The Bullerin of the lnternqtionol Lineor Algebrc Society flL lLl ns rsl Servingthe Internotionol Lineor Algebro Community

IssueNumber 22, pp. l-32, April 1999

Editor-in-Chief: George P. H. SryAN [email protected], styan@togethe r. n et Departmentof Mathematicsand Statistics tel. (1-514)398-3333 McGillUniversity, Burnside Hall, Room 1005 FAX(1-514) 398-3899 805 ouest,rue SherbrookeStreet West Montreal,Qu6bec, Canada H3A 2KG EditorialAssistants: Shane T. Jensen & EvelynM. Styan

Senior AssociateEditors: Steven J. LEoN,Chi-Kwong Lt, Simo PUNTANEN,Hans JoachimWeRrueR. AssociateEditors: S. W. Dnuny, StephenJ. KrRxllno, PeterSeunl, Fuzhen ZHANG, PreviousEditors-in4hrbf; Robert C. THoMpsoN(198H9); StevenJ. Leoru& Robert c. THoMPSoN(1989€3) stevenJ. LEoN(1993-199a); Steven J. LEoN& GeorgeP. H. Srvnn (1994-1997).

Commentsfrom Thirty Years of TeachingMatrix Algebra to AppliedStatisticians (S. R. Searle)..3 Requestfor Feedbackon LinearAlgebra Education (Frank Uhlig) ...... s DennisRayEstes:1941-1999 ...... 7 FeliksRuminovich Gantmakher and TheTheory of Matrices ...... 12 ILASPresidenWice-President's Annual Report: April 1999 (R. A. Brualdi& D. Hershkowitz).... 6 ILASTreasurer'sReport: March 1, 1998-February 28, 1999 (James R. Weaver) ...... 22 New& ForthcomingBooks on LinearAlgebra and Related Topics: 1998-1999 ...... 10 Book Reviqts DavidA. Harville:Matrix Algebra from a Statistician'sPerspective (Jiirgen GnoB) ...... 9 A. M. Mathai:Jacobians of Matrix Transformations& Functionsof MatrixArgumenfs (S. Provost)9 Report on Linear Algebra Event Attended December11-14: Fort Lauderdale, Florida ...... 15 Forthcoming Linear Algebra Events June7-8, 1999:Regina, Saskatchewan ...... 19 July15-16, 1999: Coimbra, Portugal ...... 1g Julyl*22,1999:Barcelona,Spain ...... 19 August6-7, 1999: Tampere, Finland ...... 19 June26-28, 2000: l,lafplio, Greece ....21 July12-14,2000: , ....-...-.21 December9-13,2000:Hyderabad,lndia . .._.2j IMAGEProblem Corner 18-1:5x5 ComplexHadamard Matrices ...... 25 19-3:Characterizations Associated with a Sumand Product ...... 25 19-4:Eigenvalues of Non-negativeDefinite Matrices ...... 25 19-5:Symmetrized Product Definiteness? . . -...... 26 21-1:Square Complex Matrix of Odd Order ...... 27 21-2:TheDiagonal ofan lnverse ...... 29 21-3:Eigenvalues and Eigenvectors of Two Symmetric Matrices ...... 29 21-4:Square Complex Matrix, its Moore-PenroseInverse and the LdwnerOrdering .... .30 21-5:Determinants and g-lnverses ...... 91 NewProblems ...... 92

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April 1999 page3

Commentsfrom ThirtyYears of Teaching Matrix Algebrato AppliedStatisticians

by ShayleR. SrnRLE, Cornell University

Abstrrct. A few simpleideas gleaned from teachingmatrix algebra of the term's gade; in some years nothing. Yet there was mo- aredescribed. For teachingbeginners how to provetheorems a useful tivation: an early handout explained that the term grade would 'Think mantrais: of somethingto do: do it andhope." be F if there was any failure to do homework, no matter how much help had been grven. To me, homework is for shrdents I . Introduction: Hooked on Marix Algebra As a Master's stu- to leam to do mathematics; it is not inquisitorial for assessing dent of mathematics in New Tnaland, I had a course on matri- how much they have learned. That is the purpose of exams- ces from Jim Campbell who had done his under A. and students were told that exam questions would be similar to C. Aitken at Edinbutgh. That one could have something like homework problems-indeed, most exarns included at least one - - AB - 0 without A 0 or B 0 fascinated me-and I was of the homework problems. hooked. Eight years later, as a graduate student at Cornell, I For a number of years the course was split into two back-to- gave an informal 1957 suulmer course on matrix algebra; and back 7-week modules, each of two credits and two exams. This after returning to Cornell on faculty that course developed in came about because agricultural economics graduate students 1963 into a regrrlar fall semester offering, Matrix Algebra, in asserted the second part of the course was too theoretical. So the Biometrics Unit. I taught it until 1995, when I retired; and it with the split they did only the first part and got no further than still continues with 20-30 students a year. inverting a mahix (not even to rank and linear independence). One may well "*hy ask matrix algebra when linear algebra So their attendancedwindled almost to z.ero,coinciding with in- is widely taught in Mathematics?' Severalreasons: I'm no ge- creasing enrolment of statistics undergraduates. For them I soon ometer. Dropping perpendictrlar a from 8-spaceto 4-spacehelps learnt that two modules involving four exilms was too weighty my intuition not one bit. I cannot illustrate that perpendicular on a presentation, and after 10 years the course reftrrned to a reg- a blackboard, and I cannot use my fingers to do so either. I have ular 3-lecfine, l*week routine. That was less intensive, more many kindred spirits in this regard. They come from students enjoyable for the shrdents,and under less pressureI believe they minoring in statistics for use in their down-to-earth majors such leamt more, and more easily. as agfonomy, agricultural economics, plant breeding, pomology and animal breeding. They find no joy in geometry, but they can 3. Terching How to do Mahemqtics. How do we do mathemat- get the hang of algebra-even to the stageof enjoying it, despite ics? How do we learn how to do mathematics? It seems to me job -of the 8 a.m. hour for the course! That lasted 25 years, whereupon that we don't do a good of teaching this how to do alge- bra, for I changed it to 9 a.m., grving the college curriculum cornmittee example. Many studentshave difflculty in leaming this: the reason"After 25 years,I'm tired of 8 a.m.!" th"y find even simple methods of proof difficult to assimilate. This arises, one must assume,from high school and mandatory 2. Fedures of the Course. Each lecture began with my im- college courses in mathematics paying insufficient attention to mediately launching into the day's topic, saving the end of the these matters. Students seem to have little idea as to how math- hour, when all of a day's attendeeswould be there, for announce- ematicians think, how they go about solving problems, and es- ments. And so no repetition was needed. pecially how they start tryrng to solve a problem. So the ma- An 8 a.m. class inevitably generateslate arrivals. That never trix algebra course attempted to glve students ideas about some worried me. Mathematical subjects demand using today's new of the thought processes which are helpful in proving already- knowledge for tomorrow's work, so that I insisted that late ar- establishedresults. One would hope that in learning such tech- rival was far better than not coming at all. And with the class- niques students would subsequentlyfind them very useful, both room having a rear door, which I heartily recommend, entry for for checking on the validity of results they will find in their own laggards causedno intemrption, except on the day a chronically subject-matter research literature, and also for developing their 3O-minute late arrival was greeted with the 8:30 a.m. salutation own results ab initio. And part of the neoessarylearning is: how "Good afternoon". do mathematicians motivate themselves to develop the step-by- The course always had homework-every wednesday, a step process which leads to final and useful results? The nature help sessionthe following Monday and hand it in two dayr later. of this motivation can be illustrated in the proving of a simple, By that time, one hoped it was all correct. Each week's work well known matrix result. was graded 2, 1 or 0 for satisfactory mediocre or hopeless, re- Supposewe want to prove the followiog little theorem: spectively. No student ever argued about this holistic grading. And a whole term's homework counted for no more than lOVo If A is idempotent,so isl - A. page 4 IMACE22

As we all know, the proof is easy: to successfully tackle problems. True, the manffa is not much more than formalinng the trial-and-error - -zA+ - - process, but after all (r- A), r a'2 I -2A+ A r_ A, that is precisely how mathematical progress is often made. It seldom leads to the shortened proof, but who cares about that and the proof is complete. "But", asks a student, "where does insofar as learning the process of proof is concerned. Supple- this come from?' "Why?' asks another; and "So what?' says a menting the mantra there ffe, of course, a number of techni- third. These reactions, I suggest,stem from unfamilimity of im- cal aids for helping with algebra, some of which are detailed in portant stepsin the process of proof, steps which ffie, it appears, SearleQnT. seldom emphasized early enough in one's mathematical educa- 4. Paterns md Relaionships. We also need to teach students tion. For example, two important steps for algebraic proof are that mathematicians look for patterns and relationships. For ex- (i) whenever possible convert text statementsto algebraic state- aurple, in proving that ments and, in doing so, (ii) clearly label the statementsas Given, or To Be Proved. In our case this yields ool'fl Given : A2 - A, the definition of idempotent. =P, ToBeProven' (I - A)' - I - A. ,-(abc)

Now comes the difficult part. We know what has to [: ) be proven: we must show that (I - A)' can be reduced without calculating the matrix inverse, it is the "pattern" of the to I - A. on thinking about this one soon concludes row vector also being the first row of the matrix, combined with that it seems (and indeed is) difficult to straighforwardly the thinking about the cofactors in the elements of the matrix start at I - A and get to (I - A)'. Actually, one can do inverse, that leads at once to this, as follows. Begrnning at I - A we can obsele that A(I - A) - A - A2 - A - A - 0. Therefore r-A f-(100) -p. = I-l-li;flr- zA+A2-(r-A), (l)

This is correct, but itis sort of gimmicky, and not obviously logi- 5. Two Kinds of Thinking. This "thinking about the cofactors" cat. Itis much more logical to startfrom something complicated, brings us to what I feel are two kinds of thinking in this case (I - A)', and try to reduce it to something simpler, needed for doing mathematics. They can be called "automatic thinking" (I - A). This is usually easierthan the other way round. So we and "cogitative thinkit g". Automatic thinking is the start with (I - A)' and hope to get to I - A. Bur now what? kind of (r - A)' -? straighdorward thinking involved in doing algebra, especially in simplifying algebraic expressions;e.g., I - 2 A + A - I - A. The following mantra, to be used iteratively, is what helps. The second type of thinking is that of coEtating; e.g., consider the statement "Think of something to do: do it and hop"."

