The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society. Volume 5, pp. 1-10, January 1999.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
AN EXTENSIONOFA RESULTOF LEWIS
TIN-YAU TAMy
Abstract. A result of Lewis on the extreme properties of the inner product of two vectors in a Cartan subspace of a semisimple Lie algebra is extended. The framework used is an Eaton triple which has a reduced triple. Applications are made for determining the minimizers and maximizers of the distance function considered by Chu and Driessel with spectral constraint.
Keywords. Eaton triple, reduced triple, nite reection group
AMSsubject classications.15A42
1. Mainresults. The purpose of this note is to establish an extension of a result
of Lewis [7, Theorem 3.2] and make some applications.
Theorem 1.1. (Lewis [7]) Let g be a real semisimple Lie algebra with Cartan
decomposition g = k+p where the analytic group of k is K G. For x 2 p, let
x
0 denote the unique element of the singleton set Ad(
K)x\a
+ where
a
+ is a closed
fundamental Weyl chamber. For x;y 2 p, (x;y) (x 0
;y
0), where (
;) denotes the
Killing form, with equality holding if and only if there isk2K such that both Ad(k)x
and Ad(k)y are ina +.
Lewis' result generalizes some well-known results including von Neumann's result [10] and a result of Fan [2] and Theobald [15], which are corresponding to the real simple Lie algebrasu
p;q and the reductive Lie algebra gl
n(
R). Here is a framework
for the extension which only requires basic knowledge of linear algebra. Let G be
a closed subgroup of the orthogonal group on a real Euclidean spaceV. The triple
(V;G;F) is an Eaton triple ifFV is a nonempty closed convex cone such that
(A1) Gx\F is nonempty for eachx2V.
(A2) maxg2G(
x;gy) = (x;y) for allx;y2F.
The Eaton triple (W;H;F) is called a reduced triple of the Eaton triple (V;G;F) if
it is an Eaton triple andW := spanF and H :=fgj W :
g 2G; gW =WgO(W)
[14]. Forx 2V, letx
0 denote the unique element of the singleton set
Gx\F. It is
known thatH is a nite reection group [11]. Let us recall some rudiments of nite
reection groups [4]. A reections
on
V is an element of O(V), which sends some
nonzero vectorto its negative and xes pointwise the hyperplaneH
orthogonal to
, i.e.,s
:=,2(;)=(;),2V. A nite groupGgenerated by reections
is called a nite reection group. A root system ofGis a nite set of nonzero vectors
inV, denoted by , such thatfs :
2ggeneratesG, and satises
(R1) \R=fgfor all2.
(R2) s
= for all
2.
Received by the editors on 3 August 1998. Accepted for publication on 1 January 1999. Handling Editor: Danny Hershkowitz.
yDepartment of Mathematics, 218 Parker Hall Auburn University, Auburn, AL 36849-5310, USA (tamtiny@mail.auburn.edu, http://www.auburn.edu/tamtiny). Dedicated to Hans Schneider on the occasions of his 70th birthday and retirement.
ELA
2 T.Y. Tam
The elements of are called roots. We do not require that the roots are of equal length. A root system is crystallographic if it satises the additional requirement:
(R3) 2(;) (;)
2Zfor all;2,
and the groupGis known as the Weyl group of .
A (open) chamberCis a connected component ofVn[ 2
H. Given a total order <inV [4, p.7],2V is said to be positive if 0<. Certainly, there is a total order
inV: Choose an arbitrary ordered basisf 1
;:::;mgofV and say> if the rst
nonzero number of the sequence (; 1)
;:::;(;m) is positive where=,. Now
+
is called a positive system if it consists of all those roots which are positive
relative to a given total order. Of course, = +
[
,, where ,= ,
+. Now +
contains [4, p.8] a unique simple system , i.e., is a basis forV
1 := span
V,
and each 2 is a linear combination of with coecients all of the same sign
(all nonnegative or all nonpositive). The vectors in are called simple roots and the corresponding reections are called simple reections. The nite reection groupGis
generated by the simple reections. Denote by +(
C) the positive system obtained by
the total order induced by an ordered basisf 1
;:::;mgCofV as described above.
Indeed +(
C) =f2 : (;)>0 for all2Cg. The correspondenceC7! +(
C)
is a bijection of the set of all chambers onto the set of all positive systems. The groupGacts simply transitively on the sets of positive systems, simple systems and
chambers. The closed convex cone F := f 2 V : (;) 0; for all 2 g,
i.e., F = C
, is the closure of the chamber
C which denes
+ and , is called a
(closed) fundamental domain for the action of G onV associated with . Since G
acts transitively on the chambers, givenx2V, the setGx\F is a singleton set and
its element is denoted byx
0. It is known that (
V;G;F) is an Eaton triple (see [11]).
