Metric Clifford Algebra"
V. V. Fernández1 A. M. Moya1! and W. A. Rodrigues Jr.1, 2ż
,
1
Institute of Mathematics, Statistics and Scientific Computation
IMECC-UNICAMP CP 6065
13083-970 Campinas-SP, Brazil
2
Department of Mathematical Sciences, University of Liverpool
Liverpool, L69 3BX, UK
10/30/2001
Abstract
In this paper we introduce the concept of metric Clifford algebra
C (V, g) for a n-dimensional real vector space V endowed with a metric
extensor g whose signature is (p, q), with p+q = n. The metric Clifford
product on C (V, g) appears as a well-defined deformation (induced
by g) of an euclidean Clifford product on C (V ). Associated with the
metric extensor g, there is a gauge metric extensor h which codifies
all the geometric information just contained in g. The precise form of
such h is here determined. Moreover, we present and give a proof of
the so-called golden formula, which is important in many applications
that naturally appear in ours studies of multivector functions, and
differential geometry and theoretical physics.
Contents
1 Introduction 2
"
published: Advances in Applied Clifford Algebras 11(S3), 49-68 (2001).

e-mail: [e-mail]
!
e-mail: [e-mail]
ż
e-mail: [e-mail] or [e-mail]
1
arXiv:math-ph/0212049 v1 16 Dec 2002
2 Metric Clifford Algebra of Multivectors 3
2.1 Metric Scalar Product . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Metric Reciprocal Bases . . . . . . . . . . . . . . . . . . . . . 4
2.3 Metric Interior Algebras . . . . . . . . . . . . . . . . . . . . . 6
2.4 Metric Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . 7
3 Eigenvalues and Eigenvectors 9
4 Gauge Metric Extensor 10
4.1 Gauge Metric Bases . . . . . . . . . . . . . . . . . . . . . . . . 14
5 The Golden Formula 16
6 Metric Adjoint Operators 19
7 Standard Hodge Extensor 20
8 Metric Hodge Extensor 21
9 Conclusions 24
1 Introduction
This is paper IV of a series of seven. Here, we introduce the concept of a
metric Clifford algebra for a n-dimensional real vector space V endowed with
a metric extensor g of an arbitrary signature (p, q), with p + q = n. The
novelty, regarding previous presentations of the subject as, e.g., in ([1],[3]),
is that a metric Clifford product appears as a well-defined deformation of
an euclidean Clifford product. More important, we show that associated to
any metric extensor g there is a gauge metric extensor h (defined  modulus
a gauge) such that g = h ć% · ć% h, where · is a standard orthogonal metric
extensor over V with the same signature as g. This theorem for the decom-
position of g (which is somewhat analogous to Silvester s theorem will play
a fundamental role in the intrinsic formulation of the differential geometry
on smooth manifolds. The paper ends with the proof of the so-called golden
formula, which will show worth to deserve that name. We introduce also the
concepts of standard and metric Hodge (star) operators, and find a formula
connecting them.
2
2 Metric Clifford Algebra of Multivectors
2.1 Metric Scalar Product
Let us consider V endowed with the eulidean scalar product associated to
any fixed basis for V denoted by {bk}, as in previous papers of this series
([4]-[5]), i.e., the b-scalar product on V .
Now, take a (1, 1)-extensor over V, say g, adjoint symmetric (g = g ) and
non-degenerate (det[g] = 0). It will be called a metric extensor over V.

We can define another scalar product of multivectors X, Y " V by
X · Y = g(X) · Y, (1)
g
where g is the extended of g. It will be called a metric scalar product generated
by g. Or, g-scalar product, for short.
As we can see, this scalar product is a well-defined scalar product on V .
It is symmetric, satisfies the distributive laws, has the mixed associativity
property and is non-degenerate, i.e., X · Y = 0 for all X, implies Y = 0.
g
All the properties just mentioned above are immediate consequences of
the corresponding ones for the b-scalar product. But, a g-scalar product is
not necessarily positive definite.
We present now some of the most important properties of the g-scalar
product of multivectors.
g1 For any Ä…, ² " R
Ä… · ² = Ä…² (real product). (2)
g
j k
g2 For any Xj " V and Yk " V
Xj · Yk = 0, if j = k. (3)