What comes easily to mind? Expand: AB is a matrix of columns which are linear combinations of columns of A. (r- A)'= (r- A)(r- A). Automatic (e.g., algebraic) thinking is relatively easy, onse one l'{ow what? {Jse the mantra again, and multiply out: knows the rules of algebra, for example. In contrast, cogitative thiokiog is not so easy. As soon as we see that statement about - (I A), AB we know it is tnre. But developing the statement requires cogrtating (i.e., "thinking hard", according to the dictionary) = I-A-A+A2-I-zA+A2. about the operation of matrix multiplication. Then we hit on the statement about AB. Generally speaking, it is a much harder Now comes an important step in the process of developing re- kind of thinking than that involved in I - 2A + A - I - A. sults. (iii) Use what is given: in this case A2 - A. Thus And the real difficulty is to recogmze when this kind of cogrtat- ing is required. The trick is to know when, in the presence of (r-A)'-r-2A+A-r-A. just the product slmbol AB, it will be useful to think of that product as just desqibed. At least a first step in being motivated This prosess, especially the mantra, has helped numerous towards this possibility is to recognize that there are occasions students to overcome their fear of the proof process and to go on when the "autornatic thinking" or "algebraic thinkittg" has to be April 1999 page5 replaced by the cogrtating (hard thinking). Dstinguishing be- Requestfor Feedbackon tween these two modes of thinking has proven to h one more LinearAlgebra Education aid in helping non-mathematics majors to leam a mathe,lnatical subject. A simple example involving both kinds of thinking is the following. Suppose by FrankUHLIC, Auburn University

I would like to ask our members for their ideas on IIAS educa- tional offerings: Linear Algebra Education Day in Atlantain 1995 so far was orr biggest "production". We had one 1/2 hotr educational talk in Chernnitz, and in Madison last swnmer we again offered sev- Then, with y6. - U=t AU, eral educational activities at the IIAS Conference with one ple- nary lecture, a double mini-symposium on educational matters (I* s 1;)(I* I J")y - (I* I nt'")v and the interactive Linear Algebra textbook presentations. Otu next chance for promoting educational issues is at the = n(l* S 1;)y = n{" y; }02r. II-AS Conferencesin Haifa, Israel, June 2*29,2W1, and Auburn, The first two equations here stem from automatic (algebraic) Alabama, in the summer 2C0i2. thinking; but the last comes from cogrtative thinking. For this purpose I would like to ask the ILAS membership for their input: We would like to hear your plans, desires,drems 6. Doing Mahemdics ls a Gwne. Finally, there is nothing and ideas now on what you want to see or hear at a future II-AS wrong in suggesting to students that doing mathematics is re- meeting as regards issues of teaching and leanring Linear Alge- ally a game: make the rules and play by them. For example, for bra. scalars What can we do to foster educational thoughts and progress ab:0+a-0 and/or 6=0: and ry-yr always. in our field? Who could help us? Please specify topics, direc- tions, unmet needs,and so on, to your IIAS Education Commit- But for matrices the rules are different: tee:

AB-0#A-0 and/orB-0; RichardA. BRUALDT:bruald [email protected], and XY - YX sometimes(not often). David Cenls oN: [email protected], JaneDeY : [email protected], one must, therefore, always remember which game is being GuershonHnnpl: [email protected], played. CharlesR. JouNsoN: [email protected], Hopefully some of these ideas may make a contribution to Jeff SruaRr: [email protected], reducing the fear of mathematics that we seein so many students and thence to improving their mathematical literacy! Frank Uslrc: uhI [email protected]. This paper was presented at the Seventh International Work- I will meet many of you in Barcelona. Maybe by then some shop on Matrices and Statistics: Fort l^auderdale, Florida, De- plans will have crystalizefl. cember ll-L4, 1998, and is Paper No. BIJ-776 in the Depart- ment of Biometrics, Cornell University.

Reference S. R. SrRnLE Qyn). hoof. InternationalJournal of Mathcmatics Edtrcationin ScienceandTbchnology 8, lg5-ZO2. Frank UHLrc ILAS Education Committee Chair Department of Mathematic s Auburn University, 312 Parker Hall ShayleR. SnnRLE Auburn, AL 36U9-531.0,USA Departmentsof Biometrics and Statistical Science Cornell University, Ithaca, NY 14853, USA uhI igfd@ma il.auburn.ed u b [email protected] u http ://www. a u b u rn.ed ulu h Iigfd. page6 IMACE22

ILASPresident/Vice-President's Annual Report April 1999

'organization' 1. The following have been elected to ILAS offices with (2) the same is not eligible for support more terms that began on March L, 1999. frequently than once every three years; Boud of Directors: Nicholas J. HlcHau and Pauline VAN (3) the support should be widely distributed geographically. DEN DnlnsscHE (three-yearterms ending February 28,2W2). Drring this experimental period, the II-AS Board (three ex- hesidenf; Richard A. BRUnI-ut. ecutives and six other me,mbers)will review proposals and de- The following continue in their offices to which they were cide on which, if ffiy, will receive supprt, and how much that previously elected: support will be. Next year we will review these guidelines and Vice hesident: Daniel HpnsHKowrrz (terrr ends February this may result in some changes. 28,2m1) We are now accepting requests for conferences held in 20m. Secretuy/Treasurer: James R. WeAvER (term ends Febru- Such requestsshould be submitted by September 15, 1999. Elec- ary 29,2000) tronicrequests are preferred md shouldbe sent to brualdi@math. Boud of Directors: Jane Dev (term ends February 29,2W), wisc.edu. The request should include: Volker MnSnMANN (term ends February 29,2W), Jose Dtas (1) date and place of conference, (2) sponsoring "organiza- DA SILvn (term endsFebnrary 28,2W1), Roger A. HonN (term tion," (3) organizing committee, (a) purpose of conference, (5) ends February 28,2W1). invited speakersnto the extent known, (6) expected attendance, The President's Advisory Committee consists of Chi-Kwong (7) proposed II-AS Ircturer, and brief informationabout the lec- U (chair), Shmuel Friedland, Raphi Inewy, and Frank Uhlig. ture,r, (8) amount requested. IIAS is sponsoring two l*ctrnes in 1999: Hans Schneider 2. The Nominating Committee for 1999 consists of: Ra- (University of Wisconsin-Madison) at the workshop on Applied jendra Bhatia (chair, appointed by the president), Jim Weaver Linear Algebra honoring l-udwig Elsner (Bielefeld, GermffiY, and Wayne Barrett (appointed by the president's Advisory Com- January2l-23,1999) and Gene H. Golub (Stanford tlniversity) mittee), and Tom taffey and Judi McDonald (appointed by the at the Eighth International Workshop on Matrices and Statistics Board of Directors). (tamp€f,e, Finland, August 6-7, 1999).

3. This year there are elections for two positions on the Board 6. The 1999 ILAS Linear Algebra Prize Committee (Hans that become vacant on March l, 2000, and the position of Sec- Schneider-chair, Angelika Bunse-Gerstner,Biswa Datta, Mirek retarylTreasurer being vacated by Jim Weaver. Fiedler & Russ Merris) has recommended that Ludwig Elsner pnze. pr^ze will be awarded at the 4. The following IIAS conferences are planned for the near be awarded the 1999 The July 1999. futtne: IIAS Barcelona meeting in a. The Sth IIAS Conference will be held at the Univer- 7. ILAS has enteredinto a cooperative agreementwith SIAM sitat Politecnica de Catalunya, Bmcelona, Spain, on July 19- for its next Applied Linear Algebra meeting to be held in Raleigh, 22, l99F. The organizing committee consists of R. Bru, R. A. North Carolina, October ?3-25,2000. The terms of this agree- Brualdi, L. de Alba, I. Garcia-Planas(co-chair), J. M. Gracia, V. ment are: Hernandez,N. J. Higham, R. A. Horn, T. J. I^affey (co-chair), G. (1) IIAS will nominate two plenary speakers,and IIAS will de Oliveira, F. Puerta (chair), and P. M. van Dooren. The local support the travel and expensesof these two speakers. (2) The committee consistsof J. Clotet, A. Compta, M. D. Magret, and IL"AS speakerswill be identified in publicity for the meeting and X. hrcrta. in the program. (3) SIAM will offer the two II-AS speakersfree b. The 9th IIAS Conference will be held at the Technion, registration. (4) SIAM will offer the two IIAS speakersa free Haifa, Israel,June 2Y29,2OOI. banquet ticket. (5) SIAM will offer ILAS members who are c. The 10th II-AS Conference will be held at Auburn Uni- not already a member of the SIAM/SIAG-IA the same reduced versity, Auburn, Nabama, USA, srunmer 2W2. registration fee that it offers SIAM/SIAG-LA members.

