Linear and Multilinear Algebra, 1987, Vol. 21, pp. 253-259 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 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 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 2 sij = luijl , 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 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 Cn and p c .!P n 3 :here e.xicr T F wrh that cSP = cT. Taking S = I, in Theorem 1, we have c.Yn P, ct ,jor every complex n-rj'pie c. COROLLARY1 COROLLARY2 .P3 c C 3. Proof Let c = (O,1, v/-l)Given P = (pij)E .Y3, let S = (sij)E (' satisfy cP = cS. Then for eachI j = 1,2,3, we have p2, + ,,/ - ~p~~= s2j + ls3j =.ptj=szi ad P.~~=.:.- 3~ ->PI. = 1 - pzi - p:; = 1 - s,; - S3] = Sli - ., 1 herc!'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 forsomel ,<r<s,<nandO< t,< l.LetP= (pij)~R,withpij=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 Ij) is either a pirrhing matrix or a permutation matrix. In both cases. ~(ij) E .t-, because every 1 pc;i~iiuiaiiuilib il p~udu~iUL ~1a11~po~i~ions. Let P = ipijj, Q = (qij)and R = !rij!€ R, with R = PQ and rij = 1 for some i,j. Then we have = = j) = (j). So there exists k with p, 1 qkj and R(i 1 P(i I k)Q(k To finish the proof, it suffkes to show that if P = (pi,) E .ewith pij = 1 = then ~(i1 j) .?,- ,_,.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 Cf3 and Pe.P3.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 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,"=,+, Pk are 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,, .