#### Domain wall seeds in CSO-gauged supergravity

Accepted: June
Domain wall seeds in CSO-gauged supergravity
Juan Diaz Dorronsoro 0 1 2
Harold Erbin 0 1
Thomas Van Riet 0 1 2
0 F-75231 , Paris , France
1 Celestijnenlaan 200D B-3001 Leuven , Belgium
2 Instituut voor Theoretische Fysica, K.U. Leuven
Gravitational domain wall solutions in gauged supergravity are often constructed within truncations that do not include vectors. As a consequence the gauge group is only a global symmetry of this truncation. The consistency of the truncation requires the restriction to solutions with vanishing Noether charge under this global symmetry, since otherwise vector elds are sourced. We show that this has interesting consequences for the orbit structure of the solutions under the global symmetries. We investigate this for CSO(p; q; r)-gaugings in various dimensions with scalar elds truncated to the SL(n; R)=SO(n) subcoset. We prove that the seed solution | which generates all other solutions using only global transformations | has a diagonal coset matrix. This means that there exists a transformation at the boundary of the geometry that diagonalises the coset matrix and that this same transformation also diagonalises the whole ow as a consequence of the vanishing charge.
Global Symmetries; Supergravity Models; AdS-CFT Correspondence
1 Introduction
2
3
4
5
6
CSO-gauged SUGRA
Supersymmetric domain wall ows
Noether charges
Normal forms
5.1
5.2
Normal form for SO(p; q) gaugings
Normal form for CSO(p; q; r) gaugings
Discussion
A Noether currents B
Normal forms and vanishing Noether charges
C Jordan normal forms and SO(p; q)
any solution to the equations can be mapped to another solution under the action of G.
This leads to the concept of orbits of solutions.
Studying the orbits of solutions has been a very active eld of research over the past
decades in the context of black hole solutions to ungauged supergravity, see [3, 4] for some
original references. Less well studied is the orbit structure for domain wall solutions in
gauged supergravity. This structure is expected to be somewhat more complicated since
the symmetry group in gauged supergravity is not as big as in ungauged supergravity.
The diminished symmetry is a consequence of the gauging which makes a subgroup of the
{ 1 {
isometry local and destroys the rest1 of the original global symmetry by the introduction
of a scalar potential (see for instance [5]).
Often one is interested in domain walls with maximally symmetric wall spaces and with
all elds with non-trivial spin (apart from the metric) set to zero. The resulting equations
of motion enjoy then the gauge group as a global symmetry. For this to be consistent the
scalar elds should not source any vector elds. This implies that the charge of the scalars
has to vanish. This is a necessary and su cient condition for the truncation to be valid and
has to be checked separately when a solution of the scalar-metric system is found, since this
truncation is not guaranteed to be consistent. We are therefore interested in understanding
the orbit structure of solutions with zero charge under global gauge rotations, which at rst
sight seems harder than the orbit structure under local rotations.
In this paper we initiate a rst systematic study of this in the context of a popular
class of truncations of gauged supergravity theories in D dimensions with 2 < D < 10:
the CSO(p; q; r)-gaugings of theories with as scalar coset SL(n; R)= SO(n) where n = p +
q + r. A review of these theories, their solutions, and their link with (string theory)
compacti cations can be found in [6]. We prove here that all solutions in these theories
can be found by transforming, under the global symmetry group, the solutions with diagonal
coset matrices. In other words, we show that there exists a rotation matrix inside the global
symmetry group that diagonalises the coset matrix at the boundary and that this same
matrix diagonalises the whole
ow. This is non-trivial since a priori the transformation
that diagonalises the matrix could depend on the position along the
ow. It is also not
evident that one can diagonalise the coset matrix using non-compact gauge groups. In this
paper we prove both claims.