5. The ILAS Board has agreed that, on an experimental ba- 8. The Electronic journal of Lineu Algebra(El-A): Volume sis, ILAS will consider requestsfor the sponsorship of an II-AS 1, published in 19!b, contained 6 papers. Volume 2, published I-ecturer at a conference which is of substantial interest to ILAS in 1997, contained 2 papers. Volume 3, the Hans Schneider is- members. II-AS will set aside $1,0m per year to support such sue, published in 199{3,contained 13 paprs. Volume 4, pub- conferences, with a maximum amount of $500 available for any lished in 199t3as well, contained 5 papers. Volume 5 is being one conference. The guidelines being used during this experi- publishednow. As of April lgggitcontains 6 papers. EIA's pri- mental perid are: mtry site is at the Technion. Mirror sites are located in Te'mple (t) the conferencemust be of interest to a substantialntrmber University (Philadelphia, USA), in the University of Chemnitz of II-AS members; (Germany), in the lJniversity of Lisbon (Portugal), in The - April 1999 page7

pean Mathernatical Information Service ofTeredby the European MathematicalSociety (EMIS), and in EMIS's 36 Mirror Sites. we are happy to announce the appointment of five new ad- visory editors and three new associate editors. The new advi- sory editors are: Richard A. Brualdi, Ludwig Elsner, Miroslav Fiedler, Shmuel Friedland, and Hans Schneider. The new asso- ciate editors are: Ravindra B. Bapat, StephenJ. Kirkland, and Bryan L. Shader. The entire board is shown in ELA s primary homepage:

http:/iwww. m ath.techn ion. a c. il liicl elal

and in E[A's mirror sites:

http ://www. mat h.temp Ie. ed u/i i cle Ia http:i/hermite.c i i. fc. u l.pt/i icl el al DennisRay Estes: 1941-1999 htt p ://www. t u- ch e m n itz. del ftp- h o m e/p u b/e Ia http:i/www. em is. de/j ou rna Is/E LN Dennis Ray Estes was born in Ada, oklahoma, on June 18, l94l- He received his PhD from Louisiena State University in 9. George Styan has continued as Editor-in-Chief of IM- 1965 under the direction of Gordan Pall. He was a Bateman AGE, with issues no. 2O and 21 published in April and October ResearchFellow at the California Institute of Technology from 1998, respectively. 1966 to 1968. He then moved to the University of Southern 10. A new Journals Committee has been appointed for a Califonria, where he spent 30 years, until his rurtimely death on three year term. It consists of Chi-Kwong U as chair, Hans February 1, 1999. Schneider,Jim Weaver,Danny Hershkowitz (representing E[A), He was a leading expert in the arithmetic theory of quadratic and George Styan (represenring IMAGE). The committee has forms over rings of algebraic integers and wrote more than 4O been glven the immediate charge of exploring th" possibility papers on quadratic forrns, number theory, linear and coillmu- and implications of IIAS publishing an annual (or whatever rhe tative algebra with over 20 different co-authors including kn quantity of papers suggests)hard copy of EIA, for those who Adleman (USC), Hyman Bass (Columbia), John Hsia (ohio wish to have such in addition to the free electronic availability State) and olga Thussky (caltech). He was a major speaker at of EI-A. several intenrational conferences in Germany ancl France, and 11. il.-{S-NET: As of April 21, 1999, we have circulated most recently at the meeting on Quadratic Fonns and t-attices in 836 ILAS-NET announcements. ILAS-NET currentlv has 524 Seoul, Korea, last June. subscribers. Estes had 6 PhD students over the years. He was deeply 12. I[-A'S II'.IFORMAIION CENTER (IIC) has a daily aver- involved in graduate education at USC. He served 12 years as age of 300 information requests (not counting tifP operations). Math Department Vice Chair for Graduate Studies and was a IIC's primary site is at the Technion. Mirror sites are located constant source of help and advice to our graduate students. He in Temple University, in the University of Chemnitz and in the also served several terms on the University Graduate Advisory University of Lisbon. Committee. He is survived by his wife t,ee and two daughters, Darcy and Shelly. In recogmtion of his service to the mathematical cornmunity and in particular to graduate students, the University Richard A. BRUALDI, IIA,S President has establishedthe Dennis Ray Estes Graduate Memorial Fund, Department of Mathematics to benefit graduate sfirdents in the Math Deparment. Donations University of Wisconsin-Madison may be sent tc the Department of Mathematics, University of Van Vleck Hall, 480 Lincoln Drive southem califonria, I-os Angeles, cA 90089-l I13, usA. Madison, WI 537K-,-1388,USA The photograph is courtesy Darcy Estes, via John S. Hsia. bruald [email protected] u

Robert M. GUnALMcK Daniel HgnsrrxowlTz, ILAS Vice-president Department of Mathematics Department of Mathematics University of SouthernCalifornia Technion,Haifa 32m, Israel Los Angeles,CA 90089-1113,USA [email protected] n ion.ac. i I guraI n [email protected] u BILINEARALGEBRA An fntroductionto the AfgebraicTheory of QuadraticForms Algebra,L@ic and Applications,Volume 7 Kazimierz Szymiczek Givingan easilyaccessible elementary introduction to the algebraictheory of quadratic forms,this book coversboth \Yitt'stheory and Pfister'stheory of quadraticforms. Leadingtopics include the geometry of bilinearspaces, classification of bilinear spaces up to isometrydepending on theground field, formalV real fields, ffister forms, the Witt ringof anarbiffiary field (charocteristic two included),prime ideals of theWitt ring, Braua groupof a field,Hase and Witt invarian6 of quadraticficrms, and equivalence of fields with respectto quadraticforms. Problem sections are included at theend of eachchapter. Thereare two appendices:the first gives a treatmentof Hasseand Witt irvarianG in the languageof Steinbergsymbols, and the second contains some more advanced problems in 10groups, including the u-invariant, reduced and stable Witt rings, and Witt equilolence of fields.

1997o 496pFo Cloth ISBN90-5699-076-4. US$84 lE55i ECU70.Crordon and Breach MULTILINEARALGEBRA Algebra,Logic and Applications,Volume B RussellMenis Theprototypical multilinear operation is multiplication.Indeed, every multilinear mapping can be factoredthrough a tensorproduct. Apart from iE intrinsicinter€st, the tensor product is of fundamen- tal importancein a varietyof disciplines,ranging from matrix inequalities and group representation theoryto thecombinatorics of syrnmetricfunctions, and all these subjecB app€ar in thisbook. Anotherattaction of multilinearalgebra lies in itspo^/er to unifysuch seemingly diverse topics. This isdone in thefinal chapter by meansof thentional representations of the full lineargroup. Arising as charactersof theserepresentations, theclasical Schur polynomials are one of thekeys to unification.

1997 . 396pp o Cloth ISBN90-5699-078-0 . US$90/ 059 / ECU75o Gordon and Br€ach ADVANCESIN ALGEBRAAND MODELTHEORY Algebra,Logic and Applications,Volume 9 Edited by M. Droste and R. Gobel Contains95 surveys in algebraand model theory all written by leadinge

1997. 500pp . Cloth ISBN90-5699-101-9. US$95 | L62/ ECU79r Gordonand Breach

?*;'si:tfflc'rn April 1999 page9

New & ForthcomingBooks on LinearAlgebra & RelatedTopics

We present two book reviews, a list of new and fofihcoming allowed to be eigenvalues. This approach seems to be moti- books on linear algebra and related topics, and some comments vated from the author's teaching experience. [As a matter of on Feliks Ruminovich Gantmacher and his Theory of Matrices. fact, students as well as teachers in statistical sciences some- times feel it an unnecessary effort to deepen the consequencesof the fundamental theorem of algebra, which, among other things, DavidA. HanvrLLE: implies that real matrices can have complex eigenvalueswhich Matrix Algebrafrom a Statistician'sPerspective always sum up to a real number, that triangutarization can in general only be done by u unitary transformation, or that the absolute value of an eigenvalue is usually understood as the ab- Reviewedby liirgenGnoB solute value of a complex number (so that it can happen that no eigenvalue of a real orthogonal matrix is - 1 or 1, but all eigen- Matrix Algebra from a Statistician's Perspective [Springer- values have absolute value 1).1 As a disadvantage of this ap- Verlag,New York, xvii + 630 pp., IWT,ISBN 0-387-94975-Xl proach one might regard the fact that triangularization, spectral is a detailed, essentially self-contained book, grving an abun- decomposition, or singular value decomposition sometimesglve dance of useful results for students, teachers and researchers more insight and ease proofs. Certainly, the book is a very good with an interest in linear statistical models and related topics. model for proviog results without directly using eigenvaluesand hoofs are grven for almost all results, and a lot of exercisesare related canonical forms. provided, whose solutions the author intends to make available Everyone with an interest in matrix algebra should use this on at least a limited basis. book as a standard of comparison for his own work, as a source The book contains twenty-two chapters. The first eight chap- for ideas of proofs, as a supplier of useful results, as a com- ters are an introduction to maffices and treat classical' topics panion for a course in linear models, or as a basis for teaching such as submatricesand partitioned matrices, linear dependence matrix algebra in statistics. and independence (of matrices), row and column spaoes,trace of a matrix, geometrical considerations, solution to linear systems, A. M. and inverse matrices (including orthogonal and permutation rna- MnrHAl: trices). Although not ne@ssarily required, prior knowledge of lacobiansof Matrix Transformations the basic concepts in linear algeb'ramakes these chapters easier and Functions to grasp. Chapters nine and ten introduce concepts which are of of Matrix Arguments great importance in statistics, namely generalizedinverse matri- ces and idempotent matrices. These zue used in chapters eleven Reviewedby SergeB. Pnovosr and twelve to discusssolutions to linear systemsand projections, respectively. Chapter thirteen gives basic properties and useful This excellent book lWorld Scientific, xxii + 435 pp., 1997, results on determinants. Chapters fourteen to twenty deal with ISBII 9t31-02-3095-81makes the evaluation of Jacobiansof ma- subjects which are indispensable for the understanding of linear trix transformations and of integrals involving scalar fimctions statistical models, such as quadratic forms (as well as linear and of matrix argume,lrt accessible to a host of researchers and ap- bilinear forms), matrix differentiation, Kronecker products, vec plied scientists. Reat multivariable calculus, some properties and vech operators, constrained and unconstrained minimization of rnatrices and determinants and the basic notions of complex of a second-degreepolynomial, md the Moore-Penrose inverse. variables are the only prerequisites. An algorithm for determining the Moore-Penrose inverse (e.g. Although Jacobians can be obtained from matrix derivatives Greville's algorithm) would have been a nice completion, al- which per se are discussed in several books currently avail- though a procedure for the computation of a generalizredinverse able [such as Rogers (1980), Graham (lJal), and Magnus and is given in Sec. 9a. Chapter twenty-one treats eigenvalues and Irleudecker (19SS)1,the monograph being reviewed could very eigenvectors,and chapter twenty-two is about linear spaces. well be the first book dealing directly with Jacobians. The book As visible from the title, the book is written from a statis- also distinguishes itself by the inclusion of a detailed coverage tician's perspective. This qm of course be confirmed by the of complex Gaussian, complex Wishrt and complex matrix- choice of subjects in the book, which play a fundamental role variate statistical di stributions . in linear statistical models. The author assrunesthat the reader A rather exhaustive collection of Jacobians of linear (Chap- is aware of the potentiality of applications, or becomes aware ter 1) as well as nonlinear transformations (Chapter 2) are de- of the,m when attending a course on linear models. A further rived for both the real and complex cases. To this reviewetr's indication for the statistician's pe,rspectiveis the exclusion of knowledge, the results obtained for the complex ca$e (Chryters complex mahices and numbers. Eigenvalues are treated in the 3 and 4) are new, their derivations usually paralleling their real next-to-last chapter of the book, and coryplex numbers tre not counterparts. Chapter 5 deals with real-valued scalar functions page 10 IMACE22