g
k k
g3 For any simple k-vectors v1 '" . . . vk " V and w1 '" . . . wk " V
îÅ‚ Å‚Å‚
v1 · w1 . . . v1 · wk
g g
ïÅ‚ śł
(v1 '" . . . vk) · (w1 '" . . . wk) = det . . . . . . . . . , (4)
ðÅ‚ ûÅ‚
g
vk · w1 . . . vk · wk
g g
3
where (as before) by det vp · wq we denote the classical k × k determinant.
g
Eqs.(2-4) follow without difficulties from the corresponding properties of
the b-scalar product, by taking into account that g(Ä…) = Ä… with Ä… " R,
the grade-preserving property of the outermorphisms (i.e., of the extension
operator [5]), and that g(v1 '" . . . vk) = g(v1) '" . . . g(vk) with v1, . . . , vk " V.
g4 For any X, Y " V
X · Y = X · Y , (5)
g g
X · Y = X · Y . (6)
g g
The proof is immediate and left to the reader. Hint: take into account
that g(X) = g(X) and g(X) = g(X) with X " V.
2.2 Metric Reciprocal Bases
l
Let ({ek}, {ek}) be an arbitrary pair of b-reciprocal bases of V, i,e., ek ·el = ´k.
Theorem 1. Take an invertible (1, 1)-extensor over V, say f. We can con-
struct two bases for V, say {Ek} and {Ek}, by the following formulas
Ek = f(ek), (7)
Ek = g-1 ć% f"(ek) for each k = 1, . . . , n. (8)
These bases satisfy the metric scalar product conditions
l
Ek · El = ´k. (9)
g
Reciprocally, given two arbitrary bases {Ek} and {Ek} which satisfy eq.(9),
there exists an unique invertible (1, 1)-extensor f such that the eqs.(7) and
(8) hold.
Proof. Since {ek} and {ek} are bases for V and, f and g are invertible (1, 1)-
extensors over V , it follows that the n vectors E1, . . . , En " V and the n
vectors E1, . . . , En " V must also determine two well-defined bases for V .
Now, a straightforward calculation gives
l
Ek · El = g ć% f(ek) · g-1 ć% f"(el) = ek · f ć% g ć% g-1 ć% f"(el) = ek · el = ´k,
g
4
and the first statement follows.
Now, {ek} and {ek} are bases for V, and {Ek} and {Ek} are supposed
to be also bases for V. Then, there must exist exactly two invertible (1, 1)-
extensors over V, say f1 and f2, such that
Ek = f1(ek),
Ek = f2(ek) for each k = 1, . . . , n.
It is easy to check that f1 and f2 are given by
f1(v) = (es · v)Es,
f2(v) = (es · v)Es.
But, using eq.(9) we have

l l
f1(ek) · f2(el) = ´k Ò! ek · f1 ć% g ć% f2(el) = ´k Ò! f1 ć% g ć% f2(el) = el,
g