LetV 0 :=
fx2V :gx =x for allg 2Gg be the set of xed points inV under the
action ofG. Let =f
1
;:::;ng, i.e., dimV 1=
n. Iff 1
;:::;ngdenotes the basis
of V 1 =
V ?
0 dual to the basis
fi = 2i=(i;i) :i = 1;:::;ng, i.e., (i;j) =ij,
thenF =f
Pn
i=1
cii:ci0gV
0. Thus the interior Int
F =CofF is the nonempty
set f Pn
i=1
cii : ci >0gV
0. There is a unique element
! 2G sending
+ to ,
and thus sendingF to ,F. Moreover, the length [4, p.12] of ! is the longest one [4,
p.15-16]. So we call it the longest element.
We will present two examples requiring some basic knowledge of Lie theory [5]. Letg=k+pbe a Cartan decomposition of a real semisimple Lie algebrag. Denote
the Killing form ofgbyB(;). The Killing form is positive denite onpbut negative
denite onk. LetK be an analytic subgroup ofkin the analytic groupGofg. Now
Ad(K) is a subgroup of the orthogonal group onpwith respect to the restriction of
the Killing form on psince the Killing form is invariant under Ad(K). Among the
Abelian subalgebras of gthat are contained inp, choose a maximal one a(maximal
Abelian subalgebra inp). For2a
(the dual space of
a), setg=fx2g: [h;x] = (h)xfor allh2ag. If 06=2a
and
g6= 0, thenis called a (restricted) root [5,
p.313] of the pair (g;a). The set of roots will be denoted . We have the orthogonal
direct sumg=g 0+
P
2
gknown as restricted-root space decomposition [5, p.313].
We viewaas a Euclidean space by taking the inner product to be the restriction of
B to a. The map a
! a that assigns to each 2 a
the unique element
x of a
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AnextensionofaresultofLewis 3
isomorphism to identify a with
a, allowing us, in particular, to view as a subset
ofa. The set =f2 :
1 2
2=ggenerates a nite reection groupW, i.e.,W is
generated by the reectionss (
2), which is called the Weyl group of (g;a), and
is a root system ofW. It is called the reduced root system of the pair (g;a). Now x
a simple system for the root system . Then determines a fundamental domain
a
+ for the action of
W ona. We now describe another way to view the Weyl group
W. Use juxtaposition to represent the adjoint action of Gon g, i.e., gx = Ad(g)x,
action ofK onginduces an action of the groupN K( w2W,x2a[5, p.325, p.394]. We use the isomorphism to identify these two groups
(in the literature, the Weyl group is usually dened to be N K(
automorphism of g, N K(
a) =fk 2K:ka=ag. ThusW =fAd(k)j a :
k2K ;ka= ag. Obviously a= spana
+. A theorem of Cartan asserts that Ad(
K)x\a6= [5,
p.320] for anyx2p. SinceW acts simply transitively on the (open) Weyl chambers
ina, Ad(K)x\a
For verication of (A2), see [7].
Example 1.2. (real semisimple Lie algebras) With the above notation. Now (p;Ad(K);a
+) is an Eaton triple with a reduced triple ( a;W;a
+). It is also true for
real reductive Lie algebras [3].
Remark: One may verify (A2) by Kostant's theorem. Ifx;y2p, letx
thogonal projection with respect to the Killing form in p, where y
0 = Ad (
k 0
k)y.
By Kostant's convexity theorem [6] (y
0) is in the convex hull of
Similarly we have the following example.
Example 1.3. (compact connected Lie groups) LetGbe a (real) compact
con-nected Lie group and let (;) be a bi-invariant inner product ong. Now Ad(G) is a
subgroup of the orthogonal group ong[5, p.196]. Let t
+ be a xed (closed)
funda-mental chamber of the Lie algebratof a maximal torusT ofG. Now (g;Ad(G);t +)
is an Eaton triple with reduced triple (t;W;t
+) where the Weyl group
W ofGis often
dened as N(T)=T where N(T) is the normalizer of T in G [5, p.201]. We remark
that Theorem 1.1 is also true for compact connected Lie groups.
The following lemma is a slight extension of Proposition 2.1 in [7] which is stated for Weyl groups. Now we add the lower bound and the proof is similar.
Lemma 1.4. LetH be a nite reection group acting on a real Euclidean space W. For any x;y 2 W, (x
ELA
4 T.Y. Tam
that bothhx andhy lie in a common closed chamberF (hx2F andhy2,F).