of matrix argument in the real case; hypergeometric functions Blatter, Christian (1999). Linearc Algebru fiir Ingenieufr, are defined in terms the ge,neralizrd laplace transform, zsnal Clrcmiker und Nanruissensch$tler. In German. vdf Hochschulverlag polynomials and the M-transform and scalar functions of many an der ETH Ziirich, vi + 151 pp., ISBN 3-728I-266G8. matrix arguments are introduced, iunong other topics. Real- Blondel, Vincent D.; Sontag, Edouardo D.; Vidyasagar, M. & valued scalar functions of one or more matrices in the complex Willems, Jan C., eds. (1999). Open hoblems in Malhematical ,S)s- tems md Control Theory. Communication and Control Engineering case are discussedin Chapter 6. Series,Springer, ca.30O pp., ISBN 1-85233-0l1 9 (H). The material is developedprogressively from first principles, B0ttcher, Albrecht & SilberTnann,Bernd (1999). Infioduction to lending itself to self-study. The theory is often illustrated with Larye Truncaled Toeplik Marrices. Springer, xi + 258 pp., ISBN O- examples taken from distribution theory. Interestiog applica- 387-e8s7O-OIZbl 990.0OO I 71. tions and extensions of the results are submitted as exercises Bollobds, B6la (1999). Linea, Annlysis: An Infioductory Course. at the end of each section. The book also contains a thorough Znd edition. Cambridge Mathematical Textbooks. Cambridge Univer- glossary of symbols. Those wishing to ftrther their knowledge sity Press,xi + )N pp., ISBN 0-521-65577-3(P) lzbl990.080151. of the materials discussedand relatd topics will find the repre- Bourbaki, Nicolas (1998). ElemenB oJ Muhematics: Algebm I, sentative references listed at the end of each section eminently ChaptersI-3. Translatedfrom the French. Softcover edition of the 2nd (1989). pp., ISBN 3-540-64243-9 useful. This book could well serve as a primary text for a one- printing Springer, xxiii + 7O9 IZbl 9O4.0OOOll. semester graduate course on Jacobians and functions of matrix Bykov, V. I.; Kytmanov, A. M.; I-azman, M. Z.; Passare,Mikael, argument. eds. (1998). Elimination Methods in Polynomial Computer Algebra In addition to being a valuable reference source in a vari- Mathematics and its Applications, vol. 1 18. Kluwer, xi + 237 pp., ety of disciplines such as statistics, engineering, econometrics I SBN O-7e23- 5240-8 [Z;bl 743 .6M1. md physics, this monograph should stimulate researchbased on Cameron,A. Colin & Trivedi, havin K. (1998). Regressi,onAnal- mul tivariabl e cal culus . ysis of Coun DAa Econometric Society Monographs, vol. 30. Cam- bridge University Press,xviii + 4I1 pp., ISBN 0-521-63%7-5 [CMP 1 References ffi274, ZblW.01357l. A. GnanaM (1931). KroneckerProducts and Marix Calculus: Carlson, David (1999). Linear Algebra Cogito l-earning Media, with Applications.Wiley (HalstedPress), New York. 300 pp., ISBN 1-888-X)2-76O. J. R. Macwus & HeinzNEr-rorcKER (1988). Matrix Difrerential D'Alotto, l-ouis A., Giardina, Charles R. & Luo, Hua (1ry8). A Calculuswith Applicationsin Statisticsand Econometrics.Wiley, New fJnified Signal AlgebraApproach n Two-Dimensional Parallel Digital York. (Revisededition: L99g.) Signal hocessing. Marcel Dekker, 2X) pp., ISBN 0-82+70A-252. G. S. RocBRS(1980). Matrix Derivatives.Wiley, New York. Dirschmid, Hans JOrg (1998). HAhere Mathematik: Matrizen und lineare Gleichungen. In German. Manz Verlag Schulbuch, vi + 72O pp., ISBN 3-7068-05%-6 lzbl 990.01561 1. Bools Dryden, I. L. & Mardia, K. V. (1998). Statistical Shape Analysis. New and Forthcoming on Wiley Seriesin Probability and Statistics. Wiley, xx + 347 pp., ISBN LinearAlgebra and RelatedTopics: 1998-1 999 0-471-95816-6[CMP r ffi Ir4, Zbl9or.620721. El Ghaoui, I-aurent & Niculescu, Silviu-Iulian (1999). Advances in Lineaf Mnrix Inequality Methods in ControL SIAM, c8. 475 pp., Listed here are some new and forthcoming books on linear ISBN 0-89t171-438-9(P), in press. algebra and related topics that have been published in 1998 Farebrother,Richard William (1999). Fitting Linear Relationships: or in 1999, or are scheduled for publication in 1999. This A History oI the Calculus oJ Observations I75FI9M. Springer Series list updates and augments our previous list in IMAGE 2l in Statistics: Perspectivesin Statistics. Springer,xii + T7I pp., ISBN (October 1998), pp. 8-9. The references to CMP (Cur- 0-387-98598-0. rent Mathematical Publications) and MR (Mathematical Re- Farebrother, R. William; htntanen, Simo; Styan, George P H. & eds. (1999). SeventhSpecial Issae on Linesr views) were found using MathsciNet and to Zbl (kntralblatt Werner, Hans Joachim, Algebra and Statistics. Linear Algebra and lts Applications, Elsevier, MathemotiklMATH Database) by visiting the Zbl Web site: fur vol. 289, ix + 344 pp. http://www.emis.de/cgi-bin/MATH(three free hits ta Zbl are al- Federer, Walter T. (1999), Statistical Design and Analysis for In- for non-subscribers). lowed tercropping Experiments: Volume II, Three or More Crops. Springer Series in Statistics. Springer, xxiv + 262 pp., ISBN 0-387-98533-6 Andrilli, Stephen Francis & Hecker, David (Ig99). Elementary lcMP 1 653 18U. Linear Algebm Znd edition. Academic Press,496 pp., ISBN 0-120- Fialkow, I-awrence A. & Curto, Raul E. (1993). FlaI Exteruions oI 586-908. Positive Moment Matrices: Recursively GeneratedRelations. Memoirs Anton, Howard (1993). Linearc Algebra Einfiihrung, Grund.lagen, of the AMS, vol. 136. AMS, 56 pp., ISBN 0-8218-08-699. Ubungen. In German, hanslated from English by Anke Walz. Spek- Fomin, Vladirnir (1999). Optimal Filtering, Volume I: Filtering trum Akademischer Verlag, x + 680 pp., ISBN 3-827+032+3 IZbl of Stochostic Processes.Mathematics and its Applications, vol. 457. 980.3s8011. Kluwer, xiii + 375pp., ISBN 0-7923-5286-6. Bhatia, Rajendra; Bunse-Gershner,A.; Mehnnann, V. & Olesky, D. Frazier, Michael & Meyer-Spasche,R. (1999\. An Introduction to D., eds. (1999). Special Issue Celebraing tht ffith birthday oJ Ludwig Wavelets Through Linear Algebra. Undergraduate Texts in Mathemat- Elsner. Linear Algebra and Its Applications, Elsevier, vol. 2W, x + 379 ics. Springer,536 pp., ISBN 0-387-9{36-391. pp. [CMP L 662 857]. April 1999 paSe1 1

