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Box 1.7 (Left general eigenvalue problem and right general eigenvalue problem: Ricci calcu-
lus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2
d
S
2
=d
s
2
,
λ
2
d
s
2
=d
S
2
,
Λ
2
G
MN
U
A
U
B
d
V
A
d
V
B
=
λ
2
g
μν
u
α
u
β
d
v
α
d
v
β
=
=
g
μν
u
μ
M
u
ν
N
U
A
U
B
d
V
A
d
V
B
=
G
MN
U
μ
U
ν
u
α
u
β
d
v
α
d
v
β
⇐⇒
⇐⇒
(1.54)
Λ
2
G
MN
U
B
=
c
MN
U
B
λ
2
g
μν
u
β
=
C
μν
u
β
⇐⇒
⇐⇒
Λ
2
G
MN
)
U
B
=0
,
λ
2
g
μν
)
u
β
=0
,
(
c
MN
−
(
C
μν
−
subject to
subject to
g
μν
u
μ
M
u
ν
N
U
A
U
B
=
δ
AB
.
G
MN
U
μ
U
ν
u
α
u
β
=
δ
μν
.
Box 1.8 (Left general eigenvalue problem and right general eigenvalue problem: Cayley cal-
culus).
Left eigenvalue problem:
Right eigenvalue problem:
Λ
2
d
S
2
=d
s
2
,
λ
2
d
s
2
=d
S
2
,
Λ
2
d
V
T
F
l
G
l
F
l
d
V
=
λ
2
d
V
T
F
r
G
r
F
r
d
V
=
=d
V
T
F
l
C
l
F
l
d
V
=d
V
T
F
r
C
r
F
r
d
V
(1.55)
⇐⇒
⇐⇒
(C
l
− Λ
2
G
l
)F
l
=0
,
(C
r
− λ
2
G
r
)F
r
=0
,
subject to
subject to
F
l
G
l
F
l
=I
.
F
r
G
r
F
r
=I
.
Certainly, we agree that the various transformations have to be checked by “paper and pencil”, in
particular, by means of Examples
1.2
and
1.3
.Incasethatweareledto“
non-integrable differen-
tials
” (namely differential forms), we have indicated this result by writing “d
V
”and“d
v
” according
to the M. Planck notation. In this context, the
left
and
right Frobenius matrices
,F
l
and F
r
,haveto
be seen. They are used as matrices of
integrating factors
which transform “
imperfect differentials
”
d
V
A
(namely d
V
1
,
d
V
2
, or differential forms
Ω
1
,Ω
2
)ord
v
α
(namely d
v
1
,
d
v
2
, or differential forms
ω
1
,ω
2
)to“
perfect differentials
”d
U
A
(namely d
U
1
,
d
U
2
)ord
u
α
(namely d
u
1
,
d
u
2
). As a sample
reference of the theory of differential forms and the
Frobenius Integration Theorem
, we direct the
interested reader to
De Azcarraga and Izquierdo
(
1995
),
Do Carmo
(
1994
), and
Flanders
(
1970
,
p. 97). Indeed, we hope that the reader appreciates the triple notation index notation (Ricci
calculus), matrix notation (Cayley calculus), and explicit notation (Leibniz-Newton calculus).
Thus, we are led to the general eigenvalue problem as a result of simultaneous diagonalization
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