UPPER BOUNDS ON CERTAIN FUNCTIONALS DEFINED ON
GROUPS OF LINEAR OPERATORS∗
MAREK NIEZGODA†
Abstract. The problem of estimating certain functionals defined on a group of linear operators generating a group induced cone (GIC) ordering is studied. A result of Berman and Plemmons [Math. Inequal. Appl., 2(1):149–152, 1998] is extended from the sum function to Schur-convex functions. It is shown that the problem has a closed connection with both Schur type inequality and weak group majorization. Some applications are given for matrices.
Key words. Group majorization, GIC ordering, Normal decomposition system, Cone preorder-ing, Schur type inequality, Schur-convex function, Eigenvalues, Singular values.
AMS subject classifications. 06F20, 39B62, 15A48, 15A18.
1. Introduction. Berman and Plemmons [2] proved that the functional
f(U) =
n
i=1
max
j {(U T
MjU)ii}
(1.1)
over alln×northogonal matricesU is maximized byan orthogonal matrixQwhich
simultaneouslydiagonalizes the symmetric matrices Mj, j = 1, . . . , k. An analogous
result holds for Hermitian matrices [2, Section 3].
In the present paper we studya similar problem for a general linear space endowed with the structure of normal decomposition (ND) system (to be defined below). Also, we replace the sum function in (1.1) byan increasing function with respect to certain vector orderings (see Section 2). Some applications are given for matrices in Section 3. A further extension to weak group majorization is discussed in Section 4.
2. Results. Let V be a finite-dimensional real linear space equipped with an
inner product ·,·. By O(V) we denote the orthogonal group acting on V. Let G
be a closed subgroup ofO(V). Thegroup majorization induced byG, abbreviated as
G-majorization and written asG, is the preordering onV defined by
yGx iff y∈convGx,
where convGxdenotes the convex hull of the orbitGx:={gx:g∈G} (see [18]).
Let (·)↓ : V →V be aG-invariant map, that is (gx)↓ =x↓ for any x∈ V and
g∈G. We say that (V, G,(·)↓) is anormal decomposition (ND) system (see [10, 11])
if
(A1) for anyx∈V there exists g∈Gsatisfyingx=gx↓,
(A2) max
g∈Gx, gy=x↓, y↓ for allx, y∈V.
∗Received by the editors 27 November 2005. Accepted for publication 13 June 2006. Handling
Editor: Shmuel Friedland.
†Department of Applied Mathematics, Agricultural University of Lublin, P.O. Box 158,
In this event, it can be deduced (see [10, Thm 2.4]) that the set D =V↓, the range
of (·)↓, is a closed convex cone. Under axioms (A1)-(A2), the group majorizationG
is said to be a group induced cone (GIC) ordering [4, 5, 6, 18]. The operator (·)↓
is called a normal map. See [4, 5, 6, 10, 11] for examples of ND systems and GIC
orderings. See also Section 3.
Axiom (A1) generalizes the Spectral Theorem for Hermitian (or real symmetric) matrices as well as the Singular Value Decomposition Theorem for complex (or real) matrices [4]. On the other hand, (A2) can be viewed as an extension of the funda-mental von Neumann trace result on singular values [4] and of the analogous result of Miranda and Thompson [14, 15]. This makes it possible to applyresults for a general ND system in matrix theory, convex analysis, optimization, etc.
The notion of ND system was introduced by Lewis (see [10, pp. 928-929] and [11, pp. 817-818]). The corresponding theoryof group induced cone orderings was devel-oped byEaton and Perlman [7], Eaton [4, 5, 6], Giovagnoli and Wynn [8], Steerneman [18] and Niezgoda [16, 17].
For an ND system (V, G,(·)↓) with D being the range of (·)↓, it can be shown
thatx↓ is the unique element of the setD∩Gx. Denote
W :=span DandH :={h=g|W :g∈G, g W =W}.
It is known (see [16, Thm 3.2], [16, p. 14], [18, Thm 4.1] and [9]) that the following statements (i)-(iv) are mutuallyequivalent:
(i) The group majorizationsG andH are equivalent onW, that is
yGx iff yHx for all x, y∈W.
(ii) (W, H,(·)↓) is an ND system (with the inherited normal map).
