for some permutation matrix Q and 0 < t < I , then we call P a pinching matrix. Let .Yn be the .... and D, are diagonal unitary matrices, instead). By considering the.
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Linear and Multilinear Algebra, 1987, Vol. 21, pp. 253-259
Photocopying permitted by license only 1987 Gordon and Breach Science Publishers, S.A. Printed in the United States of America
Inclusion Relations Between Orthostochastic Matrices and Products of Pinchinq -- Matrices YIU-TUNG POON Iowa State University, Ames, lo wa 5007 7
and NAM-KIU TSING CSty Poiytechnic of H W I K~~ t i g ;;e,~g , ,?my"; 2nd A ~ ~ KLI k:w e ~ i t y.,A u h r n ;
Al3b3m-l36849
Let C be thc set of ail n x n urthusiuihaatic natiiccs and ;"', the se! of finite prnd~uctpo f n x n pinching matrices. Both sets are subsets of the set of all n x n doubly stochastic
a,,
matrices. We study the inclusion relations between C, and 9 ' " and in particular we show that.'P,cC,b~t-~p,t~forn>J,andthatC~$~~fori123.
1. INTRODUCTION
An n x n matrix is said to be doubly stochastic (d.s.) if it has nonnegative entries and ail row sums and column sums are 1. An n x n d.s. matrix S = (sij)is said to be orthostochastic (0.s.)if there is a unitary matrix U = (uij) such that 2 i ,j = 1, . . . ,n. sij = luijl , Following [ 6 ] , we write S = IUI2. The set of all n x n d.s. (or 0s.) matrices will be denoted by R, (or ( I , , respectively). If P E R, can be expressed as 1-t
(1
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2 54
Y. T POON AND N K . TSING
for some permutation matrix Q and 0 < t < I , then we call P a pinching matrix. Let .Ynbe the set of all tz x tz d.s. matrices which can be expressed as a finite product of pinching matrices. For n = 2 we have
./Pz = (
2
=
R2.
I t is interesting to investigate the inclusion relations between . 3, we are going to prove that I , 3 / S is an 0,s. matrix which is not in 9". First we see from (1) that if ( P , ~F) R, is a pinching matrix, then we have
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2 58
Y T POON AND N. K. TSING
forsomel , < r < s , < n a n d O < t , < l . L e t P = ( p i j ) ~ R , w i t h p i j =lfor some i, j. We use P(i ( j )to denote the matrix obtained by deleting the i-th row and j-th column of P . Clearly, P(i Ij) E R,- If P is a pinching matrix, then from (7), we have that P(i I j ) is either a pirrhing matrix or a permutation matrix. In both cases. ~ (1 j)iE . t - ,because every pc;i~iiuiaiiuili b il p ~ u d u UL ~ i~ 1 a 1 1 ~ p o ~ i ~Let i o nPs .= ipijj, Q = (qij)and R = !rij!€ R, with R = PQ and rij = 1 for some i , j . Then we have
So there exists k with p, = 1 = qkj and R(i 1j) = P(i I k)Q(k ( j ) . To finish the proof, it suffkes to show that if P = (pi,) E with pij = 1 then ~ (1j )i .?,-,_,.Let P = P , . . . Pm for some pinching matrices , by induction on m :
.e
for some k .
w 3. SOME REMARKS R ~ m n r k1 1-Jsing the characterization of !' given .Ac-Y~.ng 2nd Poon in [I], we can show that SP is O.S.for any S E C f 3 and P e . P 3 . This also proves Corollary 2.
Remark 2 The proof in Theorem 3 actually shows that S cannot be expressed as a product of a proper pinching matrix and an 0.s. matrix. Since.4 c C 3 and C, is compact, it follows that I n - , / S cannot even be represented as a limit of a product of pinching matrices. For if
where { P , ) is a sequence of pinching matrices, then each of the first (n - 3 ) rows of ( P , . . . Pm) must contain exactly one 1, as both ( P I . . . P,,,) and n,"=,+, Pk are d.s. Using the argument developed in the p r o d of Theorem 3, for each 1 < i < n - 3, we can find a sequence
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259
ORTHOSTOCHASTIC MATRICES
{i,, i,, . . .f such that i, = 1 and (Pk)ik.ikh, = 1 for all k. Let Pk be the 3 x 3 pinching matrix obtained by deleting all the (ik)throws and the (ik+ ,)th columns From Pk (1 6 i < n - 3). Then
can be written as a product of a proper plnchmg mair~xand aii o.s matrix. This is a contradiction. R..~ m c i r k3 Let c = (c:; . . . .c,) be a complex n-tuple and A an n x n . normal matrix with eigenvalues c,, . . . . c,,. Au-Yeung and Sing 121 proved that Yi ( Aj i= cC ,j is convex if a i d on!y if c,,
. . . ,c, are colinear on the complex plane.
(8
Thus app!y:fig .. . C=rol]ary !. ve: ha1.e that for anv --J rnmnlru -----r*---n-tup!e c i l - ,. . - . . . . c,;;.iil! three condiiions i2i. (31. a : ~ !(S! are equivale~:!. -
' -
=
(.?.
We wish to thank Prufcaaor G. N.dc Olibeil-a foi- bringing this problem and !be paper 61 to our attention al rhc Auburn Matrix Theory Confeienci. l9S6. Wc also thank the referee for pointing out an error ix an earlier version of this paper.
References [i j Y . H.Au-Ysuiig and Y . T. Poon, 3 x 3 orthostochastic matrices axd :he c~nvexi!ys f genrra!ized numcrica! ranges, i.inuur A iyrhru u t d A ! I ! ~27 . 119791, 69 79.
?. H . Au-Yeung and F. Y. Sing. A remark on the generaltzcd numerical range of a normal matrix, G l u ~ y o wMuih. i. 18 (l877j, 179-iRO. [?I M. Goldbeig a d E. G. Siiaiis. Ele~qcntaryinclusion relations fcr genera!ized numerical ranges. Lineur illqehru und 4ppl. 18 (1977), 1-24. [4] G. H. Hardy, J. E. Littlewood and G. Polya, 1nequulirie.s. Cambridge University Press. 1952. [5] A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, ilrner. J , ~..r..l ~76 (1953), h2%630. .Z f -. [6] M. Marcus. K . Kidman and M. Sandy. Products of elementary doubly stochastic matrices, Lirirur- rind iLfuitihrur Alyrhi~i15 (1984), 331-340. 171 R . F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc.. Edinhuryh Muth. Soc. 21 (1903). 144 157. [8] N . K . Tsing, On the shape of the generalized numerical ranges, Linrur und Mulrilin~ar .g'..f+;.o 10 (!%?I), ! 73 !Z2. [2]
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