for each l = 1, . . . , n. Thus, f1 ć% g ć% f2 = iV .
Then, choosing f1 = f and f2 = g-1ć%f", the second statement follows.
l
Two bases {Ek} and {Ek} satisfying Ek · El = ´k are said to be a pair of
g
metric reciprocal bases, and we say that {Ek} is the metric reciprocal basis
of {Ek}.
We end this section presenting two interesting and useful formulas for the
expansion of multivectors in terms of a g-scalar product.
Proposition 1. Let ({Ek}, {Ek}) be any pair of metric reciprocal bases for
l
V, i.e., Ek · El = ´k. We have the following two expansion formulas. For all
g
X " V
n
1
1 k
X = X · 1 + X · (Ej '" . . . Ej )(Ej '" . . . Ej ) (10)
1 k
g g
k!
k=1
and
n
1
1 k
X = X · 1 + X · (Ej '" . . . Ej )(Ej '" . . . Ej ). (11)
1 k
g g
k!
k=1
The proof is left to the reader. (Hint: use eq.(1), eq.(7), eq.(8) and
some of the properties of extension operator, and take also into account
the expansion formula for multivectors in the euclidean Clifford algebra as
defined in [4] (paper I of this series) .
5
2.3 Metric Interior Algebras
We define now the metric left and right contracted products of multivectors
X, Y " V, denoted respectively by and ,
g g
X Y = g(X) Y, (12)
g
X Y = X g(Y ). (13)
g
When no confusion arises we call and the g-contracted products, for short.
g g
These g-contracted products and are internal laws on V. Both of
g g
them satisfy the distributive laws (on the left and on the right) but they are
not associative products.
The vector space V endowed with the g-contracted product either
g
or is a non-associative algebra. They are called metric interior algebras of
g
multivectors. Or, g-interior algebras, for short.
We present now some of the basic properties of the metric interior alge-
bras.
mi1 For any Ä…, ² " R and X " V
Ä… ² = Ä… ² = Ä…² (real product), (14)
g g
Ä… X = X Ä… = Ä…X (multiplication by scalars). (15)
g g
j k
mi2 For any Xj " V and Yk " V with j d" k
Xj Yk = (-1)j(k-j)Yk Xj. (16)
g g
j k
mi3 For any Xj " V and Yk " V
Xj Yk = 0, if j > k, (17)
g
Xj Yk = 0, if j < k. (18)
g
k
mi4 For any Xk, Yk " V
Xk Yk = Xk Yk = Xk · Yk = Xk · Yk. (19)
g g g g
6
mi5 For any v " V and X, Y " V
v (X '" Y ) = (v X) '" Y + X '" (v Y ). (20)
g g g
All these properties easily follow from the corresponding properties of
the euclidean interior algebras, once we take into account the properties of
extension operator [4].
Proposition 2. For all X, Y, Z " V it holds
(X Y ) · Z = Y · (X '" Z), (21)
g g g
(X Y ) · Z = X · (Z '" Y ). (22)
g g g
These properties are completely equivalent to the definitions of the right
and left contracted products given in eqs.(12-13), and can be proved without
difficulties by using the properties of extension operator [4].
Proposition 3. For all X, Y, Z " ›V it holds
X (Y Z) = (X '" Y ) Z, (23)
g g g
(X Y ) Z = X (Y '" Z). (24)
g g g
Proof. We prove only the first statement. Take X, Y, Z " V . Using the
multivector identity A (B C) = (A '" B) C and a property of extension
operator, we have
X (Y Z) = g(X) (g(Y ) Z) = (g(X) '" g(Y )) Z
g g
= g(X '" Y ) Z = (X '" Y ) Z.
g
2.4 Metric Clifford Algebra
We define a metric Clifford product of X, Y " V associated to g by the
following axioms:
A1 For all Ä… " R and X " V
Ä… X = XÄ… equals multiplication of multivector X by scalar Ä….
g
7
A2 For all v " V and X " V
v X = v X + v '" X and X v = X v + X '" v.
g g g g
A3 For all X, Y, Z " V
X (Y Z) = (X Y ) Z.
g g g g
This metric Clifford product is an internal law on V. It is associative
(by the axiom A3) and satisfies the distributive laws (on the left and on the
right) which follow from the corresponding distributive laws of the euclidean
contracted and exterior products [4].
V endowed with this metric Clifford product is an associative algebra.
It will be called a metric Clifford algebra of multivectors generated by g, or
simply, g-Clifford algebra. It will be denoted by C (V, g).
We present now some of the most basic properties which hold in C (V, g).
clg1 For any v " V and X " V
1 1
v X = (v X - X v) and X v = (X v - v X), (25)
g g g g g g
2 2
1 1
v '" X = (v X + X v) and X '" v = (X v + v X). (26)
g g g g
2 2
clg2 For any X, Y " V
X · Y = X Y = X Y . (27)
g g g
0 0
clg3 For X, Y, Z " V
(X Y ) · Z = Y · (X Z) = X · (Z Y ), (28)
g g g g g g
X · (Y Z) = (Y X) · Z = (X Z) · Y. (29)
g g g g g g
clg4 For any X, Y " V
X Y = X Y , (30)
g g
X Y = Y X. (31)
g g
8
n
clg5 Let I " V, then for any v " V and X " V
I (v '" X) = (-1)n-1v (I X). (32)
g g g
Eq.(32) will be called the metric duality identity, or g-duality identity, for
short.
3 Eigenvalues and Eigenvectors
Let t be a (1, 1)-extensor over V. A scalar  " R and a non-zero vector v " V
are said to be an eigenvalue and an eigenvector of t, respectively, if and only
if
t(v) = v. (33)
We say that  and v are naturally to be associated to each other. This
means that, if  " R is an eigenvalue of t, then there is some non-zero v " V
(the associated eigenvector of t) such that eq.(33) holds, and if a non-zero
v " V is an eigenvector of t, then there is some  " R (the associated
eigenvalue of t) such that eq.(33) is satisfied.
A scalar  " R is an eigenvalue of t if and only if it satisfies the following
algebraic equation of degree n
det[iV - t] = 0, (34)
where iV " ext1(V ) is the known identity (1, 1)-extensor over V.
1
Theorem 2. For any adjoint symmetric (1, 1)-extensor s, i.e., s = s , there
exists a set of n eigenvectors of s which is a b-orthonormal basis for V.
This means that there are exactly n linearly independent non-zero vectors
v1, . . . , vn " V and n scalars 1, . . . , n " R such that
s(vk) = kvk, for each k = 1, . . . , n
and {vk} is a basis for V which satisfies vj · vk = ´jk.
Corollary 3. All eigenvalues of a metric extensor g (i.e., g " ext1(V ),
1
g = g and det[g] = 0) are non-zero real numbers.

9
Proof. We first calculate det[g] by using {vk}, taking into account the eigen-
value equation of g and recalling that, in this case, the euclidean reciprocal
vectors vk are equal to the vectors vk,
det[g] = (g(v1) '" . . . g(vn)) · (v1 '" . . . vn)
= (1v1 '" . . . nvn) · (v1 '" . . . v1)
= 1 . . . n(v1 '" . . . v1) · (v1 '" . . . v1)
det[g] = 1 . . . n. (35)
Since det[g] = 0, all 1, . . . , n must be non-zero real numbers.

The integer number s = p-q, where p, q are non negative integer numbers,
respectively the numbers of positive and negative eigenvalues of t and p+q =
n, is called the signature of t. We already have used (and will continue to do
so) the usual convention of physicists and denote the signature of g by the
pair (p, q).
4 Gauge Metric Extensor
Lemma 4. Any b-orthogonal symmetric (1, 1)-extensor over V, say Ã, (i.e.,
à = Ã" and à = à ) can only have eigenvalues Ä…1.
Proof. If  " R is an eigenvalue of Ã, there is an non-zero v " V, the as-
sociated eigenvector of Ã, such that Ã(v) = v. And, the orthogonality and
symmetry of à yield Ã2 = iV .
Thus, we have that v = 2v. Since v = 0, it follows that 1 - 2 = 0, i.e.,