(iii) Schur type inequality holds, that is
P xHx↓ for all x∈V,
(2.1)
whereP is the orthogonal projection from the spaceV onto the spaceW.
(iv) H is a finite reflection group.
LetC⊂W be a convex cone. Thecone preordering C induced byCis defined
as follows:
yCxiffx−y ∈C
forx, y∈W. A functionϕ:W →Ris said to beC-increasing (resp.H-increasing) if
yCx(resp.yH x) impliesϕ(y)≤ϕ(x) forx, y∈W. The functionϕis said to be
CH-increasing if it is both C-increasing andH-increasing. We saythat C hasmax property if for anyvectorsa1, . . . , ak ∈W there exists a maximal vector max
j aj with
respect toC. We call a linear operatorL:V →V C-positive ifLC ⊂C. Observe
that theC-positivityofL impliesLyCLxwheneveryCx.
Theorem 2.1. Let (V, G,(·)↓) be an ND system satisfying any of the above
(I) Let w, w1, . . . , wk be vectors in W. If wj C w for j = 1, . . . , k, then for each
g∈G
P gwjCP gwHw for j= 1, . . . , k,
(2.2)
max
j ϕ(P gwj)≤ϕ(P gw)≤ϕ(w).
(2.3)
(II) If, in addition, C has max property, then for eachg∈G
max
j P gwjCP gwH w,
(2.4)
ϕ(max
j P gwj)≤ϕ(P gw)≤ϕ(w).
(2.5)
(III) In particular, if C has max property and w= max
j wj, then for eachg∈G
ϕ(max
j P gwj)≤ϕ(maxj wj),
(2.6)
i.e., the functional
f(g) :=ϕ(max
j P gwj), g∈G,
(2.7)
is maximized byg=id, the identity operator on V.
Proof. Fix arbitrarily g ∈ G. We shall prove that the linear operator P g is C
-positive. Denote D = V↓. Recall that {x↓} =D∩Gx for x∈V. Using (2.1) and
theG-invariance of the normal map (·)↓:V →D, we obtainP gzH (gz)↓=z↓ for
z∈W.
On the other hand, employing condition (A1) for the ND systems (V, G,(·)↓)
and (W, H,(·)↓), for each z ∈ W we get z = g0z↓ and z = h0d for some g0 ∈ G,
h0 ∈H and d∈D. By[16, Lemma 2.1], we derived=z↓. Therefore z =h0z↓. In
consequence,Hz=Hz↓ forz∈W. It now follows that P gz∈convHz↓ = convHz,
sinceP gzH z↓. Hence
P gz H zforz∈W. (2.8)
SinceCisH-invariant, we obtainHz∈Cand convHz⊂Cforz∈C. Therefore
P gz ∈ C forz ∈C, since P gz ∈convHz by(2.8). This yields the C-positivityof
P g, as required.
(I).Sincewj C wforj= 1, . . . , k, we getP gwjCP gwbytheC-positivityof
P g. Additionally,P gwH wby(2.8). This completes the proof of (2.2).
Applying (2.2) and the fact that ϕ is CH-increasing, one obtains ϕ(P gwj) ≤
ϕ(P gw)≤ϕ(w) forj= 1, . . . , k. This proves (2.3).
(II). Suppose that C has max property. Then there exists max
j P gwj. Using
(III). Substituting w := max
j wj into (2.5) yields (2.6). Since wj ∈ W and
P wj = wj, (2.6) means that the functional defined by(2.7) is maximized bythe
identityoperatoridonV.
Corollary 2.2. Let (V, G,(·)↓) be an ND system satisfying any of the
equiva-lent conditions (i)-(iv). Assume C⊂W is an H-invariant convex cone having max property. Suppose thatϕ:W →Ris aCH-increasing function.
Let v1, . . . , vk be vectors in V such that g0v1, . . . , g0vk ∈ W for some g0 ∈ G.
Then g0 maximizes the functional
f(g) :=ϕ(max
j P gvj), g∈G.
Proof. ApplyTheorem 2.1, part (III), forwj := g0vj ∈ W, j = 1, . . . , k, and
w:= max
j g0vj.
Theorem 2.1, part (I), can be modified. See Theorem 3.2 for application of the result below.