Ge, Liming; Lin, Huaxin; Ruan, Zhong-Jin; Zhang, Dianzhou; Lyons, Louis (1993). All YouWantedTo Know About Malhemat- Zhang, Shuang,eds. (1998). Operalor Algebras and Operalor Theory : ics But Were Aftaid To Ask: Mahematics for Science Students, Vol- Proceedings oJ the International ConJerence,Shanghqi, China, July 4- ume 2. CambridgeUniversity Press,xv + 382 pp, ISBN 0-521-4346i6-L 9, 1997. ContemporaryMathematics, vol. 228. AMS, xx + 389 pp., (H), O-52L-43601-X(P). rsBN o-82I8-I W3-6 [Zbl eO8.OO02A]. Magnus, Jan R. & Neudecker, Heinz (1999). Mdlrix Differential Ghosh, Subir, ed. (1999). Maltivariare Analysis, Design of F;- Calculus with Applications in Statistics wtd Econometrics. Revised periments, and Sumey Sampling. A Tribute to Jagdish N. Srivastava, Edition. Wiley, xviii + 395 pp., ISBN A-47t-98632-l (H), 0-471- in Celebrationof his 65th Birthday. Statistics:Textbooks and Mono- e8633-X (P) tzbl ee0.080161. graphs Series,vol. 159. Marcel Dekker, ca. 696 pp., ISBN 0-8247- Martin, Richard Kipp (1999). Larye Scale Linear md Integer Opti- 0052-X (H), in press. mization : A Unifi,edApprcach. Kluwer, xvii + 74Opp., ISBN 0-7923- Golubitsky, Martin & Dellnitz, Michael (1999). Linear Algebra 82A2-r. utd Dillerential Equations using Matlab. Brooks/Cole, ISBN O-53+ Mathews, P. C. (1998). Vector Cahulas. Springer,ix + 182 pp., 3*725-4, O-53+363ffi-7, in press. ISBN 3-f,O-76180-2. Goode, StephenW. (1999;. An Introduction to Differential Equa- Mayer, Janos (1998). StochasticLinear ProgmnnmingAlgorithms: tions and Linear Algebra. 2nd edition. Prentice Hall, 72O pp., ISBN A Comparison Based on a Model Mmagement System, voL /. Gordon -757-y^. 0-13-263 & Breach, 1.53pp., ISBN 9-056-99I-M2. Gray, JeremyJ., Scheer,U. & Nover, L., eds. (1999). Linear Dif- Mazliak, l.aurent, hiouret, Pierre & Baldi, Paolo (1998). Mqrtin- Jerential Equations and Group Theory Jrom Riemann to Poincar4. 2nd gales et chainesde Markov. Hermann, viii + 215 pp., ISBN 2-7056- edition. Birkhiiuser,36O pp., ISBN 0-8176-3837-7 . 6382-7 Green, Edward L. & Huis gen-Zimmerrnann,Birge, eds. (19%). McQuarrie, Allan D. R. & Tsai, Chih-Ling (1998). Regressionand Trends in the RepresentationTheory of Finite Dimensional Algebras. Time SeriesModel Selection. World Scientific, xxii + 455 pp., ISBN ContemporaryMathematics, vol. 229. AMS, xiii + pp., 356 ISBN 0- 981-02-3242-X [CMp L &r 582,2b1980.483201. 8218-0928-8. Melnikov, Yuri A. (1998). Influence Functions and Matrices. Me- Hausner,lv{elvin (1998). A Vector Sparc Approach to Geometry. chanicalEngineering, vol, 119. Marcel Dekker, 488pp., ISBN 0-8247- Dover, xii + 397 pp., ISBN 0-486-4M52-8 51-01 1 6517321. [CMP I94I-'.7,ISSN 0025-6sOL [Zbl 990.00433]. (Reprint of the 1965original.) Merlier, Thdrbse (1998). Exercices conigds sur lesformes quadm- ( Heumann,Christian 1998). Likelihoodbasiertemarginale Regres- tiques et groupes classiques. In French. PresseslJniversitaires de sionsmodelle konelierte kuegoriale Dolen, vol. 2. In German. fiir Pe- France,iv + 182 pp., ISBN 2-13-M5796-7 [Zbl98O.54155]. ter l-ang, l8O pp., ISBN 3-63I-32671-8 [CMP I 638 t++]. Nakos, George (1999). Linear Algebra. PWS hrblishing, ISBN Hoffman, Kenneth (1999). Linear Algebra. Prentice-Hall, ISBI{ O-53+9 55-2X), i n press. O-13- 181496-6, in press. Nelsen, Roger B. (1999). An Introduction to Copulas: Properties Jensen,Shane Tyler (1999). The Laguene-SamuelsonInequatity and Applications. Lecture Notes in Statistics,vol. 139. Springer,xii + with Ertensions srd Applications in Statisticsand Marix Theon. MSc 216 pp., ISBN O-387-9t]623-5ICMP I 653 2O3,Zbl9X).Al219l. thesis,Dept. of N{athematics& Statistics,N{cGill University,Montr6al, Neville, A. M. Ghali, (1998). vi + 65 pp. & A. StructuralAnalysis: A Unified Classical md Morrix, Approach. Revised Edition. Chapman & Hall, Kailath, Thomas& Sayed,Ali H., eds. (1999).Fast Reliable Atgo- 848 pp., ISBN O-419-212-W. rithmsfor Matrices with Structure. SIAM, ca. 330 pp., ISBN 0-89871- Nipp, Kaspar & Stoffer, Daniel (1998). 131-1(P), in press. Lineare Algebra: Eine EinJilhrung fur Ingenieure unter besonderer Befi)cksichtigung nlt- Kaplan, Wilfred (1999). ltlaxima wtd Minima with Applications: merischerAspekte. In German. vdf HochschulverlagAG an der ETH Practical Optimization utd Dualiry. Series in Discrete Mathematics & Ztinch, vi + 25L pp.,ISBN 3-728I-2&9-7 [Zbl9S0.54181]. Optimization. Wiley',x + Z%pp., ISBN 0-.X71-25299-1. Parshall, Karen Hunger (1998). JamesJoseph Sylvester. Life and Katz, Nicholas N{. & Sarnak, Peter (1998). Random Marrices, Wo* in Letters. Oxford University Press,368 pp., ISBN{0-19-850391- Frobenius Eigenvalues, wrd Monodromy. Colloquium publications, r [zbte9o.o4o16]. vol. .45AMS, xi + 419 pp., ISBN 0-8218-1OI7-A. Rrntanen, Simo (1999). RegressioanalyysiI & il. In Finnish. Re- Keller, H. A., Kuenzi, LI.-M. & wild, (199s). M., eds. orthogonal ports 848 &B,49, Dept. of Mathematics,Statistics & Philosophy,Univ. Geometry in Infinite Dimensional Vector Sparcs. Bayreuther Mathe- of Tampere,2 vols., viii + 590 pp. matischeSchriften vol. 53. 326 pp.,ISSN 0172-rc62IZbl9O5.110011. Rawlings, John O., Pantula,Sastry G. & Dickey, David A. (1998). Kitchen, Joseph (1999:). Applied Lineqr Algebra hentice Hall, Applied RegressionAnalysis: A ResearchTool. 2nd edition. Springer ISBN 0-13-4893A6-7,in press. Texts in Statistics.Springer, xviii +65'7 pp., ISBN 0-387-9545/-2 tMR Knowles, JamesK. (1998). Linear Vector Spacesand Cartesian 99d: 620'70,Zbl 980.2fr321. Tenson oxford UniversityPress, viii + LZapp., ISBN 0-19-51 r2*7 Schwartz, I-aurent (l99S). Les tenseurc. With Toneun sur un es- 8e6.1s0201. lzbt pace affine by Yves Bamberger& Jean-PierreBourguignon. In French. Kostrikin, Alexei I. & Manin, Yu I. (199s). Linear Algebra and Actualit6s Scientifiqueset Industrielles,vol. 1376. Hermann, 203 pp., Geometry,vol. /. Gordon & Breach,308 pp., ISBN 9-056-99{J-4y7. r sBN 2-7 056-1376-5 [Zbt e05. 1sO I 61. Kuipers, Jack B. (1998). Quaterniorn and Rotation Sequences:A Srivastava,Hari M. & Rassias,Th. M., eds. (1999). Analytic and Primer with Applications to Orbits, Aerospate, and Virtuqt Reality. Geometric Inequalities andTheir Applications. Kluwer, in press. PrincetonUniversity hess, xxii + 3'71pp., ISBN 0-691 -osg7z-S. stark, Henry & Yang, Yongyi (1998). vector space hojections: A Kunz, Herv6 (1998). Malrices aldaoires en physique, vol. i. In Numerical Approrch to Signal and Image Processing,Neural Nets,and French. PressesPolytechniques et UniversitairesRomandes, viii + 84 Optics. Wiley Series in Telecommunications and Signal Processing. pp., ISBN 2-88074-373-7ICMP I 610 74O,Zbl894,1_5009]. Wiley, xvi + 408 pp., ISBN 0-471-2411A-7 [Zbl903.65O0U. page1 2 IMACE22