The following is an extension of Theorem 1.1 except we add an assumption for the upper (lower) bound attainment.
Theorem 1.5. Let(V;G;F) be an Eaton triple with a reduced triple (W;H;F).
the interval[(x 0
F which denotes
the relative interior ofF in W, then the upper (lower) bound is achieved if and only
if there exists g2Gsuch that bothgxandgy are inF (gx2F andgy 2,F).
0) by (A2). We notice that the inequality is true
without using reduced triple.
For the lower bound, notice that (x;y) = (x 0
;gy) = (x 0
;(gy)) where:V !W
is the orthogonal projection ontoW. According to a result of Niezgoda [11, Theorem
3.2],(gy) =
obviously and (,y 0)
Now we are going to handle the attainment of the upper bound. Suppose x
0 or y
0 2 Int
W
F. For deniteness, let x 0
because otherwiseP
h2H
0) by (A2), a contradiction. In
consequence, there existsg h
h=his a product of simple reections
xingy
0and hence
(gy) =y
0. However
gyandy 0have
the same norm (induced by the inner product) sincegis an element of the orthogonal
group onV; thusgy=y
The proof of the attainment of the lower bound is similar. We remark that
it is not known whether the condition x
0 or y
0 2 Int
W
F can be removed. For
the semisimple Lie algebra case [7], no such assumption is made (same for compact connected Lie groups) and is mainly due to the richer structure of the Lie framework.
Applying Theorem 1.5 on the reductive Lie algebrasgl n(
F), F = R, C, and H
respectively, we have the following result.
Corollary 1.6. (Fan [2], Theobald [15]) Let A and B be real symmetric
(Hermitian, quaternionic Hermitian) matrices with eigenvalues
1
n, respectively. The set
ftrUAU
B :U 2SU n(
ELA
AnextensionofaresultofLewis 5
isthe interval [a;b]where
symmetric groupS
n. The longest element sends the elements diag( a
1 ;:::;a
n) in F
to its reversal diag(a n SL(2n;R) which leaves invariant the quadratic form,x
2
has two components (usuallySO
0(
n;n) refers to the identity component) and hence
is not connected. It is also noncompact. It is well known that [5]
so
The Killing form is
B(
Now the adjoint action ofK onpis given by
nn and thus
a will then be
iden-tied with real diagonal matrices. We may choose a
+ =
jg. The action of K on p is then orthogonal equivalence, i.e.,
H 7! UHV where U;V 2 SO(n). The action of the Weyl group W on ais given
symmetric group) and the number of negative signs is even. The longest element !
ELA
6 T.Y. Tam
Applying Theorem 1.5 on the simple Lie algebraso
n;n, we have the following result. Corollary1.7. (Miranda-Thompson [9], Tam [13]) LetAandB bennreal
matrices with singular values 1
n
0,
1
n
0 respectively, the
set
ftrUAVB:U;V 2SO(n)g
is the interval [a;b], where
a=
, P
n,1 i=1
s
i+ sign det( AB)s
n if
nis odd ,
P n,1 i=1
s i
,signdet(AB)s n if
nis even,
b= n,1 X
i=1 s
i+ sign det( AB)s
n ;
wheres i=
i
i,
i= 1;:::;n.
Proof. Recall thata + =
fdiag(a 1
;:::;a n) :
a 1
a
n,1 ja
n
j 0g. Any
realnnmatrixAis special orthogonally similar to diag(a 1
;:::;a n,1
;[sign detA]a n)
ina
+ where
a 1
a
n
0 are the singular values ofA.
2. Applications to least squares approximations with orbital
con-straint.
In [1] Chu and Driessel considered the following two least squaresprob-lems with spectral constraints. Let S(n) be the space of n n real symmetric
matrices. Given H 2 S(n), the isospectral surface O(H) is dened to be the set O(H) = fQHQ
T :
Q 2 O(n)g. It is simply the manifold of all real symmetric
matrices which are co-spectral withH, by the well-known spectral theorem.
Problem 1. GivenX 2S(n). Find the minimizerL2O(H) for
min
L2O(H)
kL,Xk 2
;
where the norm is the Frobenius matrix norm, i.e.,kYk 2= tr
YY .
GivenH 2R
pq, set O
0(
H) =fUHV :U 2O(p);V 2O(q)g. SoO 0(
H) is simply
the manifold of all real matrices which have the same singular values of H, by the
singular value decomposition.
Problem 2. GivenX 2R
pq. Find the minimizer L2O
0( H) of
min
L2O 0
(H)
kL,Xk 2
;
where the norm is also the Frobenius matrix norm.