Since these statements can be confusing, let us consider a reasoning that can be found
(indirectly stated) in the literature on susy domain walls for SO(n)-gaugings, see for
instance [7]. The argument is that prior to truncating the vectors, one can use the local
rotation symmetry to diagonalise the coset matrix everywhere. Hence it is su cient to
look at solutions with diagonal matrices. These solutions do not source vectors since their
SO(n)-charge vanishes. The susy solution then obeys the typical ow equation
where i are the scalars, Gij the metric on eld space, W a function of the scalars called
the real superpotential and z is the coordinate transverse to the wall. For diagonal coset
matrices only n 1 scalars are switched on and this equation is easily solved [7]. But imagine
now that we started solving (1.1) for a general coset matrix with all o -diagonal elements
switched on and that the solutions have zero charge, as required for consistency. Then the
resulting solution is expected to be diagonalisable using a local SO(n)-transformation. Such
a transformation will switch on vectors AI that are pure gauge, such that the eld strengths
still vanish. Hence one would expect the diagonal solution to solve the ow equation with @z
replaced by a covariant derivative that includes the vectors. But since the charge vanishes
1Not necessarily all the rest of the global symmetry is destroyed. What is left is the normaliser of
by the following action
V ( ) ;
where Gij denotes the metric on the scalar manifold X and V is the scalar potential. All
maximal and half-maximal gauged supergravities allow a further consistent truncation of
the scalars such that the scalar manifold X is
X =
SL(n; R)
SO(n)
or
X =
GL(n; R)
SO(n)
:
For example in N = 8; D = 4 maximal gauged SUGRA the scalar manifold is E7(7)= SU(8),
describing 70 real scalars. Those 70 can be split into 35 scalars and 35 pseudo-scalars. The
35 scalars span SL(8; R)= SO(8). Maximal supergravities in D = 7 or higher do not need a
truncation in order to be of the kind (2.2) (or be products of such manifolds).
We are particularly interested in CSO(p; q; r) gaugings as they form a \canonical" set
of gaugings in maximal supergravity [6].
Well-known examples are the SO(n) gaugings
originating from sphere compacti cations of type II supergravity in D = 10 or of D = 11
supergravity. For instance SO(8) comes from 11D sugra on S7, SO(6) from IIB on S5,
SO(5) from 11D sugra on S4, SO(3) from IIA on S2 or 11D on S3 (as a group space,
not as a coset). The SO(p; q) gauge groups can be found from generalised hyperbolic
compacti cations [8].
Consider the coset representative L of SL(n; R)= SO(n) in its fundamental
representation. It is multiplied by the left with the SL(n; R) isometry transformations and from the
right with the SO(n) isotropy group:
L !
Lh
2 SL(n; R) ;
h 2 SO(n) :
It turns out useful to work with the symmetric coset element, M
unit determinant, is positive de nite, and invariant under the isotropy group. Under the
= LLT , which has
SL(n; R) isometry group it transforms as a bilinear form
(2.1)
(2.2)
(2.3)
(2.4)
The metric on SL(n; R)= SO(n), which has SL(n; R), as an isometry group is given by
The coset can be extended to GL(n; R)= SO(n) with an extra scalar ' as follows
1
2
1] :
HJEP06(217)9
The scalar potential for CSO-gaugings can be written as
where a is a number that depends on the dimension
and the matrix
is diagonal
where p + q + r = n. The combinations of n and D for which a vanishes are also theories
for which the scalar ' is absent.
The global symmetry group, in presence of the scalar potential, is the group that leaves
the bilinear form
invariant. When r > 0 this is larger than the CSO(p; q; r)-group, as we
will explain below.
3
Supersymmetric domain wall
ows
The metric ansatz for Minkowski-sliced domain wall solutions is
ds2 = f 2(z)dz2 + g2(z)ds2D 1
;
where ds2D 1 is the metric on Minkowski space. The warpfactors g; f and the scalars i
are only functions of z. The warpfactor f (z) is pure gauge and can be chosen freely. From
the action (2.1) one can derive the following e ective action for domain-wall solutions:
Z
Se =
dz
"
2
1 _2 +
1
4
Tr(M_ 1M_ ) + (D
1)(D
2)
g_ 2
g
g2(D 1)ea Tr( M
M) +
ea [Tr( M)]2 :
g2(D 1)
2
Varying (3.2) gives the general second order equations of motion for M:
1
d
f gD 1 dz
f 1gD 1
M
1 _
M
= ea' 4( M)
2 2Tr( M)
M
Tr( M)2 +
Tr M
4
n
{ 4 {
#
2
n
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(3.1)
(3.2)
2 :
(3.3)
Supersymmetric solutions solve rst-order ow equations derived from a speci c real
superpotential W :
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
is equivalent to (3.2).