 = Ä…1.
Lemma 5. Let {bk} be the fiducial basis for V (i.e., bj · bk = ´jk). We can
b
construct a fiducial b-orthogonal metric extensor over V, say · " ext1(V ),
1
(i.e., · = ·" and · = · , det[·] = 0) with signature (p, q) and which b-

orthonormal basis of V, made of the eigenvectors of ·, is exactly {bk}.
Such a (1, 1)-extensor over V is given by
p p+q
·(v) = (v · bj)bj - (v · bj)bj. (36)
j=1 j=p+1
10
Proof. We first shall prove that · has p eigenvalues +1 with associated
eigenvectors b1, . . . , bp and q eigenvalues -1 with associated eigenvectors
bp+1, . . . , bp+q.
Take bk with k = 1, . . . , p we have
p p+q p p+q
·(bk) = (bk · bj)bj - (bk · bj)bj = ´kjbj - 0bj = bk,
j=1 j=p+1 j=1 j=p+1
and, for bk with k = p + 1, . . . , p + q, it yields
p p+q p p+q
·(bk) = (bk · bj)bj - (bk · bj)bj = 0bj - ´kjbj = -bk.
j=1 j=p+1 j=1 j=p+1
Then, we have
bk, k = 1, . . . , p
·(bk) = . (37)
-bk, k = p + 1, . . . , p + q
Now, we shall prove that · as defined above is a metric extensor over V,
i.e., · = · and det[·] = 0.

Take v, w " V then
p p+q
· (v) · w = v · ·(w) = v · ( (w · bj)bj - (w · bj)bj)
j=1 j=p+1
p p+q
= (v · bj)(w · bj) - (v · bj)(w · bj)
j=1 j=p+1
p p+q
= ( (v · bj)bj - (v · bj)bj) · w = ·(v) · w,
j=1 j=p+1
i.e., · = ·.
We calculate the determinant of · by using the fundamental formula with
{bk}, i.e., det[·] = ·(b1) '" . . . ·(bn) · (b1 '" . . . bn). Recall that, in this case, the
b-reciprocal basis vectors bk coincide with bk for k = 1, . . . , n.
det[·]
= (·(b1) '" . . . ·(bp) '" ·(bp+1) '" . . . ·(bp+q)) · (b1 '" . . . bp '" bp+1 '" . . . bp+q)
= (b1 '" . . . bp '" (-1)qbp+1 '" . . . bp+q) · (b1 '" . . . bp '" bp+1 '" . . . bp+q),
11
i.e., det[·] = (-1)q.
Next, we shall prove that ·2 = iV , i.e., ·-1 = ·.
Take v " V then
n
· ć% ·(v) = (v · bk)· ć% ·(bk)
k=1
n
·(bk) k = 1, . . . , p
= (v · bk)
-·(bk) k = p + 1, . . . , p + q
k=1
n
= (v · bk)bk = v,
k=1
i.e., ·2 = iV .
This lemma allows us to construct, associated to the fiducial basis {bk}, a
fiducial b-orthogonal metric extensor · " ext1(V ) with signature (p, q). Such
1
a (1, 1)-extensor over V has p eigenvalues +1 and q eigenvalues -1, and their
corresponding associated eigenvectors are the vectors of {bk}.
Theorem 6. For any metric extensor g " ext1(V ) whose signature is (p, q),
1
there exists an invertible extensor h " ext1(V ) such that
1
g = h ć% · ć% h, (38)
where · " ext1(V ) is just the fiducial b-orthogonal metric extensor with sig-
1
nature (p, q), as considered in eq.(36).
Such a (1, 1)-extensor over V is given by
n
h(a) = |j|(a · vj)bj, (39)
j=1
where 1, . . . n " R are the eigenvalues of g and v1, . . . , vn " V are the
corresponding associated eigenvectors of g.
Proof. First we need calculate the adjoint extensor of h.
Take a, b " V then
n
h (a) · b = a · h(b) = a · ( |j|(b · vj)bj)
j=1
n
= ( |j|(a · bj)vj) · b,
j=1
12
n
i.e., h (a) = |j|(a · bj)vj.
j=1
Now, let a " V. A straightforward calculation yields
n n
h ć% · ć% h(a) = |jk|·(bj) · bk(a · vj)vk
j=1 k=1
p p
= |jk|·(bj) · bk(a · vj)vk
j=1 k=1
p p+q
+ |jk|·(bj) · bk(a · vj)vk
j=1 k=p+1
p+q p
+ |jk|·(bj) · bk(a · vj)vk
j=p+1
k=1
p+q p+q
+ |jk|·(bj) · bk(a · vj)vk,
j=p+1
k=p+1
and, by taking into account eq.(37) we have
p p
h ć% · ć% h(a) = |jk|´jk(a · vj)vk + 0
j=1 k=1
p+q p+q
+ 0 - |jk|´jk(a · vj)vk
j=p+1
k=p+1
p p+q
= |j| (a · vj)vj - |j| (a · vj)vj
j=1 j=p+1
p p+q n
= j(a · vj)vj + j(a · vj)vj = j(a · vj)vj.
j=1 j=p+1 j=1
On the last step we have used that the signature of g is (p, q), i.e., g has p
positive eigenvalues and q negative eigenvalues.
And, by using the eigenvalues equation of g, i.e., g(vj) = jvj for each
j = 1, . . . , n, we have
n n
h ć% · ć% h(a) = (a · vj)g(vj) = g( (a · vj)vj) = g(a),
j=1 j=1
13
i.e., h ć% · ć% h = g.
Finally, since
det[g] = det[h ć%·ć%h] = det[h ] det[·] det[h] = det[·] det2 [h] = (-1)q det2 [h],
and det[g] = 0, then det[h] = 0, and so h is an invertible (1, 1)-extensor.

