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Unformatted text preview: 8.6. RATIONAL CANONICAL FORM 397 and only if, the KŒx–modules determined by these linear transformations
are isomorphic as KŒx–modules.
Let V1 denote V endowed with the KŒx–module structure derived
from T1 and let V2 denote V endowed with the KŒx–module structure
derived from T2 . Suppose U W V1 ! V2 is a KŒx–module isomorphism;
then U is a vector space isomorphism satisfying T2 .U v/ D x.U v/ D
U.xv/ D U.T1 v/. It follows that T2 D U T1 U 1 .
Conversely, suppose that U is an invertible linear transformation such
that T2 D U T1 U 1 . It follows that for all f .x/ 2 KŒx, f .T2 / D
Uf .T1 /U 1 ; equivalently, f .T2 /U v D Uf .T1 /v for all v 2 V But this
means that U is a KŒx–module isomorphism from V1 to V2 .
I Rational canonical form for matrices
Let A be an n–by–n matrix over a ﬁeld K . Let T be the linear transformation of K n determined by left multiplication by A, T .v/ D Av for
v 2 K n . Thus, A is the matrix of T with respect to the standard basis of
K n . A second matrix A0 is similar to A if, and only if, A0 is the matrix
of T with respect to some other ordered basis. Exactly one such matrix is
in rational canonical form, according to Theorem 8.6.4. So we have the
Proposition 8.6.6. Any n–by–n matrix is similar to a unique matrix in
rational canonical form. Deﬁnition 8.6.7. The unique matrix in rational canonical form that is similar to a given matrix A is called the rational canonical form of A.
The blocks of the rational canonical form of A are companion matrices
of monic polynomials a1 .x/; : : : ; as .x/ such that ai .x/ divides aj .x/ if
i Ä j . These are called the invariant factors of A.
The rational canonical form is a complete invariant for similarity of
Proposition 8.6.8. Two n–by–n matrices are similar in Matn .K/ if, and
only if, they have the same rational canonical form. Proof. There is exactly one matrix in rational canonical form in each similarity equivalence class, and that matrix is the rational canonical form of ...
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