and is given by:2
where
The matrix form of the rst-order equations has not appeared earlier in the literature to
our knowledge. The proof proceeds in the usual way by squaring the action. In the gauge
f = gD 1, we nd that the following squared e ective action,
The diagonal solution was constructed in [9] (see also [7, 10]) in the gauge f = g3 D
1
2(D
2)
W :
V =
2
(D
4(D
1)
2)
W 2 :
a gD 1Tr( M) :
The potential can be written in terms of W as follows:
For the class of theories discussed here we have W = ea'=2 Tr[ M]. Both V and W are
manifestly invariant under the CSO gauge group. The ow equation for the scalars (3.4)
can be written as an equivalent rst-order equation for the matrix M as follows:
HJEP06(217)9
ds2 = h(3 D)=(2D 4)dz2 + h1=(2D 4)ds2D 1
;
e' = h a=4 ;
M = h1=ndiag(1=h1; : : : ; 1=hn) ;
hi = 2 iz + `i2 ;
h = h1 : : : hn ;
with i the i'th diagonal entry of (hence
1 or 0) and `i an arbitrary integration constant.
In the context of maximal supergravity, these ows were shown to preserve half of the
supersymmetries [7, 9, 10].
In the next sections, we will argue that all susy Minkowski-sliced domain wall solutions
are related to this diagonal solution through a global transformation. The same also holds
for all non-susy solutions with vanishing Noether charges: the seed will again be diagonal.
2In the case n = 3 the solution with all axions turned on was found in [11], but we do not require it here
since we only care about seed solutions.
{ 5 {
The e ective action (3.2) for nding domain walls corresponds to a normal Hamiltonian
system in classical mechanics with the spatial coordinate z playing the role of time. Since
the e ective action has the gauge group G = CSO(p; q; r) as a global symmetry, there
must be a set of conserved charges equal to the dimension of G, which remain constant
along the ow in z. These conserved charges can always be written in terms of momenta
and generalised coordinates and hence provide a set of rst-order equations. We now show
that for supersymmetric solutions in our models these rst-order equations contain no new
information since they are implied by (3.7).
To gain some intuition we rst consider the case without potential. The equations of
motion for the scalars in the gauge f = gD 1 decouple from the metric and are derived
from the geodesic action
The geodesic equations for the scalars in M are summarized as
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
Sg =
Z
dz Gij _i _j :
d
dz (M
1M_ ) = 0 :
M
1M_ = Q ;
M(z) = M(0)eQz :
{ 6 {
This can be integrated once
with Q any traceless matrix. This equation is further integrated as
So remarkably all geodesics on SL(n; R)= SO(n) can be found in the language of the
symmetric coset matrix in a trivial manner.3 From (4.3) one nds that Q is in the Lie algebra
of SL(n; R) since it is traceless and corresponds to the matrix of Noether charges.
Once we deform the geodesic motion by the potential V , only those Noether charges
corresponding to the symmetry preserved by V should survive. Hence the Noether charge
matrix corresponding to the CSO gauged supergravity theories should be related to the
projection of the general Q 2 Lie(SL(n)) to Lie(CSO(p; q; r)). An explicit computation of
the Noether charge carried out in appendix A yields
Q =
M_ M
1
M
1 _
M
:
Substituting the rst-order equation (3.7) into (4.5) one nds that supersymmetric solutions
have vanishing charge. Hence the supersymmetric solutions source no vectors, consistent
with our truncation. Moreover, one can explicitly check that Q
(which is also conserved)
is an element of the CSO algebra.
We already explained in the Introduction that we expect solutions with zero Noether
charges to behave in a special way under global transformations of the gauge group. Namely
3This is in sharp contrast with the algorithms that are developed to integrate the equations at the level of
the scalars [12{14] where the solutions can be found algorithmically but the computations and expressions
are complicated.
a solution whose coset matrix can be diagonalised at some value for z using a gauge
transformation will be diagonal for all values of z. We have included a computationally
explicit proof of this in appendix B for SO(p; q)-gaugings. For gaugings with contracted
gauge groups the story complicates somewhat in the sense that a bigger transformation is
needed to diagonalise a solution. But again the vanishing of the Noether charge will turn
out su cient to argue that all solutions are global transformations of the solution with
diagonal coset matrix.
The reasoning uses the equations of motion (3.3), which can be schematically written as
M = F (M) :
(4.6)
Since the matrix F (M) is diagonal for diagonal M, if both M and its rst derivative are
diagonal at some point, the whole ow will remain diagonal. We will show that if we require
the Noether charge to vanish, we can diagonalize M and M_ at a reference value (e.g. z = 0)
using a transformation inside the global symmetry group of the action. This ensures that
all solutions with vanishing Noether charge (and hence all the solutions consistent with the
truncation) can be obtained through a global symmetry transformation from the diagonal
seed solutions.