It should be noted that h satisfying eq.(38) is not unique. If there is some
h " ext1(V ) which satisfies eq.(38), then h a" ›ć%h, where › is a ·-orthogonal
1
(1, 1)-extensor over V (i.e., › ć% · ć% › = ·)1, also satisfies eq.(38).
Indeed, we have h ć% · ć% h = (› ć% h) ć% · ć% › ć% h = h ć% › ć% · ć% › ć% h =
h ć% · ć% h = g.
In general, an invertible extensor h " ext1(V ) which satisfies eq.(38) will
1
be said to be a gauge metric extensor for the metric extensor g.
4.1 Gauge Metric Bases
Let {ek} and {ek} be two b-reciprocal bases to each other for the vector space
l
V, i.e., ek ·el = ´k. Since h, a gauge metric extensor over V, is non-degenerate,
i.e., det[h] = 0, it follows that the n vectors h(e1), . . . , h(en) " V and the n

vectors h"(e1), . . . , h"(en) " V will be also well-defined bases for V.
As the reader can easily prove ({h(ek)}, {h"(ek)}) is also a pair of b-
reciprocal bases of V, i.e.,
l
h(ek) · h"(el) = ´k. (40)
Two other remarkable properties of these bases are:
h(ej) · h(ek) = g(ej) · ek a" gjk, (41)
·
h"(ej) · h"(ek) = g-1(ej) · ek a" gjk. (42)
·-1
These bases {h(ek)} and {h"(ek)} will be said to be a pair of gauge metric
bases for V.
1
As the reader can prove without difficulties, a ·-orthogonal (1, 1)-extensor › preserves
the ·-scalar products, i.e., for all v, w " V : ›(v) · ›(w) = v · w.
· ·
14
Theorem 7. Given n non-zero real numbers Á1, . . . , Án and a b-orthogonal
(1, 1)-extensor l over V (i.e., l = l"), we can construct an invertible (1, 1)-
extensor h over V using the following formula
n
h(v) = Áj(l(v) · bj)bj. (43)
j=1
Then, the (1, 1)-extensor g over V defined by
g = h ć% · ć% h, (44)
where · " ext1(V ) is just the fiducial b-orthogonal metric extensor over V
1
with signature (p, q), as considered in eq.(36).
The p positive real numbers Á2, . . . , Á2 are the eigenvalues of g with the
1 p
associated eigenvectors l (b1), . . . , l (bp) of g, and the q negative real numbers
-Á2 , . . . , -Á2 are the eigenvalues of g with the associated eigenvectors
p+1 p+q
l (bp+1), . . . , l (bp+q) of g.
The set of n non-zero vectors {l (b1), . . . , l (bp), l (bp+1), . . . , l (bp+q)} is
a b-orthonormal basis of V, made of the eigenvectors of g.
The signature of g is also (p, q).
Proof. We first must check that g is symmetric and non-degenerate. Using
eq.(44) we have
g = h ć% · ć% (h ) = h ć% · ć% h = g,
thus, g = g , i.e., g is symmetric.
We now calculate det[g],
det[g] = det[h ] det[·] det[h] = det[h] det[·] det[h] = (-1)q det2 [h].
But, it is possible to calculate det[h] by using a trick. We shall evaluate
det[h ć% l ] in two different ways.
First, using eq.(43) and the b-orthogonality of l, i.e., l-1 = l , we have
n
h ć% l (v) = Áj(a · bj)bj.
j=1
Sencondly, using the fundamental formula for the determinant of a (1, 1)-
extensor (see [5]) we have,
det[h ć% l ] = (h ć% l (b1) '" . . . h ć% l (bn)) · (b1 '" . . . bn)
= (Á1b1 '" . . . Ánbn) · (b1 '" . . . bn)
= Á1 . . . Án.
15
Now, taking into account that det[l] = Ä…1 and property (d1) of the
determinant (see [5]) we get,
det[h ć% l ] = det[h] det[l ] = det[h] det[l] = ą det[h].
Thus, we have det[h] = Ä…Á1 . . . Án. And, therefore det[g] = (-1)qÁ2 . . . Á2 .
1 n
Since Á1, . . . , Án are non-zero real numbers, det[g] = 0, i.e., g is non-degenerate.

