The associated norm is often referred to as the energy norm. Sometimes, it is possible to
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Projection operators or projectors play an important role in numerical linear algebra, par
ticularly in iterative methods for solving various matrix problems. This section introduces
these operators from a purely algebraic point of view and gives a few of their important
properties.
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A projector is any linear mapping from to itself which is idempotent, i.e., such that
A few simple properties follow from this de¬nition. First, if is a projector, then so is
3B , and the following relation holds,
¤ C B
C
C
) B (
3 §0
B¢ ¤ ¤
)
B B¢)
(
In addition, the two subspaces and intersect only at the element zero.
C q
C
)
(
¦ ¦ ¦
Indeed, if a vector belongs to , then , by the idempotence property. If it
qC
)
B
7
7
¦ ¦ ¦
is also in , then . Hence, which proves the result. Moreover,
3 B
j C
‘ — ‘
¦ ¦ ¦
every element of can be written as . Therefore, the space can
C
be decomposed as the direct sum
j C B¢ ¤
¤
)
‘ B )
(
C
D
‘
Conversely, every pair of subspaces and which forms a direct sum of de¬nes a
¤
)
B¢) B
(
unique projector such that and . This associated projector
C C
‘
0 0
¦ ¦ ¦
maps an element of into the component , where is the component in the
0‚
— 6
¦ ¦ ¦
unique decomposition associated with the direct sum.
In fact, this association is unique, that is, an arbitrary projector can be entirely
determined by the given of two subspaces: (1) The range of , and (2) its null space
3 ¦ ¦
which is also the range of . For any , the vector satis¬es the conditions,
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3
¦ ¦
¦
The linear mapping is said to project onto and along or parallel to the subspace .
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