The intuition behind this is simple. Consider for instance a spherically symmetric
system in classical mechanics for which the angular momenta vanish. Since angular
momentum is linked to rotation, there exists a frame in which the system has no rotation and
all angular variables vanish at all times. Let us now extend this logic to our context. If it is
possible to diagonalise M(0) using SO(p; q) then the general coset element can be written
as L = P
D, with D diagonal and P in SO(p; q). The degrees of freedom are explicitly
separated into `radii' sitting in D and `angular variables' residing in P. From the point of
view of the scalar potential the variables P are cyclic. If the variables in P are then zero
in a certain frame (basis) at z = 0 they will remain zero throughout the ow. Hence the
transformation that brings one to the frame of vanishing angles at the boundary brings us
to vanishing angles along the ow.
5
Normal forms
We now set out to prove the main claim made in the introduction that all solutions of
the CSO-gaugings with zero charge can be found performing a global transformation on
solutions with diagonal coset matrices. We rst discuss SO(p; q) gaugings and then we
treat the contracted CSO algebras.
5.1
Normal form for SO(p; q) gaugings
We start by arguing that any coset matrix M evaluated at the reference value z = 0 can be
diagonalised using an SO(p; q) transformation. Using the result form the previous section
this then implies that the solution will be diagonal globally.
It is well-know that M(0) can be diagonalised using an SO(n)-transformation since it
is a symmetric bilinear form. We want to generalise this to SO(p; q)-transformations with
p + q = n. The statement is not true for general symmetric bilinear forms, but relies on
{ 7 {
M(0) being positive de nite. Our proof uses a detour via the normal form of the coset
element of SL(n; R)= SO(p; q) constructed in [
15, 16
]. To make this connection, note that
any symmetric matrix M can be written as
with
the invariant bilinear form under SO(p; q) de ned earlier and A some generalised
symmetric matrix, which means
Such a matrix A can be regarded as being in the Lie algebra of SL(n; R)= SO(p; q). It was
shown in [
15, 16
] that one can bring A into its Jordan normal form J through a similarity
The proof of this proceeds by showing that one can always nd a basis for in which the
normal form of a generalised symmetric matrix is itself generalised symmetric: J
This means that the similarity transformation is inside SO(p; q). The detailed proof of this
T =
J
.
statement can be found in [
15, 16
]. In appendix C we show that, due to M being positive
de nite, the normal form J has to be strictly diagonal. After this we get
M = A
=
J
1
=
J
T ;
and we therefore see that we can indeed diagonalize M with an SO(p; q) transformation.
5.2
Normal form for CSO(p; q; r) gaugings
For the CSO(p; q; r) gaugings the global symmetry group is larger than the gauge group.
The CSO group can be obtained by exponentiating the Lie algebra elements gij given by [6]
whereas the global symmetries of the action are larger and contain all the transformations
that leave
invariant
This second group is larger, since even though one has
for any choice of constants ij , not all the matrices
form exp
ij gij . Indeed, (5.6) only implies that
which leave
invariant are of the
(gij )kl = [ki j]l ;
=
T
:
exp
ij giTj
exp
ij gij =
=
Ar (p+q)
2 SO(p; q), A is an arbitrary r by (p + q) matrix and B is an arbitrary r by
{ 8 {
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
Here, M1 has size (p + q)
(p + q) and M3 has size r
previous section, we know that there is a matrix
is diagonal. Let us transform M with the matrix
r. From the arguments of the
2 SO(p; q) such that
M1(0) T = D1
where for the moment we keep B arbitrary. Then we get
M(0) T =
D1
The matrix S is symmetric, and moreover it is positive de nite.4 From the properties of S
we know that there exists an SO(r) transformation O which brings it into a diagonal form:
We will argue now that we can diagonalize any susy ow through a global transfor
mation. To this aim, let us split M(0) into blocks as follows:
M(0) 1 =
( ) ( )
( ) S 1
!
{ 9 {
M(0) =
M1(0) M2(0)!
M2T (0) M3(0)
with di > 0. If we then let
we see that
This shows that using the transformation we just described, we get
block of the inverse matrix M(0) 1. Indeed,
We now argue why this transformation will diagonalise the coset matrix at all values for
z if the Noether charge vanishes. This result is not implied from the results in the previous
section since we only require that the Noether charge under CSO rotations are zero, whereas
the -transformation above is typically inside a larger symmetry group. Hence we have to
prove that the -transformation will diagonalise the solution for all values of z.