Corollary 2.3. Under the hypotheses of Theorem 2.1, assume that ϕ is
H-increasing on some set W0 (in place of W) such that W↓ ⊂W0 ⊂W. Suppose that
there exists a subgroup H0 ⊂H satisfying (i) for eachh0 ∈ H0 there existsg0 ∈G such that h0P =P g0, and (ii) for each w ∈ W there exists an h0 ∈ H0 such that h0w∈W0.
Then
max
j ϕ(P g0gwj)≤ϕ(P g0gw)≤ϕ(w↓),
(2.9)
whereh0P gw=P g0gw∈W0.
Proof. Applying (i)-(ii) we take an h0 ∈ H0 and g0 ∈ G such that h0P gw ∈
W0 and h0P = P g0. Since C is H-invariant, H0 ⊂ H and H is H-invariant, it
follows from (2.3) thath0P gwjCh0P gwandh0P gwH w↓ withw↓∈W↓ ⊂W0.
Therefore (2.9) holds.
3. Applications. In this section we interpret the results of Section 2 in matrix setting. We consider two special cases. The first leads to a result generalizing a theorem of Berman and Plemmons [2] (see Corollary3.1).
To do this, we set
V := the linear spaceSn ofn×nreal symmetric matrices,
G:= the group of operators of the formX→U X UT,X ∈V, where U runs over
the groupOn ofn×northogonal matrices,
X↓:= diagλ(X) forX ∈Sn, whereλ(X) denotes the vector of theeigenvalues ofX
arranged in decreasing order on the main diagonal, and
diagx stands for the diagonal matrix with x∈ Rn. We adopt the convention that
the members ofRn are rown-vectors. Then (V, G,(·)
↓) is an ND system by virtue of
Also, it is well known thatD=V↓is the convex cone ofn×nreal diagonal matrices
with decreasinglyordered diagonal entries.
In addition, (W, H,(·)↓) is an ND sy stem for
W := the linear space ofn×ndiagonal matrices,
H := the group of operators of the formX→U X UT,X ∈W, where U varies over
the groupPn ofn×npermutation matrices.
The orthoprojectorP fromV ontoW is given by
P(X) = diag ∆(X) forX ∈Sn,
(3.1)
where the symbol ∆(·) means “the diagonal of”.
Conditions (i)-(iv) of Section 2 are fulfilled (see [16, Example 4.1] for details). In
particular, inequality(2.1) takes the form of theclassical Schur inequality:
∆(U X UT)mλ(X) forX ∈Sn andU ∈On.
(3.2)
Here the relationmis the ordinary majorization onRn defined as follows (see [13,
p. 7]). Given two vectorsa, b∈Rn, we say thatamajorizes b if
m i=1
b[i]≤
m i=1
a[i] form= 1, . . . , n,
(3.3)
with equalityfor m = n. By c[i] we denote the ith largest entryof a vector c =
(c1, . . . , cn)∈Rn. A functionψ:Rn→Ris said to be Schur-convex ifψ(b)≤ψ(a)
wheneverbmafora, b∈Rn (cf. [13, p. 54]).
It is known thatmis the group majorization onRninduced bythe permutation
groupPn (see [4, p. 16]). There is a closed connection between theH-majorization
H onW and the ordinarymajorizationmonRn. Namely ,
diagyH diagx iff ymx forx, y ∈Rn.
(3.4)
To see this, employthe formula
U(diagx)UT = diag (xUT) forU ∈Pn andx∈Rn.
(3.5)
Applying Corollary 2.2 for
C:= the convex cone ofn×ndiagonal matrices with nonnegative diagonal entries,
C:= the entrywise ordering onW,
max := the maximum operator with respect to the entrywise orderingC onW,
we get
Corollary 3.1. Let Mj, j = 1, . . . , k, be a collection of pairwise commuting
n×n symmetric matrices, and let U0 be an n×n orthogonal matrix such that the similarity operatorg0:=U0(·)UT
Let ψ : Rn → R be an Rn
+-increasing function. If ψ is Schur-convex then U0 maximizes the functional
f(U) :=ψ(max
j ∆(U MjU T))
over alln×northogonal matricesU.
Proof. Apply(3.1)-(3.5) and Corollary2.2 for the functionϕ(X) :=ψ(x), where
X := diagxforx∈Rn.
