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ξ
∈
n
w
ξ
=
0
. For any unit vector
T
w
∈
S
n
−
1
,theFrechet derivative of
r
(w
,
R
)
is the linear functional
Dr
(w
,
R
)
w
:
T
w
S
n
−
1
→
defined by
Dr
(w
,
R
)
w
(ξ )
=
2
R
w
,
ξ
=
2
w
T
R
ξ
To define the gradient on the sphere, the standard Euclidean inner product on
T
w
S
n
−
1
is chosen as a Riemannian metric on
S
n
−
1
[i.e., up to a constant,
T
η
∈
T
w
S
n
−
1
]. The gradient
ξ
,
η
≡
2
ξ
η
∀
ξ
∇
r
(w
,
R
)
is then determined
uniquely if it satisfies the tangency condition and the compatibility condition:
,
Dr
(w
,
R
)
|
w
(ξ )
= ∇
r
(w
,
R
)
,
ξ
r
T
T
w
S
n
−
1
=
2
∇
(w
,
R
) ξ
∀
ξ
∈
)
w
(ξ )
=
T
R
]
T
which since
Dr
(w
,
R
2
w
ξ
implies that [
∇
r
(w
,
R
)
−
R
w
ξ
=
0.
From the definition of tangent space, it follows that
∇
r
(w
,
R
)
=
R
w
+
λw
with
T
R
T
λ
=−
w
0 to satisfy the tangency condition. Thus,
the gradient flow for the Rayleigh quotient on the unit sphere
S
n
−
1
w
,sothat
w
∇
r
(w
,
R
)
=
is
d
w (
t
)
=−
[
R
−
r
(w
,
R
)
I
n
]
w (
t
)
(2.43)
dt
which is eq. (2.21) (i.e. the ODE of OJAn).
The Rayleigh quotient gradient flow restricted to the unit sphere extends to a
flow on
n
− {
0
}
. Deriving the expression of the RQ with respect to
w
gives the
Frechet derivative of
r
(w
,
R
)
:
n
− {
0
} →
,whichis
2
w
T
Dr
(w
,
R
)
|
w
(ξ )
=
2
(
R
w
−
r
(w
,
R
) w)
ξ
(2.44)
2
n
−
{
0
}
)
Define a Riemannian metric on each tangent space
T
w
(
as,
∀
ξ
,
η
∈
n
−
{
0
}
)
,
T
w
(
2
w
d
w (
t
)
(
)
⇒
2
.
42
T
2
2
ξ
,
η
≡
2
ξ
η
=−
w
∇
(
w
)
(2.45)
2
dt
ODE LUO
n
The gradient of
r
(w
,
R
)
:
− {
0
} →
with respect
to this metric is
n
characterized
by
grad
r
(w
,
R
)
∈
T
w
(
− {
0
}
)
and
Dr
(w
,
R
)
|
w
(ξ )
=
n
grad
r
(w
,
R
)
,
ξ
∀
ξ
∈
T
w
(
− {
0
}
)
. The first condition imposes no
constraint on grad
r
(w
,
R
)
, and the second one is easily seen to be equivalent to
grad
r
(w
,
R
)
=
R
w
−
r
(w
,
R
) w
n
Then eq. (2.43) also gives the gradient of
r
(w
,
R
)
:
− {
0
} →
.
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