4The way to see this last point is by remarking that S is the combination that appears on the lower-right
whenever S is not singular. The matrix S is also known as the Schur complement of M1(0), and the matrix
M(0) is positive de nite if and only if M1(0) and its Schur complement are both positive de nite.
After performing a transformation as the one we described, we can write
M =
D1 0 !
The Noether charge is therefore
Q =
M =
M_ 2T (0) s
sM_ 1(0)D1
D1M_ 1(0) s sM_ 2(0)!
D1 0 !
where s is a diagonal matrix with p times +1 and q times
know that M_ 2(0) must vanish, and therefore we have
1. If this charge vanishes, we
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
Suppose therefore that M is a solution of the second order equations of motion
such that
Let us expand such a solution around the reference value z = 0 as
Q =
M_ M
1
M
z + O(z2) :
0
;
After applying the global transformations we described above, we have shown that both
the coset matrix and its rst derivative can be made diagonal at the reference value z = 0.
Due to the structure of the second order equations of motion, the matrix will preserve its
diagonal form along the whole ow. This concludes the proof that any consistent solution
(therefore with vanishing Noether charge) can be brought into a diagonal form through a
global symmetry transformation.
In this frame, the equations of motion for the upper-left diagonal part of the matrix M
are precisely the same equations of motion of an SO(p; q) gauging, and we can use the
throughout the ow. This implies that M_ 1(0) = D_ 1 is of course diagonal.
same arguments of the previous section to show that this part of M will remain diagonal
As for the lower-right component of M, we remark that since M_ 3(0) is symmetric,
we can diagonalize it through an SO(r) transformation Or such that OrM_ 3(0)OrT = D_ 3.
Transforming now M as
we get around z = 0
We have investigated the space of domain wall ows in CSO(p; q; r)-gauged supergravities
with scalar
elds truncated to the coset SL(n; R)= SO(n) (or GL(n; R)= SO(n)), where
n = p + q + r. We have emphasized how the gauge group is a global and not a local
symmetry once the vectors are truncated and that this requires the CSO(p; q; r)-charge to
be zero. We then showed that all the zero-charge solutions of the theory can be found
by letting the global symmetry act on solutions with diagonal coset matrices. As the
vanishing of the Noether charge is a necessary consistency condition for any solution of the
truncated theory, we have proved that all the consistent solutions of CSO gaugings (within
our truncation) can be obtained through a global rotation of a diagonal solution. All
supersymmetric domain walls with diagonal coset matrices were constructed in [6, 7, 9, 10].
For supersymmetric ows the results we found were implicitly known for SO(n) gaugings.
For all other CSO(p; q; r)-gaugings our
ndings were to our knowledge not explained in
the literature.5
We emphasize that this result is more surprising for the gaugings with
contracted gauge groups (r > 0). In that case the global transformation that diagonalises
the
ows are outside of the gauge group, but still within the global symmetry of the
action. In the extreme case r = n, there is no gauging and hence no scalar potential
so that the ows describe geodesic curves on SL(n; R)= SO(n). Those curves were known
to be diagonalisable using SL(n; R) (see for instance [
16
]). The reason the
ow can be
diagonalised when 0 < r < n is a mixture between the di erent reasonings used for the
extremes r = 0 and r = n.
This work is a rst step towards classifying (supersymmetric) domain wall ows in
gauged supergravity. To achieve that goal, one should go beyond the SL(n; R)=
SO(n)truncation.6 We then expect that a very non-trivial orbit structure should arise.
Acknowledgments
We thank Nikolay Bobev, Fridrik Gautason for useful discussions and especially Adolfo
Guarino and Mario Trigiante for discussions and essential feedback on a earlier draft.
The work of JDD and TVR is supported by the FWO odysseus grant G.0.E52.14N and
by the C16/16/005 grant of the KULeuven. We furthermore acknowledge support from
the European Science Foundation Holograv Network and the COST Action MP1210 `The
String Theory Universe'.
A
Noether currents
tions M !
M
T , with
Consider the e ective action (3.2). It is invariant under global CSO(p; q; r)
transformaan element of the CSO(p; q; r) group such that
T
= .
5Aside some comments in [11] about the case n = 3 where the claim can be found that \the change of
SL(n; R)-frame" diagonalises the solution.