FeliksRuvimovich Cantmakher and TheTheory of Matrices

\\nth the reputilication [EI-IR, El -IIRI of the English translation by K. A. Hirsch of Gantmacher'swell-known work The Theory of Marices, we would like to present here a brief biographical sketch of Gantrnacher and some bibliographical information on The Theory of Morices. Feliks Ruvimovich Ganhnakher [GantmacherJ( 1908- 1964) was a research scientist whose work in both pure and applied mathematics yielded major contributions of wide-ranging theo- retical, engineering, and pedagogical importance. [For his re- search in rocketry, he was awarded the Military Order of the Red Star and for his researchin external ballistics, a State Prize of the first degree.l Gantmacherwas born in Odessaon February ?3,1908. Af- ter completing gymnasium, his serious education in mathemat- ics began at home under "experienced tutors". At the age of 15, he entered the OdessaInstitute of National Education, where he I Stoppel,Hannes & Griese,Birgit (1998). UbungsbuchzurLinea,ren I of Algebra. In German. Vieweg Studies:Basic MathematicsCourse, vol. was impressed with the clear, elegant and expressivelecfures 88. Vieweg & Sohn,x * 27Opp., ISBI{ 3-528-07288-1. S. O. Shatunovski.At the sameInstitute, he was also impressed 1 Stuart, Alan; Ord, J. Keith & Arnold, Steven (1999). Kendall's with the lechres of the farnous engineer G. K. Suslov in applied Advanced Theory of Statistics: Volume 2A, Classical Inference and mathematics and It{. G. Chebotarevin pure mathematics: infinite the Linear Model. Sixth Edition. Arnold, & Oxford University groups and the theory of analytic functions. Suslov helped Gant- Fress,New York, xxii + 885 pp., ISBN 0-340-66234-1 ff). macher to obtaitr a research fellowship in the Applied Math- Styan,George P H. (with Giilhan Alpargu, Mylbne Dumais, Shane ematics section of the Scientific Research Department of the T. Jensen & Simo hrntanen), ed. (1998). Three Bibliographies md Odessa Institute and he began teaching there in 1927, the year the Seventh International Workshop on Matri- a Guide. Prepared for his research fellowship began. Here began his lifelong project ces and Statistics (Fort l;uderdale, Florida, December 1998). Dept. of developing a course of theoretical applied mathematics, the of Mathematics & Statistics, McGill University, Montr6al, 100 pp. main subject of his distinguished teaching career. Ohe bibliographiescover the Craig-sakamototheorem, the Laguerre- Samuelsoninequality, and the Frucht-Kantorovich inequality, while the During the summer of 1926 (at the age of 18), Gantmacher guide is to books on matricesand books on inequalities.) wrote his fust researchpap€r. This paper "was devoted to a con- Tenenhaus,Michel (1998). La rigression PLS, thiorie et pratique. cise exposition of a number of basic relations in affine differen- In French. gditions Technip, x + 254 pp., ISBN 2-7108-0735-1IMR tial geometry on n-dimensional surfacesin (t * 1)-dimensional 99e.62119,7.,bl 980.293 461. spa@s". Beginning in l929,he published a mrmber of research Tian, Yongge (1999). Rutk Equalities Related to Generalized In- papers and monographs, co-authored with M. G. Krein, on the andTheir Applicalio,rus.MSc thesis, Dept. of Math- versesof Matrices theory of matrices and integral equations. These papers intro- ernatics& Statistics,Concordia University, Montrdal, iv + I29 pp. duced and studied an inrportant class of matrices and kernels, Torrence,Bruce F, & Torrence,Eve A. (1999). The Student'sIn- called oscillatory. In 1934 Gantmacher enteled the Steklov truduction to Malhematica: a Handbook for Prccalculus, Calculus, antd where he presented his doctoral thesis Linear Algebra CambridgeUniversity Press,ISBN O-52I-59+4%. Mathematical Institute, the theory of semi-simple Lie groups (1938). By 1947, he Vein, Robert & Dale, Paul (L999). Determinants and their Appli' on 1953 he was Head of the De- cations in Malhematical Physics. Applied Mathematical Sciences,vol. had become a Professor,and from L34. Springer, xiv + 376 pp., ISBN 0-387-98558-1 [CMP 1 654 M9, partment of Theoretical Applied Mathematics of the Moscow zblego.0135e1. Physi cat-Techni cal Insti tute. Wackernagel, Hans (1993). Multivariate Geostatistics: An Intro- His research in the theory of matrices alone would have d,uctionwith Applications. 2nd completely revised edition. Springer, earned for him a place of distinction in mathematical history. pp.. ISBN 3-54A-6472I-XlZbl990.013831. xiv + 29I The fundamental monograph Teoriya Mdrits (The Theory of (1998). Vectors. Chicago lectures Weinreich, Gabriel Geometrical Mdrices), first published in Russianin 1953 [R1], wff basedon in Physics. Press,x * 115 pp., ISBN 0-226- research that took place over many years. This work is widely 89M7-3, 0-226-89048-1 [CMP | 625 58O,Zbl98O.3559O]. recogqi zaa as a classic in the field and has been translated into Yixian, Yang, ed. (1998). Theory utd Applications of Higher Di- English, French, and German. In his detailed review of [R1], mensionalHadamard Mmrices. SciencePress (Beijing), zMpp., ISBN 16:438 (1953)] wrote that "The ntrmber 7-03 -00 67 82-7 [Zbl e8O.54550]. Joel L. Brenner [MR treats well is great. Since many of Zhang, Fuzhen (1999). Marrirc Theory: Basic Results and Tech- of subjects which the book niques. Springer,ca. 370 pp., ISBN 0-387-986-960,in press. these subjects are in fields commonly cla^ssifiedas applied math- April 1999 page1 3

ematics, n early translation of the book would appeal to a wide [E2-II] Applications of the Theory of Matrices. In English. Trans- audience". In fact Brenner himself translated the second part lated from the Russian by Joel L. Brenner, with the assistanceof D. of [R-l] in 1959 "adding references, an index and four appen- W. Bushaw & S. Evanusa.Interscience, New York, 1959,ix + 317pp. dices; also correcting misprints, simplifyiog a few proofs and IMR 2I:6372b1. (2nd English translationof Chapters11-14 of tRU.) The Theory of Matices: Volume I . In English. Reprint making the Second Part of [R- U a self-contained whole. The [El-IR] Edition. Translatedfrom the Russianby K. A. Hirsch. AMS Chelsea translation by K. A. Hirsch is of the complete Russian text with Rrblishing, Providence,RI, 1998, x + 374 pp., ISBNI 0-8218-1376-5 new versions of several paragraphs cornmunicated by the au- (2-vol. set ISBN 0-8218-1393-5)[MR 99f:15001, Zbl 99O.O14O7} thor. The work is an outstanding contribution to matrix theory (Reprintof [El-I].) anclcontains much material not be found in any other text" IMR [EI-IIR] The Theory oJ Matrices: Volume2. In English. Reprint 2l :6372c (1959), unsignedl. Edition. Translatedfrom the Russianby K. A. Hirsch. AMS Chelsea The supplementsin the Second Supplemented Edition tR2l Rrblishing, Providence,RI, 1998, x + 276 pp., ISBN 0-8218-0133-0 comprise, as Brenner noted in his review 34:2585 (1966)1, (2-vol. set ISBN 0-8218-1393-5).(Reprint of [El-IIl.) (( [MR a chapter on nonsingularity conditions and eigenvalue loca- tFll Thiorie des matrices: Tome I, Thdorie gdndrale. In French. tion theorems, and an appendix by V. B. Lidskii on inequali- Translated from the Russian by Ch. Sarthou. Collection Universi- taire de Mathdmatiques,vol. 18, 1966, xiii + 370 pp., Dunod ties concerning singular and proper values. Several other chap- IMR 37: 1381a1.(1st part of trvo-volumeset; translation of Chaptersl-10 of ters have also been amplified. The appearanceof this edition tRU.) posthumously-Gantmacher died on May 16, 1,9e+-is due to tF2l Thdorie des matrices: Tome2, Questions spdciales et appli- Lidskii's initiative ... The appendix by Lidskii is a beautilully cations. In French. Translatedfrom the Russianby Ch. Sarthou. Col- written, self-contained exposition of many important results." lection Universitaire de Math6matiques,vol. L9, 1966, xii + 268 pp., The only translation of [R2l appearsto be the (now out-of-print) Dunod IMR 37:L38lb, Zbl 136.0M10]. (2nd part of two-volume set; 1986 translation tc4l in Gennan by Hehnut Boseck, Detmar 1sttranslation of Chapters1L-I4of [RU.) soyka and Klaus stengert, with a preface by D. p. futobenko. tcl-Il Matrizenrechnung: Teil I, Allgemeine Theorie In Ger- man. Edition.] Translatedfrom the Russian Much of the information above comes from the obituary ar- [1st by Klaus Stengert. Hochschulbticherfiir Mathematik, vol. 36. VEB Deutscher Verlag ticle by I\'f. A. Aizerman, I\{. G. Krein, L. A. Lyusrenik, M. der Wissenschaften,Berlin, 1958, xi + 324 pp. IMR 20:3884, Zbl A. Naimark, L. C. Pontryagrn, C. L. Sobolev, C. A. Khris- 079.0110617l. (1st part of two-volume set; translationof the first ten tianol'ich, and Ya B. Shor (in English, translatedfrom the Rus- chaptersof tRll.) sian by F. W. Ponting, Russian Mahemrtical Survqts 20, 143- [Gl-II] Matrizenrechru*tg: Teil II, SpezielleFragen und Anwen- l5l, 1965). We are also grateful to John Kimmel, Erkki Liski, dungen In German. [st Edition.] Translated from the Rus- Heinz Neudecker and Sanjo Tlobec for their help. The photo- sian by Klaus Stengert. Hochschulbiicherfiir Mathematik, vol. 37. graph is reproduced from the dust-jacket of tR2l. vEB Deutscherverlag der Wissenschaften,1959, vii + 244 pp. [MR 2l:6372a, Zbl085.009041. (2nd part of two-volume set; translationof We now presenta (possibly) complete listing of all versions Chapters11-14 of [R1].) Matrizenrechnung: Teil I, Allgemeine of The Theon of Mafices by F. R. Ganrmacher (only tc4l has [G2-Il Theorie In German. Znd Corrected the author listed as Felix R. Gantmacheron the title page). Edition. Translatedfrom the Russian by Klaus Stengert. HochschulbUcherfUr Mathematik, vol. 36. VEB DeutscherVerlag der tRll The Theoryof h'Iatrices. In Russian.Gosudarstv. Izdat. Wissenschaften,Berlin, 1965,xi + 324pp. tMR 34:58381. Tehn.-Teor.Lit., N{oscow,1953, 19r pp. tMR 16:438,Zbloso.z1f;CFll. tG2-IIl Matrizenrechnrung: Teil il, Spezielle Fragen und Anwen- tR2l TheTheory of Matrices.InRussian. With an Appendix by V. dungen In Germant. Znd Corrected Edition. Translated from the B. Lidskii. 2ndSupplemented Edition. Izdat. "Nauka", Moscow, 1966, Russianby Klaus Stengert.Hochschulbticher fUr Mathematik, vol. 37. 576 pp. FUR34 #2585,Zbl 145.036041.(The "supplements"com- VEB DeutscherVerlag der Wissenschaften,Berlin, 1965,vii + 244pp. prise"a chapteron nonsingularityconditions and eigenvalue location [G3-I] Matrizenrechnung: Teil I; Allgemeine Theorie In German. theoremsand the Appendix by V. B. Lidskiion inequalitiesconcerning 3rd Edition. Translatedfrom the Russianby Klaus Stengert. [lst Vol. singularand proper values".) of Two-Volume Set.l Hochschulbticher ftir Mathematik, vol. 36. VEB tR3l TheTheory oJ Matrices.In Russian. With an Appendixby V. DeutscherVerlag der Wissenschaften, Iy7o, xi + 3z4pp. tMR 43:834, B. Lidskii.3rd SupplementedEdition. Izdat. "Nauka", Moscow, 1967, zbl223.1500U. (Apparentlyjust a correctedreprint of tG2-Il.) 576pp. (Apparentlyjust a correcredreprinr of tR2l.) [G3-II] Matrizenrechnung: Teil II, Spezielle Fragen und Anwen- tR.ll rhe TheoryoJ Manices. In Russian.with an Appendixby dungen In German. 3rd klition. Translated from the Russian by v. B. Lidskii. 4th SupplementedEdition. Izdat "Nauka",Moscow, Klaus Stengertand edited by Helmut Boseck. HochschulbUcherftir 1988,549 pp. ("Basicallya reprintof [R2]" [Zbl 66.15002]but with Mathematik, vol, 37. VEB Deutscher Verlag der Wissenschaften, apparentlyfewer pages. ) Berlin, r97a, vii + 244 pp. [Zbl 322.150M]. (Apparently just a cor- [El-Il TheTheory ol Matrices:Volume I. In English.Translated rectedreprint of [G2-II].) from theRussian by K. A. Hirsch.Chelsea, New york, 1959,x + 374 lc4l Matrizentheorie by Felix R. Gantmacher. In German. With pp.,ISBN 0-82t3+0133-0 IMR 2I:6372c].(lst partof two-volumeset; an Appendix by V. B. Lidskij & a heface by D. p. Zelobenko. [4rh translationof Chapters1-10 of [R11.) klition.l Translatedfrom the RussiantR2l by Helmut Boseck,Dietmar [El-II] TheTheory of Matrices:Volume 2. In English.Translated Soyka & Klaus Stengert, Hochschulbi.icherfUr Mathematik, vol. 86, VEB Deutscher from theRussian by K. A. Hirsch.Chelsea, New york, rgsg,x + TT6 Verlag der Wissenschaften,Berlin & Springer-Verlag, Berlin, 6Y pp., pp.,ISBN o-82t3+0131-4 [MR 2L:6372c,zbl085.01001J. (2nd part of ISBN 3-326000l-4 [MR 87i: 15001, Zbl588.1500U & ISBN 3-ffi-r6582-7 87k lsooil. (out of print.) two-volumeset; 1st English translation of ChaptersI l-t4of tRll.) IMR APP!.ITD DI }ATNI S I ONIA!. ANIALYS I S ANID },\ODT!.INI C by ThomasSzirtes McGraw-Hill,New York, 1998;ISBN: 007-062811-4 815pages, 200 illustrations, 250 t worked-outexamples and case studies; US$99.95

