Unformatted text preview: a linear operator
under .
Proof. Suppose . so that . Since and is invariant commute .□
Let and be vector spaces. The direct sum
is the vector space with elements
, where
and
and operations of vector addition and scalar multiplication
defined by
and
, where also
and
and
. For example,
. Theorem 2.6 (Primary Decomposition). Let be a linear operator on a complex vector space
with distinct eigenvalues
and let be the generalized eigenspace of with
eigenvalue . Then
is the algebraic multiplicity of
generalized eigenspaces, i.e.
. and is the direct sum of the 5
Chapter 2 part B Proof. This is proved in Hirsch and Smale.
Remark. We can choose a basis
combined to obtain a basis for each eigenspace. By theorem 2.6, these can be
for Warning. The labeling for generalized eigenvectors given above is Meiss’ notation. Note that
the eigenvectors are relabeled to give the basis for . This keeps the notation simple but the
labels must be interpreted correctly depending on context. SemisimpleNilpotent Decomposition
Shift notation from as linear operato...
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 Fall '14
 MarcEvans
 Linear Algebra, Complex number, Floquet Theory

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