Linear and Multilinear Algebra, 1987, Vol. 21, pp. 253-259

Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 March 20 00:16 At: Chi-Kwong] [Li, By: Downloaded Photocopying permitted by license only 1987 Gordon and Breach Science Publishers, S.A. Printed in the United States of America

Inclusion Relations Between Orthostochastic Matricesand Productsof Pinchinq-- Matrices

YIU-TUNGPOON

Iowa State University, Ames, lo wa 5007 7 and NAM-KIU TSING CSty Poiytechnic of HWI~K~tig, ;;e,~g ,?my"; 2nd A~~KI:Lkwe~ity, .Auhrn; Al3b3m-l36849

Let C be thc set of ail n x n urthusiuihaatic natiiccs and ;"', the se! of finite prnd~uctpof n x n pinching matrices. Both sets are subsets of a,, the set of all n x n doubly stochastic 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 sij = luijl2 , i,j = 1, . . . , n. 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 2 54 Y. T POON AND N K. TSING Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 for some permutation matrix Q and 0 < t < I, then we call P a pinching matrix. Let .Yn be 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.

It is interesting to investigate the inclusion relations between .<;. c ,, and R, for higher n. Given a subset -'/' c R, and an n-tuple c of complex numhcrs. let " Y' - .. '. - ,LS:' S E .If;. Clcarly. cR, contains both cC,, and c.'P,. On the other hand, if all components of c are real then a result of Horn [5] says that

d',= cR,. (2) Also, Goldberg and Straus [31 proved that

then one would be able tt? p!-!?ve !3! frr,rr! i?! H!lt thrv, ~c>n?in::~cj~ ------t~ sh~u.

jr_._ ^ .- tii~!ji~i: is LLVLP~~~bi~ii because i4i does not hold [(I: n ;: 4. in this p:tyr. we are going to prove that for arbitrary complex TI-tupiec we have (see Corollary 1) c.'P, c cl',. (5)

Thus (2)can actually be derived from (3) Althnugh (5) gives #3 c !' (Corollary 2), we are going to prove that ..P, g C, for n 3 4. In [6Ij- - the authors gave an example of a 3 x 3 d.s. matrix S which is not in ./P,. Howeverj S is no? 0.s. ~ln.~ of whether ! . c .Y3was ripen in [6]. We will answer this by showing that -6 ('?. This also gives examples to show that (4) does not hold for all n 3 3.

2, INCLUSION RELATIONS

Yt (A)= {Diag UAU*: U is unitary) ORTHOSTOCHASTIC MATRICES 2 55 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 March 20 00:16 At: Chi-Kwong] [Li, By: Downloaded

Given a complex n-tuple c = (c,,. . . , c,), let A be a normal matrix with eigenvalues c,. . . . , c,. Then it can be easily shown that ff '(A)- cf,. Applying 1,emma 1 to # '(A).we have

THFOREM1 Let c be a complex n-tuple. Therl !or every S E Cnand

p c .!Pn 3 :here e.xicr T F wrhthat cSP= cT. Taking S = I, in Theorem 1,we have COROLLARY1 c.Yn P, ct , jor every complex n-rj'piec. COROLLARY2 .P3 c C3. Proof Let c = (O,1,v/-l) Given P = (pij)E .Y3, let S = (sij)E (' satisfy cP = cS. Then for each j = 1,2,3, we have

p2, + ,,/ I - ~p~~= s2j + ls3j =.ptj=szi ad P.~~=.:.- 3~

->PI. 1 pzi p:; 1 s,; S3] Sli - ., = - - = - - = 1herc!'crre P = .?' 0

Naturally. Corollary I and 2 suggest the posbibiiity of .lp,c (. ,. Bul. we have

THEOREM2 Fur n 2 4, .Y,, @ (',. Proof Consider the 4 x 4 matrix r3 3 4 21 2 56 Y. T. POON AND N. K. TSING Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 Here all the unspecified entries are zero. Clearly T % I, -,E .<. Suppose T / In-, is also O.S.Then there exists a 4 x 4 unitary matrix U = (uij) such that T = lUI2. Without loss of generality. we may assume ui, and I*,,be rea! pcsiti:re for 1 6 i,,j < 4 (otherwise consider D, LrD,. u here D, and D, are diagonal unitary matrices, instead). By considering the II:I:G:I):VL~ULIulr ti:^ Ls: and third coiumns of i;. we have ju2,. u33j =