The proof of the first statement is then complete.
In order to prove the second statement, related to the eigenvalues and
eigenvectors of g, we shall use the following equations: h ć% l (bk) = Ákbk (just
used above), h (bk) = Ákl (bk) (obtained from eq.(43)) and the eigenvalue
bk, k = 1, . . . , p
equation of ·, i.e., ·(bk) = .
-bk, k = p + 1, . . . , p + q
We have
g ć% l (bk) = h ć% · ć% h ć% l (bk) = h ć% ·(Ákbk) = Ákh ć% ·(bk)
Ákh (bk), k = 1, . . . , p
=
-Ákh (bk), k = p + 1, . . . , p + q
Á2l (bk), k = 1, . . . , p
k
g ć% l (ek) = .
-Á2l (bk), k = p + 1, . . . , p + q
k
This establishes the second statement.
żFrom the euclidean b-orthogonality of l it follows easily that the eigen-
vectors of g, are b-orthonormal. It is also obvious that the signature of g is
also (p, q). Thus, the third and fourth statement are proved.
5 The Golden Formula
Proposition 4. Let h be any gauge operator for g, i.e., g = h ć% · ć% h, and
let " mean either '" (exterior product), · (g-scalar product), (g-contracted
g g g g
products) or (g-Clifford product), and analogously for ". The g-metric
g ·
products " and the ·-metric products " are related by the following remarkable
g ·
formula. For all X, Y " V
h(X " Y ) = h(X) " h(Y ), (45)
g ·
where h denotes the extended of h. Eq.(45) will be called the golden formula
16
Proof. By recalling the fundamental properties for the outermorphism of an
operator: t(X '"Y ) = t(X)'"t(Y ) and t(Ä…) = Ä…, we have that the multivector
identity above holds for the exterior product, i.e.,
X '" Y = h-1[h(X) '" h(Y )] (46)
and for the g-scalar product and the ·-scalar product, i.e.,
X · Y = h-1[h(X) · h(Y )]. (47)
g ·
By using the multivector identities for an invertible operator: t (X) Y =
t-1[X t(Y )] and X t (Y ) = t-1[t(X) Y ], and the gauge equation g = h ć%·ć%h
we can easily prove that the multivector identity above holds for the g-
contracted product and the ·-contracted product, i.e.,
X Y = h-1[h(X) h(Y )] (48)
g ·
X Y = h-1[h(X) h(Y )]. (49)
g ·
To prove eq.(48) see that we can write
X Y = h ć% · ć% h(X) Y = h-1[· ć% h(X) h(Y )] = h-1[h(X) h(Y )],
g ·
where the definitions of and have been used. The proof of eq.(49) is
g ·
completely analogous, the definitions of and should be used.
g ·
In order to prove that the multivector identity above holds for the g-
Clifford product and the ·-Clifford product, i.e.,
X Y = h-1[h(X) h(Y )], (50)
g ·
we first must prove four particular cases of it.
Take Ä… " R and X " V. By using the axioms of the g and · Clifford
products: Ä… X = X Ä… = Ä…X and Ä… X = X Ä… = Ä…X, we can write
g g · ·
Ä… X = Ä…X = h-1[Ä…h(X)] = h-1[Ä… h(X)],
g ·
i.e.,
Ä… X = h-1[h(Ä…) h(X)]. (51)
g ·
17
Analogously, we have
X Ä… = h-1[h(X) h(Ä…)]. (52)
g ·
Take v " V and X " V. By using the axioms of the g and · Clifford
products: v X = v X + v '" X and v X = v X + v '" X, and eqs.(48) and
g g · ·
(46) we can write
v X = v X + v '" X = h-1[h(v) h(X)] + h-1[h(v) '" h(X)],
g g ·
i.e.,
v X = h-1[h(v) h(X)]. (53)
g ·
żFrom the axioms of the g and · Clifford products: X v = X v + X '" v and
g g
X v = X v + X '" v, and eqs.(49) and (46) we get
· ·
X v = h-1[h(X) h(v)]. (54)
g ·
Take v1, v2, . . . , vk " V. By using k - 1 times eq.(53) we have indeed
v1 v2· · ·vk = h-1[h(v1) h(v2· · ·vk)]
g g · g
= h-1[h(v1) h(v2)· · ·h(vk)],
· ·
v1 v2· · ·vk = h-1[h(v1) h(v2)· · ·h(vk)]. (55)
g g · ·
Take v1, v2, . . . , vk " V and X " V. By using k - 1 times eq.(53) and
eq.(55) we have indeed
(v1 v2· · ·vk) X = h-1[h(v1) h( v2· · ·vk X)]
g g g · g g g
= h-1[h(v1) h(v2)· · ·h(vk) h(X)],
· · ·
(v1 v2· · ·vk) X = h-1[h(v1 v2· · ·vk) h(X)]. (56)
g g g g g ·
We now can prove the general case of eq.(50). We shall use an expansion
n
1
1
formula for multivectors: X = X0 + Xj ...jkej · · ·ej , where {ej} is a
1 k
g
k!
k=1
18
basis of V, eq.(51) and eq.(56). We can write
n
1
1
X Y = X0 Y + Xj ...jk(ej · · ·ej ) Y
1 k
g g g g
k!
k=1
n
1
1
= h-1[h(X0) h(Y )] + h-1[ Xj ...jkh(ej · · ·ej ) h(Y )]
1 k
· g ·
k!
k=1
n
1
1
= h-1[h(X0 + Xj ...jkej · · ·ej ) h(Y )],
1 k
g ·
k!
k=1
X Y = h-1[h(X) h(Y )].
g ·
Hence, eq.(46), eq.(47), eqs.(48) and (49), and eq.(50) have set the golden
formula.
6 Metric Adjoint Operators
Let g be a metric operator on V, i.e., g " ext1(V ) such that g = g and
1
det[g] = 0. To each t " 1-ext( V ; V ). We define the metric adjoint