6Another extension could be to look at the !-deformations of the standard gaugings [17] within the
SL(n; R)= SO(n)-truncation.
Using the results of [
18
] it seems that the
ndings of this paper still go
through such that again the diagonal solutions are the seed solutions.
Normal forms and vanishing Noether charges
Here we prove explicitly for SO(p; q) gaugings that the matrix that diagonalises M at z = 0
diagonalises M throughout the ow if the Noether charge Q vanishes. Inserting
into (4.5) and requiring Q = 0 yields
2! = D !D
1 + D
1 !D;
2 so(p; q):
If we let D = diag(d1; : : : ; dn), the previous equations becomes in components
(A.1)
ij gij .
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(B.1)
(B.2)
(B.3)
(B.4)
In nitesimally we can write
Here, gij label the generators of the algebra and ij are transformation parameters, which
are taken to be in nitesimal. In order to nd the conserved charges we allow
and follow the standard Noether procedure. To simplify the notation, we write h
ij = ij (z)
The two last terms of the action remain invariant even when h = h(z), so that we only
consider the kinetic term and nd:
Integrating by parts, throwing away a total derivative and by the standard argument of
letting ij be constant, we get a set of conserved charges
The generators of the CSO group are [6]
A short computation then reveals that
Qij = Tr
M_ M
1gij ;
_
Qij = 0 :
(gij )kl = [ki j]l :
Q =
M_ M
1
M
We recall that for ! 2 so(p; q), all the diagonal elements of ! vanish. If all the eigenvalues
of M are di erent (so that di 6= dj for all i; j), then (B.3) further implies that !ij = 0, so
that ! = 0 and
What happens when some of the eigenvalues of M are equal? Suppose for instance
that we are dealing with the extreme case in which all the eigenvalues are the same, so
that also di = dj for all i; j. Then we get
M =
D
T =
D( T
so that M is diagonal for any , and there is no need for diagonalizing it in the rst place.
We could then trivially keep constant in this case too.
The argument for the constancy of
when only some of the eigenvalues of M are
HJEP06(217)9
equal, follows from realizing that the only components !ij which can di er from zero are
those for which di = dj . However, in that case, the transformation
will leave D invariant
anyway: it will correspond to a trivial reshu ing of the equal di's in D. Indeed, if without
loss of generality we arrange the elements of D in k groups
D = (d1; : : : ; d1; d2; : : : ; d2; : : : ; dk; : : : ; dk) ;
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
! will be block diagonal:
and we get
with
c being constant and
::: CC
C ;
C
A
:
:
: 0 C
0 !k
(z) =
c !(z) ;
0
0
This z-dependent part of the transformation acts however trivially on D:
We see therefore that even when there are repeated eigenvalues, M can be diagonalised
with a constant SO(p; q) transformation throughout the ow.
Jordan normal forms and SO(p; q)
Let us construct the basis for the bilinear form
that makes the standard Jordan normal
form J generalised symmetric. This standard form is given by
0
J = BBBBB 0::: J2
::: CC
C
C
A
0
i 1
then from the generalised symmetric condition we see that
must be of the form
where i has the same size as the Jordan block to which it corresponds and is equal to
1,
if its size is 1, or
with
Ji = BBB :::
B ::
i = BBB :
B0 : :
1 0
:
:
:
0
otherwise. Using all the previous, we can write [
16
]
We will now argue that J is strictly diagonal, and not just block diagonal. Since
M = LL
T is positive de nite, we must have that J
=
1
M(
1)T , is also positive
de nite. Clearly, the matrix J
is block diagonal, and in order to be positive de nite, each
of its blocks Ji i must be positive de nite. Suppose that there is at least one non-diagonal
block Ji in the Jordan normal form for A. The product Ji i is then
Ji i = BBB 0 :
: :
: : :
: : : ::: CCC :
i 0 CC
0
C
C
A
De ning the vector7 v = (1; 0; : : : ; 0; 1= i), we have
vT Ji iv =
(
1
2
if Ji has size 2
if Ji has size r
2 ;
r with r > 2 ;
7We recall that i 6= 0 since det(J ) =det[ 1M(
1)T ] 6= 0.
from which we deduce that Ji i is not positive de nite. We have hence shown by
contradiction that J must be diagonal. To unify this discussion with the one of the previous
subsection, we will write D = J . We know then that we can always diagonalize M using
an SO(p; q) transformation:
M(z) =
D(z)
takes the standard diagonal form.
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U.K. (2012).
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