots in the dot diagram for . Then also for any
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(Friedberg 7.23.1) The rational canonical form of a linear operator is unique up to the arrangement of the irreducible monic divisors of the characteristic polynomial.
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(Friedberg 7.24) Let
be a linear operator on an -dimensional vector space with characteristic polynomial Where the
’s are distinct irreducible monic polynomials and each . Then (see here) - If
is the restriction of to and is the rational canonical form of then Is the rational canonical form of.
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(Friedberg 7.25) Let
be a linear operator on a finite dimensional vector space , then is a direct sum of -cyclic subspaces where each for some irreducible monic divisor of the characteristic polynomial of .