1 2
operator t (g) " 1-ext( V ; V ) by
2 1
t (g) = g-1 ć% t ć% g. (57)
As we can easily see, t (g) is the unique extensor from V to V which
2 1
satisfies the following property: for any X " V and Y " V
1 2
X · t (g)(Y ) = t(X) · Y. (58)
g g
This is the  metric version of the fundamental property given by the formula
t (X) · Y = X · t(Y ) in the paper II of this series.
Finally, we notice the very important formula that
det[t (g)] = det[t ] = det[t]. (59)
19
7 Standard Hodge Extensor
Let ({ej}, {ej}) be a pair of b-reciprocal bases to each other for V, i.e.,
k
ej · ek = ´j . Associated to them we define a non-zero pseudoscalar
b
"
Ä = e'" · e'"e'" (60)
n n
where e'" a" e1 '" . . . '" en " V and e'" a" e1 '" . . . '" en " V. Note that
e'" · e'" > 0, since the scalar product is positive definite. It will be called a
standard volume pseudoscalar for V. It has the fundamental property
Ä · Ä = Ä Ä = ÄÄ = 1, (61)
it follows from the equation e'" · e'" = 1.
żFrom eq.(61), we can get an expansion formula for pseudoscalars
I = (I · Ä)Ä. (62)
The extensor " ext(V ) which is defined by : V V such that
X = X Ä = XÄ, (63)
will be called a standard Hodge extensor on V.
p n-p
It should be noticed that if X " V, then X " V. It means that
can be also defined as a (p, n - p)-extensor over V.
-1
The extensor over V, : V V such that
-1
X = Ä X = ÄX (64)
is the inverse extensor of .
-1 -1
Indeed, take X " V. Eq.(61) gives ć% X = ÄÄX = X, and ć% X =
-1 -1
XÄÄ = X, i.e., ć% = ć% = i V , where i V " ext(V ) is the so-called
identity function for V.
Let us take X, Y " V. Using the multivector identity (XA) · Y =
X · (Y A) and eq.(61) we get
( X) · ( Y ) = X · Y. (65)
It means that the standard Hodge extensor preserves the euclidean scalar
product.
20
p
Let us take X, Y " V. By using eq.(62) together with the multivector
identity (X '" Y ) · Z = Y · (X Z), and eq.(65) we get
X '" ( Y ) = (X · Y )Ä. (66)
This identity is completely equivalent to the definition of standard Hodge
extensor given by eq.(63).
p n-p
Take X " V and Y " V. By using the multivector identity
(X Y ) · Z = Y · (X '" Z) and eq.(62) we get
(( X) · Y )Ä = X '" Y. (67)
8 Metric Hodge Extensor
Let g be a metric extensor over V of signature (p, q), i.e., g " ext1(V ) such
1
that g = g and det[g] = 0. It has p positive and q negative eigenvalues.