((I /,/j)w, (l/,,6)w2).where (i, is the primitive cube root of unity. Then by considering the inner product of the second and third columns of Li ? 1 wc gci {zL,~,u3,) = {)) or {$B,iw-j. But then the inner product of the first and second columns of U cannot be zero. Hence U does not exist. In [6],examples arc givei: to show that en @ .<$ for n 2 4. In fact we

Prooj We consider the case n = i first L~e!

Then S = /UI2 where U is the unitary matrix Ti I 27

and hence S is 0s. To show that S+.Y3, suppose the contrary that

for some picching rr,a!rices PI.. . . , P,. A p;n&ing matrix Pis said to be proper if it can be expressed as in (1) with 0 < t < 1. Let m be the smallest number such that P, is proper. This m must exist since S is not a permutation matrix. Then ORTHOSTOCHASTIC MATRICES 257 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 for some permutation matrix Q and

for some O < t < 1. From the choice of' m, R = PI , . . Pm-,Q is a permutation mail is. Thiia wz hzve

and hence ~'4~~~~-..PkPI... PmlQ= itl, IS

5;- 4 5 - 8: - 47

- ,L, :*..\-I As the matrix (6)is a praduct oforder .3 pinching matrices. it mast he i:.;. . -. -..- . . <,ii3i-j,-i.- i, t!tt- inv e<~rr,~c!,,;tgi-v ; ! J- . . . >. ti , - ! L ! .b. ~;xt;ix 0' = (14,~).'l'hcn we havc

which is a contradiction to O < r < 1. Thus, 5'4 P3. / S is an 0,s. For the case n > 3, we are going to prove that I,3 matrix which is not in 9". First we see from (1) that if (P,~)F R, is a pinching matrix, then we have 2 58 Y T POON AND N. K.TSING Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 March 20 00:16 At: Chi-Kwong] [Li, By: Downloaded forsomel ,

So there exists k with p, = 1 = qkj and R(i1 j) = P(iI k)Q(k(j). To finish the proof, it suffkes to show that if P = (pi,) E .ewith pij = 1 then ~(i1j) .?,- ,_,.Let P = P, . . . Pm for some pinching matrices , by induction on m:

for some k. w

3. SOME REMARKS

R~mnrk1 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 Cf3and Pe.P3.This also proves Corollary 2. Remark2 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 C3and C, is compact, it follows that In-,/ 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 (PI. . . P,,,)and n,"=,+, Pkare d.s. Using the argument developed in the prod of Theorem 3, for each 1 < i < n - 3, we can find a sequence ORTHOSTOCHASTIC MATRICES 259 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 {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~mcirk.. 3 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 aid on!y if c,, . . . , c, are colinear on the complex plane. (8

app!y:fig.. . - v:e anv rnmnlru c = Thus C=rol]ary !. ha1.e that for --J -----r*--- '-n-tup!e il-., . (.?.- . . . . 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!ysf genrra!ized numcrica! ranges, i.inuur A iyrhru utd A!I!~.27 11 9791, 69 79. [2] ?. H. Au-Yeung and F. Y. Sing. A remark on the generaltzcd numerical range of a normal matrix, Glu~yowMuih. i. 18 (l877j, 179-iRO. [?I M. Goldbeig ad 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 .Zf,~rl~ -.. . .. 76 (1953), h2%630. [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"L,L 10 (!%?I), ! 73 !Z2.