The above result extends the mentioned theorem of Berman and Plemmons [2].
To see this, use Corollary3.1 for the functionψ(x) :=
n i=1
xiforx= (x1, . . . , xn)∈Rn.
We are now interested in using Theorem 2.1 and Corollary2.3 to obtain a result
for singular values of matrices inMn(see Theorem 3.2) under the action of the group
G0 of orthogonal equivalences. However, there are no G0-invariant convex cones in
Mn (exceptMn and{0}). To avoid this difficulty, we use embedding of the spaceMn
inM2n. Namely, letV be the linear space of all 2n×2nmatrices of the form
αI X
XT αI
,
whereX is ann×nreal matrix,Iis then×nidentitymatrix andαis a real number.
PutGto be the group of all linear mapsg fromV toV of the form
αI X
XT αI → U1 0 0 U2 αI X
XT αI
U1 0
0 U2
T
=
αI U1XUT
2
(U1XUT
2)T αI
,
whereU1 andU2 run over the group ofn×n orthogonal matrices. Then the
Singu-lar Value Decomposition Theorem and the von Neumann trace inequalityfor n×n
matrices implythat (V, G,(·)↓) is an ND system for the normal map (·)↓ defined on
V by
αI X
XT αI
↓
:=
αI diags(X)
diags(X) αI
,
wheres(X) := (s1(X), . . . , sn(X)) is then-vector of thesingular values ofXarranged
in decreasing order (cf. [1, p. 106], [4, pp. 17-18]). That is, the numberss1(X)≥. . .≥
sn(X) are the nonnegative square roots of the eigenvalues of the positive semi-definite
matrixXTX. The range of the normal map (·)
↓ is the convex cone
D=
αI diagx
diagx αI
:α∈R, x∈Rn
+↓
,
whereRn
+↓ is the set of nonnegative realn-vectors with decreasinglyordered entries.
The system (W, H,(·)↓) is given bythe space
W :=
αI diagx
diagx αI
:α∈R, x∈Rn
and bythe groupH of linear operators fromW toW of the type
αI diagx
diagx αI
→
αI U1diagx UT
2 U1diagx UT
2 αI
,
whereU1 andU2 run over the group ofn×ngeneralized permutation matrices (i.e.,
matrices with exactlyone nonzero entry±1 in each column and each row) such that
U1diagx U2T = diag (±xi1, . . . ,±xin) (3.6)
for anychoice of±signs and of permutationxi1, . . . , xin of the entries ofx. Therefore
βI diagy
diagy βI
H
αI diagx
diagx αI
iff β=α and yaw x.
(3.7)
Here the relationy aw x means that (3.3) holds fora := (|x1|, . . . ,|xn|) and b :=
(|y1|, . . . ,|yn|) (cf. [4, p. 16]). Observe that
yawx iff (|y|↓,−|y|↑) m (|x|↓,−|x|↑)
(3.8)
iff (α1n+|y|, α1n− |y|)m(α1n+|x|, α1n− |x|)
(cf. [1, p. 107]), where|x|denotes the vector of the moduli of the entries ofx, and the
vector |x|↓ (resp. |x|↑) consists of the entries of |x| arranged decreasingly(resp.
in-creasingly).
The orthoprojectorP fromV ontoW is given by
P
αI X
XT αI
=
αI diag ∆(X)
diag ∆(X) αI
,
where ∆(X) stands for the diagonal ofX.
Denote by L2n the Loewner cone of all 2n×2n positive semidefinite matrices.
Notice thatL2n isG-invariant. Define
C:=L2n∩W.
Evidently,Cis anH-invariant convex cone. In addition,C is the cone preordering
onW such that
βI diagy
diagy βI
C
αI diagx
diagx αI
(3.9)
implies
(β1n+y, β1n−y)↓ ≤ (α1n+x, α1n−x)↓,
(3.10)
where ≤is the entrywise ordering on R2n and then-vector 1
n consists of ones. To
Take H0 to be the subgroup of H consisting of all operators of the form h0 =
U0(·)I, where U0= diag (±1, . . . ,±1) is a sign change matrix.