With a most creative use of linear algebra - specifically matrix arithmetic - this book presentsthe "dimensional method', which is /'PP[-fl=lj an enorrnously potent tool in design, analyses and tests of engineering and scientific systems and products. The method is also efficient in forming relations among variables by non-analytic arfF means, in veriffing analyical formulas, and in the logistics of graphical presentationsof multivariable relations. NG The most important application of the dimensional method, howeverois model experimentation, for which the determination of case-specificModel l^ovts and Scale Factors are mandatory. These laws and factors are all based on dimensionless variables. The book presents a highly elegant and fast method, based on matrix algebra, by which dimensionlessvariables are generated,and the relevant Model Laws and ScaleFactors easily determined. This in-depth landmark reference work also includes over 250 numerical examples and case-studiesdrawn from the theory and practical utilisation of dimensional method in applied science, engineering,biomechanics, astronomy, geometry and economics.

Did you know that a man 100 % Did you know that the more the air in a soap bubble, the taller than a child, wall$ 'most smaller its internal pressure? comfortablyt a mere 41.4 "/" faster? Did you know that the distance'a' of a material point to a large plate has no effect on the gravitational force F that the plate imparts on the material point?

materialpoint

Thesestartling factsare all true, asyou can easilyconvince yourself without any analysis,only by somenice matrix arithmetic explainedfully in A??[IEDDIil\ENSIONAT ANATYSIS AND I'[ODE!.ING.

ILAS5.SAI\{ April1999 page1 5

Seventhlnternational Workshop on Matricesand Statistics, in Celebrationof T. W. Anderson'sSOth Birthday

FortLauderdale, Florida, USA: December 11 -14,199S

This Workshop was organized by N. D'Alessio, R. W. Fare- Lynn R. I-aMOTTE, Louisiana StateUniversity, Baton Rouge brother, W. D. Hamlnack, S. hrntanen, D. S. Simon, G. P. H. ZhongshanLI, Georgia StateUniversity, Atlanta Styan (chair), H. J. Werner (vice-chair) , F. nrang (local chair), Erkki P. LISKI, University of Tampere,Finland and supported, in part, by the Intemational Linear Algebra So- Asmad MANSOUR, McGill University,Montr6al, Canada ciety, Nova Southeastern University, and the Statistical Society Augustyn MARKIEWICZ, Agricultural Univ. of Poznari,Poland of Canada. The photograph on pp. lGlT is by Simo hrntanen. Jyrki MOTTONEN, Tampre Univ. of Technology,Finland Anne MULDER, Nova SoutheasternUniversity The following persons participated: Irja & Tuulia NUMMI, Tampere,Finland YasuoAMEMIYA, Iowa StateUniversity, Ames Tapio NUMMI, University of Tampere,Finland Dorothy ANDERSON, Stanford, California Ingram OLKIN, Stanford LJniversity,California Theodore w. ANDERSON, Stanford university, california MatjaZ OMLADId, University of Ljubljana, Slovenia Jerry BA RTOLOMEO, Nova SoutheasternUniversity VesnaOMLADId, University of Ljubljana, Slovenia Philip v. BERTRAND, sheffield Hallam universiry England Yuriko OSHIMA-TAKANE, Montr6al, Canada Barbara BONDAR, Ottawa, Canada Christopher C. PAIGE, McGill University, Montr6al, Canada James V. BONDAR, Carleton University, Ottawa, Canada Marianna PENSKY, University of central Florida, orlando [-a.uraBORY sEwICZ, Nova SoutheasternUniversity Friedrich PUKELSHEIM, Universitiit A ugsburg, Germany N. Rao CHAGAI{TY, old Dominion university, Norfolk, vA Monika PUKELSHEIM, Augsburg, Germany Xiao-Wen CHANG, McGill University,Montr6al, Canada Simo PUNTANEN, University of Tampere,Finland Pinyuen CHEN, Syracuse[Jniversity, New york Lianfen QIAN, Florida Atlantic University, Boca Raton John s. CHIPMAN, university of Minnesota, Minneapolis Bhargavi RAO, University Park, Pennsylvania Roy S. CHOUDHURY, Orlando, Florida C. RadhakrishnaRAO, PennsylvaniaState University Ka t-ok CHU, McGill University, Montrdal, Canada Albert SATORRA, Universitat Pompeu Fabra, Barcelona, Spain Knut CONRADSEN, TechnicalUniversity of Denmark, Lyngby Elke & Siegmar SCHLEIE, Dortmund, Germany Carles M. CUADRAS, ,Spain Shayle R. SEARLE, Cornell University, Ithaca, New York Naomi D'ALESSIO, Nova SoutheasternUniversity Peter SgUru-, University of Ljubljana, Slovenia Maryse DANSEREAU, Montrdal, Canada Mohammed SHAKIL, Pembroke Plains, Florida Anne DLIRBIN, [.ondon, England David S. SIMON, Nova SoutheasternUniversity JamesDURBIN, London school of Economics,[,ondon, Englancl Bimal Kumar SINHA, university of Maryland-Baltimore Friedhelm EICKER, universitiit Dortmund, Germany T. N. SRIVASTAVA, concordia university, Montrdal, canada R. william FAREBROTHER, University of Manchester,England Michael A. STEPHENS, simon Fraser univ., Burnaby, BC, canada Sheila FAREBROTHER, Shrewsbury England Evelyn Matheson STYAI\T,Montr6al, CanadaI Franklin, Vermont Nancy FLOLIRNOY, American university, washington, Dc George P. H. STYAN, McGill University, Montr6al, Canada Bernhard D. FLURX Indiana university, Bloomington Gerald E. SUBAK-SHARPE, City College, New York I-eah FLURY, Bloomington, Indiana Yijun SUN, University of Maryland-Baltimore SteveK. GALITSKY, Spr Inc., Oak Brook, Illinois Yoshio TAKANE, McGill University, Montrdal, Canada A lic ia GIov INA zzo, Nova southeasternunive rsi tv Yongge TIAN, Concordia University, Montr6al, Canada Patricie GOMEZ, Cali, Colombia Imbi TRAAT, , Estonia Jtirgen GROB, Universiftit Dortmund, Germany Birgit TRENKLER, Dortmund, Germany Frank J. HALL, Georgia StateUniversity, Atlanta GOtz TRENKLER, Universitiit Dortmund, Germany william D. HAMMACK, Nova southeasternuniversity Michel vAN DE VELDEN, Univ. Amsterdam,The Netherlands Greta HANDING, Phoenix, Arizona Hrishikesh D. vINoD, Fordham University, Bronx, New york Rebekka HANDING, Phoenix, Arizona Jrilia voLAUFovA, Louisiana StateUniversity, Baton Rouge Robert E. HARTWIG, North Carolina StateUniversity, Raleigh Heinrich vo Ss, Tech.Universitiit Hamburg-Harburg, Germany David A. HARVILLE, IBM-watson Res.center,yorktown Hghts Cheng WANG, Davie, Florida B erthold HEI LIGERS, univers i tiit Ma gdeburg, Germany Tonghui WANG, New Mexico Stateuniversity, Las cruces Harri HIETIKKO, University of Tampere,Finland William E. WATKINS, California StateUniversity, Illorthridge Kaisa HUUHTANEN, Tampere,Finland Hans Joachim WERNER, Universittit Bonn, Germany Pentti HUUHTANEN, University of Tampere,Finland Haruo YANAI, center for univ. Entrance Exam., Tokyo, Japan ShaneT. JENSEN, McGill university, Montr6al, canada Qiang YE, University of Manitoba, Winnipeg, Canada Andr6 KLEIN, , The Netherlands Fuzhen ZHANG, Nova SoutheasternUniversity T6nu KOLLO, University of Tartu, Estonia Yijun ZUO, Arizona State University, Tempe

