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Unformatted text preview: r and refer to matrix instead. Let
or
and
. Let
be the diagonal matrix with the eigenvalues of repeated
according to multiplicity. Let
be a basis for of generalized eigenvectors of .
Consider the transformation matrix
and define
.
is a semisimple matrix. Multiply by on the right to obtain
this result is
, where
are the distinct eigenvalues of
as the diagonalizable part of .
Consider an arbitrary
vectors for : Then . The i^th component of
and
. Think of can be expressed as a linear combination of the basis . We then have
. Within , acts as a multiple of the identity operator. In particular,
action of . Lemma 2.7 Let
with order at most
Remark 1. is invariant under the , where
. Then
commutes with
, the maximum of the algebraic multiplicities of . is nilpotent of order means the same thing as Remark 2. Since the generalized eigenspace
, it is also invariant under the action of . of and is nilpotent has nilpotency . is invariant under the action of both Proof. See the text; plan to give it in class. Not...
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 Fall '14
 MarcEvans

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