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##

Invariant Subspaces

A *(right) invariant subspace*
of satisfies for all .
We also write this as
.
The simplest example is when is spanned by a single eigenvector of .
More generally an invariant subspace may be spanned by a subset of
the eigenvectors of , but since some matrices do not have eigenvectors,
there are invariant subspaces that are not spanned by eigenvectors.
For example, the space of all possible vectors is clearly invariant, but
it is not spanned by the single eigenvector of
in (2.3). This is discussed further in
§2.5.4 below.

A *left invariant subspace*
of analogously satisfies
for all ,
and may be spanned by left eigenvectors of .

Susan Blackford
2000-11-20