With the above notation, and from Theorem 2.1 and Corollary2.3, we obtain Theorem 3.2. Let αj and Mj for j = 0,1, . . . , k be, respectively, real numbers
andn×n real matrices, and let
Mj:=
αjI Mj
MT j αjI
be the corresponding matrices in V such thatMjL2n
M0. Assume that there exists an operatorg0 in Gsimultaneously sending theMj intoW, that is
g0Mj =
αjI diagσ(Mj)
diagσ(Mj) αjI
for j= 0,1, . . . , k,
where σ(Mj) := (±si1(Mj), . . . ,±sin(Mj)) ands1(Mj)≥. . . ≥sn(Mj)≥0 are the singular values of Mj with any choice of ± signs and any permutation i1, . . . , in of
1, . . . , n.
Let ψbe a real function which is Schur-convex and entrywise increasing on R2n.
Then
max
1≤j≤kψ(αj1n+ ∆(U1MjU T
2)U0, αj1n−∆(U1MjU2T)U0)
(3.11)
≤ψ(α01n+s(M0), α01n−s(M0))
for any orthogonal n×nmatrices U1 andU2, and for someU0= diag (±1, . . . ,±1)
such that ∆(U1M0U2T)U0=|∆(U1M0U2T)|. Proof. Consider the functions
ϕ(X) :=ψ(α1n+x, α1n−x) and ϕ(X) :=ψ(α1n+|x|, α1n− |x|)
(3.12)
forX=
αI diagx
diagx αI
withx∈Rn andα∈R. It is obvious thatϕ(X) =ϕ(X)
for nonnegativex.
It follows that ϕis C-increasing, because (3.9)-(3.10) are met, and ψ is
Schur-convex, permutation-invariant and entrywise increasing. Using (3.6) and (3.12) we
obtain that ϕ is H-invariant. Applying (3.7)-(3.8) and (3.12) we deduce that ϕ is
H-increasing. ThusϕisH-increasing onW0:={X∈W :x∈Rn
+}.
We have that
P gMj =
αjI diag ∆(U1MjU2T)
diag ∆(U1MjU2T) αjI
for anyg=g(U1, U2)∈G. Using Corollary2.3 and (2.9) withg0=h0=U0(·)I, one
obtains
max
1≤j≤kϕ(
αjI diag ∆(U1MjU2T)U0
diag ∆(U1MjU2T)U0 αjI
)
≤ ϕ(
α0I diags(M0)
diags(M0) α0I
),
4. Concluding remarks. A central role in the proof of Theorem 2.1 is played
bySchur type inequality(2.1) and bythe double inequalitiesP gwj C P gwH w
with the mediate vector P gw. Such kind of a relation is known in the literature as
weak group majorization (see [8, p. 120]).
To be more precise, assume ≺≺ is a preordering on W which is H-compactible,
i.e.,≺≺is invariant under finite convex combinations of elements ofH:
y≺≺x implies
i
αihiy≺≺
i
αihix
for hi ∈ H and αi >0 with iαi = 1. In particular, if≺≺ is a cone preordering
induced byanH-invariant convex cone, then ≺≺is H-compactible (see [8, p. 120]).
Given vectorsx, y∈V, we say thatyisweakly H-majorized byx(in symbol,yH,w
x), if there existsz∈W such that y≺≺z andzH x.
It is clear that if a functionϕ:W →Ris both≺≺-increasing andH-increasing,
thenϕ(y)≤ϕ(x) wheneveryH,wx. In particular, this is valid ifϕis≺≺-increasing,
H-invariant and convex [8, Thm 3].
Summarizing, our results can be extended to anH-compactible ordering≺≺
in-stead of the cone preordering C, provided the linear operator P g is ≺≺-isotone
for each g ∈ G. Then Theorem 2.1 remains true for any H,w-increasing function
ϕ:W →R.
Also, further generalizations are possible forH-stochastic operatorsLin place of
P g. Recall that a linear operatorL:W →W is said to beH-stochastic ifLxH x
for x∈W (cf. [17, Thm 3.3]). Namely, if wj ≺≺ w then ϕ(Lwj)≤ϕ(Lw) ≤ϕ(w)
providedL is both≺≺-isotone and H-doublystochastic, and, in addition,ϕ is both
≺≺-increasing andH-increasing. Moreover,ϕ(Lwj)≤ϕ(w) and max
j ϕ(Lwj)≤ϕ(w)
for anyH,w-increasing functionϕ.
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