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Linear and Multilinear Algebra, 1994, Vol. 37, pp. 221-223
Reprints available directly from the publisher
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@ 1994 Gordon and Breach Science Publishers S.A.
Printcd in Malaysia
On the Convex Hull of the Multiform
Numerical Range
YIU TUNG POON
Department of Mathematics, Iowa State University, Arnes, IA 5001 1
(Received May 6, 199_1)
Let A1,.. . , A m be n x n hermitian matrices. Define W(A1,...,.4,,) = { ( x A l x * , . ,r A , r ' ) : x 6. Cn;
x x * = 1). We wlil show rhar every point in tile convex huii o i ii'(Al,.. . , A m ) caii bc repiesemed as a
convex combination of not morc than k(m:n) points in W ( A I ...,
, A m ) ,where k(mln) = minjn, [,GI
+
1.
INTRODUCTION
Let Al, ...,Am be n x n herrnitian matrices. Define the multifarm numerical range
where vectors in Cn are taken as 1 x n matrices and x* denotes the complex conjugate transpose of x . Then W(A1,. ..,Am) is a compact, pathwise connected subset of
Rm.For m = 2, W(AI,A2) is the classical numerical range introduced by Toeplitz
[8]. Hausdorff [6] proved that W(Al,A2) is convex and remarked that for n = 2,
W(Al, A2,A3) need not be convex. A result of Au-Yeung and Poon [I] shows that
if n > 3, W(A1,A2,A3) is always convex. However, for n 2 2 and m 2 4, there exist n x n hermitian matrices A1,. ..,A, such that W(A1,. ..,A,) is not convex. The
study of multiform numerical ranges is closely related to the computation of structured singular value in control theory [2, 51.
The concept of structwed singular value was introduced by Doyle [2] to study certain feedback systems. Fan and Tits [5] have given an algorithm for computing an
upper bound for the structured singular value p for any complex n x n matrix A
with a given block-structure of size m. (See [2] and [5] for details.) Their algorithm
depends on the analysis of the convex hull of W(A1,. ..,Am) for a family of multiform numerical ranges W(AI,. ..,A,). When the block-structure is of size _< 3, the
vaiue obtained from their afpiithiii is actually equal ?a p hecmse w(A1,. ..,Am)
is convex. The equality may not hold when m 2 4 [5]. To date, the computation
of the structured singular value remains an open problem [3, 51. We will study the
geometry of conv W(A1, ...,A,), the convex hull of W(A1,. ..,Am).
Since conv W(A1,. ..,A,) is a convex subset of Em, by the Carathedory's Theorem [7], every point in conv W(A1,. ..,A,) can be expressed as a convex combina-
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tion of at most m + 1 points in W ( A i ,...,A,). In a recent article, Fan [3]showed
that every point in conv W ( A 1 ,.,.,A,) can be expressed as a convex combination
of m - i points in W ( A 1 ,...,A,) and used this result to prove a iifting theorem. in
[a],Fan and Lin lower the bound to m - 2 for m > 3 and m 3 for in = 5 6 . Thus.
a natural question is to find the smallest number k ( m , n ) such that every point in
conv W(.41,. . . , A m )can be expressed as a convex combination of k ( m ,n ) points in
W ( A 1 , . .,A,). The main result in this paper is:
-
THEOREM1 Let m be a positive integer and A1,...,A, be n x n hermitian matrices. Then every point in conv W ( A 1 , . .,A,) can be represented as a convex combination o f k ( m ,n ) points from W ( A 1 , . . ,A,), where k ( m ,n ) = min{n, [ f i ]+ 6,2,,
}.
Here, [ x j denotes the greatest znteger less than or equal to x and d,,, is the Kronecker
delta.
+,
2.
PROOF OF THEOREM 1
LEMMA2 A p l t t i w ~ t lR m i ~ t be
l t ' ~ p ~ t ' ~ >rc>e dic iutil t ' ~i u ~ l b i n ~ i ~UJ
u t tt p01tzi~
from W(.41,...,.A,,, ) zf and on& ij there exist X I , .. ., x, c C"' F U C that
~
w=
xiam*;
i=l
and
~ x i x f = l .
(1)
i =l
Proof Let w be a convex combination of r points from W ( A 1 , . .,A,). Then
there exist unit vectors PI,. ..;yr E C n and numbers a!, .. .; ar 1 0; C:;i ai -, 1 such
that w = C:=l
ai(yiAly;,. ..,ylAmyi).Let xi = &yi for i = 1, .. . , r , then we have
(1).
The proof for the other direction is similar.
Let W ( ' ) ( A 1 ,. ..,A,) be the set of w which can be expressed in (1). In [ I ] , AuYeung and Poon gave the following generalization of the Toeplitz-Hausdorff's theorem on the convexity of W ( A l ,A2):
THEOREM^ Let l I r < n - l a n d m < ( r + 1 ) 2 - 6 n , r + 1 a n dA1,...,A , b e n x n
hermitian matrices. Then W ( ' ) ( A ~
. .,. , A,) is convex.
Proof of Theorem 1 It is easy to see that w can be expressed in ( 1 ) if and
only if there exists an n x n positive semidefinite matrix X of rank 5 r such
that w = (Tr A I X ,...,Tr A,X) and TrX = 1, where TrB denotes the trace of B.
Hence, W(')(A1,.. .,A,) is always convex for r = n. Therefore Theorem 1 follows
from Lemma 2, Theorem 3 and the fact that the smallest integer r satisfying
rn
m < ( r + 1)2 - 6n,,+1 is equal to [ f i l + 6n2m+1.
Remarks
1. The bound min{[fi] + 6nz,m+l) in Theorem 1 is best possible. See [ I ] .
2. Suppose A l , . . . , A , are n x n real matrices and we use x, xi E Rn in the defini,
Then W ( ' ) ( A .~. .,A,)
,
is convex
tion of W ( A 1 , . .,A,) and W ( ' ) ( A .~. .,A,).
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MULTIFORM NUMERICAL RANGE
for
ji]. Hence, Theorem 1 also holds with
ACKNOWLEDGMENT
The author wants to thank Professor Michael Fan for bringing the problem to
his attention and the preprints [3] and 131, and Professor C. K. Li for some helpful
suggestions.
References
!. Y. H. Au-Yeung and Y. 1: Poon, A remark on the. convexity and positive definiteness concerning
hermitian matrices, Southeusl Asian Bull. Matk 3 (1979), 85-Y2.
2. J. C. Doyle, Analysis of feedback systems with structured uncertainties, Proc. IEEE-D 129, 6 (1982),
242-250.
3. M. K. H. Fan, A lifting result on structured singular values, preprint.
4. M. K. H. Fan and C. Y . Lin, A gecmetric chracteriza!ien of multifnrm numerical range, preprint.
5. M. K. Wn and A. L. Tits, m-form numerical range and the computation of the structurd singular
value, IEEE Trans. Auiomar. Control AC-33 (1988), 284-289.
6. E :p ~ ~ d o i fDei
f , 'JJe.":v,::l;.orra(
Bi!inear[crm, :vg!!~.Z 1 (I?!?), 314-716.
7. R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1971).
8. 0 . Toeplitz, Das algebraishe Analogon zu einem Satze von Fejtr, Math. Z. 2 (1918), 187-197.