The minimizerL2O(H) provides the shortest distance between the given X 2 S(n) and the isospectral surfaceO(H). Notice that
kL,Xk 2= (
L,X;L,X) = (L;L) + (X;X),2(X;L);
where (X;L) := trXL, X;L 2 S(n). Since (L;L) = (H;H) for all L 2 O(H) and
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AnextensionofaresultofLewis 7
maximizerL2O(H) for (X;L). We can assume that trH = trX= 0. It is because
n. So the simple Lie algebra sl
n( R)
with Cartan decompositionsl n(
R) =so(n)+pcomes into the scene. Though we have SO(n) instead ofO(n), the orbits of H are the same.
Theorem 2.1. Letgbe a real semisimple Lie algebra with Cartan decomposition g=k+pwhere the analytic group of kisKG. Forx2p, letx
0 denote the unique
element of the singleton set Ad(K)x\a
+ where
a
+ is a closed fundamental Weyl
chamber. Givenx;y 2p, ifz2Ad(K)y, thenkx ! is the longest element of the Weyl group of (g;a). The lower (upper) bound is
achieved if and only if there is k2K such that both Ad(k)x and Ad(k)z are ina
Proof. Just notice that
kx,zk
0 and then apply
Theorem 1.1.
Applying Theorem 2.1 on sl
n(
R) we have the following solution to Problem 1
(Chu and Driessel made the assumption that the eigenvalues ofx andy are distinct,
but it is not necessary. That x has distinct eigenvalues means that x
0 is in Inta a
+
(the relative interior of a + in
a) and x is called a regular element in Lie theoretic
language. See [1, Theorem 4.1]).
Corollary 2.2. Let X;H be nn real symmetric matrices with eigenvalues x
n respectively. Then for any
Q2SO(n)
T. The lower (upper) bound is achieved if and only if there is
a Q 2 SO(n) such that QXQ
One can deduce the solution for Problem 2 (Chu and Driessel [1] also made the assumption that the singular values of x and y are distinct, but again it is not
necessary. See [1, Theorem 5.1]) by applying Theorem 2.1 onso
p;q. However, special
care needs to be taken whenp=q. Let us proceed for the case p=q=n.
We have the following renement of the result of Chu and Driessel by applying Theorem 2.1 onso
n;n.
ELA
T. The lower (upper) bound is achieved if and only if there are U;V 2SO(n) such that
Corollary2.4. LetX;H bennreal matrices with singular valuesx 1
T. The lower (upper) bound is achieved if and only if there
are U;V 2 O(n) such that UXV = diag(x
Proof. Notice that we can assume thatH andX are of nonnegative determinants
sinceL2O 0(
H) andkkis invariant under orthogonally similarity. LetK=SO(n) SO(n) be the group in the discussion ofso
and only if detH 6= 0. So the problem is reduced to the optimization ofkX,Lkwhere
ELA
AnextensionofaresultofLewis 9
The minimizersLare in Ad(K)H since
min
L2O 0
(H)
kX,Lk
2= min
f min
L2Ad(K)H
kX,Lk 2
; min
L2Ad(K)(DH)
kX,Lk 2
g= n X
i=1
(x i
,h i)
2 :
The maximizers are either in Ad(K)H or Ad(K)(DH), dependingnis even or odd.
Nevertheless,
max
L2O 0
(H)
kX,Lk
2= max
f max
L2Ad(K)H
kX,Lk 2
; max
L2Ad(K)(DH)
kX,Lk 2
g= n X
i=1
(x i+
h i)
2 :
We remark that when p6=q, sayp<q, then the action of the Weyl group ona
is given by diag(d 1
;:::;d p)
7!diag(d (1)
;:::;d
(p)), diag( d
1 ;:::;d
p)
2a, 2S p
(the symmetric group) and there is no restriction on the signs, under the appropriate
identications. Nowa
+ =
fdiag(a 1
;:::;a n) :
a 1
a
n
0g. The orbit Ad(K)H
is equal toO 0(
H). Thus we can apply Theorem 2.1 directly to arrive at the results of
Chu and Driessel whenp6=q, i.e., Corollary 2.4 is true for realpqmatrices. When
one considerssu
p;q, the complex case will then be obtained and the treatment of the p =q case is the same as p6= q case. Needless to say, application of Theorem 2.1
on various real simple Lie algebras yields dierent results, e.g., real (complex) skew symmetric matrices, complex symmetric matrices, etc.
The setting for Eaton triple with reduced triple is similar and the proof of the following is omitted.