Associated to a pair ({ej}, {ej}) of b-reciprocal bases we can define another
non-zero pseudoscalar
Äg = e'" · e'" e'" = |det[g]|Ä. (68)
g
It will be called a metric volume pseudoscalar for V. It has the fundamental
property
Äg · Äg = Äg Äg = Äg Äg = (-1)q. (69)
Ü Ü
g-1 g-1 g-1
It follows from eq.(61) by taking into account the definition of determinant
of a linear operator on V, and recalling that sgn(det[g]) = (-1)q.
An expansion formula for the pseudoscalars can be obtained from eq.(69),
i.e.,
I = (-1)q(I · Äg)Äg. (70)
g-1
The extensor " ext(V ) which is defined by : V V such that
g g
X = X Äg = X Äg, (71)
g
g-1 g-1
will be called a metric Hodge extensor on V. It should be noticed that the
definition of needs the use of both the g and g-1 metric Clifford algebras,
g
a non trivial fact.
21
p n-p
It is clear that if X " V, then X " V.
g
-1
The extensor over V, : V V such that
g
-1
X = (-1)qÄg X = (-1)qÄg X, (72)
g
g-1 g-1
is the inverse extensor of .
g
-1
Indeed, take X " ›V. By using eq.(69), we verify that ć% X =
g g
-1 -1
(-1)qÄg Äg X = X, and ć% X = (-1)qX Äg Äg = X, i.e., ć% =
Ü Ü
g g g
g-1 g-1 g-1 g-1
-1
ć% = i V .
g g
Take X, Y " V. The identity (X A) · Y = X · (Y A) and eq.(69)
g-1 g-1 g-1 g-1
yield
( X) · ( Y ) = (-1)qX · Y. (73)
g
g-1 g g-1
p
Take X, Y " V. Eq.(70), the identity (X '" Y ) · Z = Y · (X Z)
g-1 g-1 g-1
and eq.(73) allow us to get
X '" ( Y ) = (X · Y )Äg. (74)
g
g-1
This remarkable property is completely equivalent to the definition of the
metric Hodge extensor.
p n-p
Take X " V and Y " V. The use of identity (X Y ) · Z =
g-1 g-1
Y · (X '" Z) and eq.(70) yield
g-1
(( X) · Y )Äg = (-1)qX '" Y. (75)
g
g-1
It might as well be asked what is the relationship between the standard
and metric Hodge extensors as defined above.
Take X " V. By using eq.(68), the multivector identity for an invertible
(1, 1)-extensor t-1(X) Y = t (X t"(Y )) and the definition of determinant of
a (1, 1)-extensor, we have
X = g-1(X) |det[g]|Ä = |det[g]|g(X g-1(Ä))
g
|det[g]| sgn(det[g])
= g(X Ä) = g ć% (X),
det[g]
|det[g]|
22
ie.,
(-1)q
= g ć% . (76)
g
|det[g]|
Eq.(76) is the formula which relates2 with .
g
We recall that for any metric operator g " ext1(V ) there exists a non-
1
degenerate operator h " ext1(V ) such that
1
g = h ć% · ć% h, (77)
where · " ext1(V ) is an orthogonal metric operator with the same signature
1
as g. Such a h is called a gauge operator for g.
The g and g-1 metric contracted products and are related to the
g
g-1
·-metric contracted product (recall that · = ·-1) by the following formulas
·
h(X Y ) = h(X) h(Y ), (78)
g ·
h"(X Y ) = h"(X) h"(Y ). (79)
·
g-1
We can also get a noticeable formula which relates a g-metric Hodge
extensor with a ·-metric Hodge extensor.
Now, take X " V. By using eq.(79), eq.(68), the definition of determi-
nant of a (1, 1)-extensor, eq.(77) and the equation Ä· = Ä, we have
X = h (h"(X) h"(Ä)) = |det[g]|h (h"(X) det[h"]Ä)
g · g ·
= |det[h]| det[h"]h (h"(X) Ä) = sgn(det[h])h ć% ć% h"(X),
· · ·
i.e.,
= sgn(det[h])h ć% ć% h". (80)
g ·
Eq. (80) is the formula which relates with .
g ·
2
It is a very important formula and good use of it will be done in our theory of the
gravitational field to be presented in another series of papers.
23
9 Conclusions
We showed that any metric Clifford product on C (V, g) can be considered as
deformation of the euclidean Clifford product on C (V ), induced by the met-
ric extensor g. We also proved that any metric extensor g is decomposable
in terms of a gauge metric extensor h and a fiducial b-orthogonal extensor ·
which has the same signature as g. Although h is not unique, since two h s
satisfying that property differ only by a composition with a general trans-
formation › which is a ·-orthogonal (1, 1)-extensor. For the case that V
is 4-dimensional and · is a Lorentzian metric extensor (i.e., with signature
(1, 3)), › is just a general Lorentz transformation. The paper contains a
proof of the non trivial golden formula, which as the future papers will show,
really deserves its name. Indeed, the formula is a key in our theory of the
intrinsic formulation of differential geometry on arbitrary manifolds that we
will present in future papers, and also find applications in some some prob-
lems of Theoretical Physics as, e.g., in geometric theories of gravitation and
Lagrangian formulation of the theory of multivector and extensor fields.
Acknowledgement: V. V. Fernández is grateful to FAPESP for a pos-
doctoral fellowship. W.A. Rodrigues Jr. is grateful to CNPq for a senior
research fellowship (contract 201560/82-8) and to the Department of Math-
ematics of the University of Liverpool for the hospitality. Authors are also
grateful to Drs. P. Lounesto, I. Porteous and J. Vaz, Jr. for their interest in
our research and useful discussions.
References
[1] Lounesto, P., Clifford Algebras and Spinors, London Math. Soc., Lecture
Notes Series 239, Cambridge University Press, Cambridge, 1997.
[2] Porteous, I. R, Topological Geometry, Van Nostrand Reinhold, London,
1969, 2nd edition, Cambridge University Press, Cambridge, 1981.
[3] Porteous, I. R., Clifford Algebras and the Classical Groups, Cambridge
Studies in Advanced Mathematics vol.50, Cambridge University Press,
Cambridge, 1995.
24
[4] Fernández, V. V., Moya, A. M., and Rodrigues, W. A. Jr., Euclidean
Clifford Algebra (paper I in a series of seven), Adv. Appl. Clifford Algebras
11(S3),1-21 (2001), and http://arXiv.org/abs/math-ph/0212043
[5] Fernández, V. V., Moya, A. M., and Rodrigues, W. A. Jr., Extensors
(paper II in a series of seven), Adv. Appl. Clifford Algebras 11(S3),23-40
(2001), andhttp://arXiv.or/abs/math-ph/0212048
25