Theorem 2.5. Let(V;G;F)be anEatontriple with areduced triple(W;H;F). Given x;y 2V, if z2Gy, thenkx
0 ,y
0
kkx,ykkx 0+ (
,y 0)0
kwherekkis induced by the inner product. If, in addition, x
0 or y
0 2 Int
W
F which denotes the relative interior of F in W, then the lower (upper) bound isachievedif andonly if thereexistsg2Gsuchthat bothgx andgy arein F (gx2F andgy2,F).
We remark that the inequalitykx 0
,y 0
kkx,ykis known and is true for an
Eaton triple without reduced triple [12, p.85].
3. Remarks.
In [8], Lewis introduced the notion of normal decomposition sys-tem (V;G;) for the study of optimization and programming problems, whereV is areal inner product space andGis a closed subgroup ofO(V) and the map:V !V
has the following properties: 1. is idempotent, i.e.,
2= .
2. isG-invariant, i.e.,(gx) =(x) for allx2V.
3. for anyx2V, there isg2Gsuch thatx=g(x).
4. ifx;y2V, then (x;y)(x;y) with equality if and only ifx=g(x) and y=g(y) for someg2G.
By Theorem 2.4 (the proof does not use equality attainment property) of [8], the range
of is a closed convex cone and we denote it by F. Thus a normal decomposition
system (V;G;) gives an Eaton triple (V;G;F). Conversely if (V;G;F) is an Eaton
ELA
10 T.Y. Tam
conditions [11, p.14] except the equality case. The equality case holds if we assume thatx
0 or y
0 are in IntW
F by Theorem 1.5.
The reduced triple is almost identical to the normal decomposition (sub)system [8, Assumption 4.1] while the dierence is only the equality case. It is interesting to notice that the Eaton triple arose from probability while normal decomposition arises from the consideration of optimization and programming.
The condition for attainment in the upper bound in Theorem 1.5 is proved by using the nite reection group structure which is an algebraic approach via a result of Niezgoda [11]. In contrast, the corresponding result [7, Theorem 3.2] of Lewis for Cartan subspace p has an analytic proof (without the restriction that x
0 or y
0 2
Inta a
+ for the equality attainment).
Acknowledgement.
The author gives thanks to an anonymous referee for pointing out that the assumption thatxory2IntW
F needs to be made in Theorem 1.5 and
for bringing [12] to the author's attention. He also thanks Professor A.S. Lewis for pointing out the relationship between Eaton triple and normal decomposition system. He thanks Professor M.T. Chu for the reprint of the reference [1].
REFERENCES
[1] M.T. Chu and K.R. Driessel. The projected gradient method for least squares matrix approxi-mations with spectral constraints. SIAM J. Numer. Anal., 27:1050-1060, 1990
[2] K. Fan. On a theorem of Weyl concerning eigenvalues of linear transformations.Proc. Nat. Acad. Sci. U.S.A., 35:652-655, 1949.
[3] R.R. Holmes and T.Y. Tam. Distance to the convex hull of an orbit under the action of a compact Lie group. J. Austral. Math. Soc. Ser. A, to appear.
[4] J.E. Humphreys.Reection Groups and Coxeter Groups, Cambridge University Press, 1990. [5] A.W. Knapp.Lie Groups Beyond an Introduction. Birkhauser, Boston, 1996.
[6] B. Kostant. On convexity, the Weyl group and Iwasawa decomposition. Ann. Sci. Ecole Norm. Sup., (4) 6:413-460, 1973.
[7] A.S. Lewis. Convex analysis on Cartan subspaces. Nonlinear Anal., to appear.
[8] A.S. Lewis. Group invariance and convex matrix analysis. SIAM J. Matrix Anal. Appl., 17:927-949, 1996
[9] H. Miranda and R.C. Thompson.A supplement to the von Neumann trace inequality for singular values. Linear Algebra Appl., 248: 61-66, 1994.
[10] J. von Neumann. Some matrix-inequalities and metrization of matrix-space.Tomsk. Univ. Rev.
1:286-300, 1937; in Collected Works, Pergamon, New York, Vol. 4, p.205-219, 1962. [11] M. Niezgoda. Group majorization and Schur type inequalities.Linear Algebra Appl., 268:9-30,
1998.
[12] M. Niezgoda. On Schur-Ostrowski type theorems for group majorizations. J. Convex Anal., 5:81-105, 1998.
[13] T.Y. Tam. Miranda-Thompson's trace inequality and a log convexity result. Linear Algebra Appl., 272:91-101, 1997.
[14] T.Y. Tam. Group majorization, Eaton triples and numerical range, Linear and Multilinear Algebra, to appear.