F 'tr{j:.:i:. .:::']. i:.,,.. page1 B IMACE22

SelectedForthcoming Linear Algebra Events

Wehave selected seven forthcoming linear algebra events, sched- The aims of the sessionare to highlight the interaction between uled asfollows: statistics and matrix theory, and to promote dialogue between o June7-8,1999: Regina,Saskatchewan researchersin both areas. For furttrer information please contact:

o July15-1 6, 1999:Coimbra, Portugal EjaaAHtu gn: ahmed@math. uregi na.ca,

o July19-22,1999: Barcelona, Spain Doug FengNICK:faren [email protected] . August6-7,1999: Tampere, Finland SteveKIRrI-AND: [email protected]. o June26-28, 2000: Nafplio,Creece For the complete SSC Annual Meeting schedule please visit the o July12-14, 2000: Leuven, Belgium SSC Conference Web site: o December9-1 3, 2000: Hyderabad,India. http://www.math. u regi na. calSSCg 9/. For information on other linear algebraevents please visit the II-ASIIIC Website:

http:II www.m ath . tec h n i on . ac. i I liic I confere n ces. htm I Workshopon Geometric& Combinatorial SpecialSession on the Interaction Methodsin the HermitianSum Spectral Problem BetweenStatistics and Matrix Theory Coimbra,Portugal: f uly 15-16,1999 ReginorSaskatchewan: une 7-8,1999 f A problem in matrix theory which has interestedmathematicians for many years is the following: Given two Hermitian matrices Matrix methods in statistics have received some attention in re- A and B, describe the spectnrm of A+ B in terms of the spectra cent years; witness the ongoing series of Intemational Work- of A and B. Recently there were decisive developmentsin work shops on Matrices and Statistics, which has been nrnning since on this problem, with contributions from algebraic geometry, 19m. At the 1999 Annual Meeting of the Statistical Society of representation theory, combinatorics and harmonic analysi s. Canada (SSC), which will be held at the Univ. of Regina, June G9,l999,there will be a Special Sessionon the Interaction Be- This Workshop on Geometric & Combinatorial Methods in tween Statistics and Matrix Th*ry. This Sessionwill feature six the Hermitian Sum Spctral Problem will gather experts from 30-minute lectures on a broad spectrum of topics, as follows: different fields who have worked on this problem and will take placejust before the BarcelonaIL-A.S meeting, July l9-22,I999. o S. Ejaz Aguno (Univ. of Regina): "Irast sqwres, pre- The provisional list of spakers is as follows: liminary test and Stein-type estimation in general vector AR(p) models" (with A. K. Basu) o Jane Day, San Jose State University, San Jose,Calif.

o Kjell A. DoKSUM (Univ. of CalifonrirBerkeley): "Par- o Shmuel FnIEoLAND, Univ. of lllinois, Chicago tial regression culyes" . Alexander KI-vacHKo, Bilkent Univ., Ankara, Ttrrkey o Carl D. MnyER, Jr. (North Carolina State University): ' Aggregation/disaggregation error for Markov chains" o Allen KNursoN, Brandeis University, Waltham, Mass.

o Frangois PnRRow (Univ. de Montr6al): 'oAn exponential o Norman WILonERGER, IJniv. New South Wales, Sydney ' bound for a 0 - 1 function of a reversible Markov chain o Andrei Zw-gvINSKy, NortheastemUniversity, Boston. o SergeB. Pnovosr (Univ. of Western Ontario): "Hilbert matrices and density estimation" (with Y.-H. Cheong) Organizing committee: E. Marques de 56, Joflo F. Queir6, Ana P. Santana. Sponsors: CIM, CMUC, Praxis XXI. For more in- o George P. H. Sryax (McGill lJniversity): "Some com- formation e-mail: [email protected]/or visit our Web site: ments and a bibliography on the Craig-Sakauroto Theo- rem" (with S. \M. Dnry & Mylbne Dmais). htt p ://www. m at. u c . pt/lfq u e i rolwrksh p2 . ht m I April 1999 page1 9

The EighthILAS Conference The EighthInternational Workshop on Matricesand statistics Barcelona,Spain: f uly 19-22, 1999 Tampere,Finland: August 1999 You are kindly invited to attend the 8th Conference of the Inter- 6-7, national Linear Algebra Society (IIAS), which will be held at The Eighth lntemational Workshop on Matrices and Statistics the Escola Tdcnica Superior d'Enginyers Industrials in Barce- will be held at the Univ. of Tampere, Finland, on Friday, August lona (E-[SEIB). The subject of the Conference is Linear Algebra 6 and Satrnday,AugustT,1999 (starting with a reception on the in a broad sense,including applications. Thursday evening, August 5). This Workshop is a Satellite of Founded in the third century BC, Barcelona today is a large, the 52nd Sessionof the International Statistical Institute (ISI) to thriving and important city and port with a population of about be held in Helsinki from Tuesday, August l0 through Wednes- 2 million people. In the center of Barcelona is the Plaza de day, August 18, 1999. The sponsors are: Academy of Finland, Catalufra, a square-the largest in Spain-rimmecl with trees Finnair-The Official Carrier of the Workshop, Foundation of and highlighted by sculptures and fountains. Rarnble along La the Promotion of the Actuarial Profession, Intenrational Un- Runbla" the tree-lined boulevard that strechesfor 2 kimometres ear Algebra Society (support awarded for the IIAS kcturer), from the Plaza to the port-it is the very pulse of Barcelona. Nokia, and the Thmpere University Foundation. The chair of the Organinng Committee is Ferran hrerta of The Keynote Speakerfor this Workshop is: Barcelona: [email protected] co-chair for programs is Tom I-affey and the co-chair for local iurangementsis Mo Isabel >kANnERSoN, T. W., StanfordUniversity, USA: Garcia-Planas. The other members of the Organizing Comrrit- "Canonical analysis and reduced rank regression .' tee are: Raphael Bru, Luz DeAlba" J. M. Gracia, V. Hernr{ndez, in autoregressivemodels". Nick Higham, RogerA. Horn, Paul M. Van Dooren, and Richard A. Brualdi -6 The L,,ocalCommittee consists of: J. fficio. The Invited Speakersare: Clotet, A. Compta, M. D. Magret, and X. Rrerta. The program will include invited talks of 50 and 30 minutes. o ANuo, T., Hokusei Gakuen University, Sapporo,Japan: n The following speakershave agreed to participate: Z. Bai, J. "Ari thmetic- geomeri c mean inequali ties for matri@s' Ferrer, D. Hinrichsen, V. Kaashoek, S. KirHand, Chi-Kwong o CuntsrENSEN, Ronald, Univ. of New Mexico, Albu- Li, N. Mackey E. Marques de 56, K. Murota, V. Ptfk, F. Silva querque, USA: "Checking the independenceassumption Lrite, A. Urbano, and I. Zaballa. in linear models by control charting residuals" The program will also include several minisymposia. Their titles and organizers are as follows: o Golus, Gene H. (IIAS lrcnrer), Stanford University, USA: "Inverting shapefrom moments" o Linear Systemsand Polynomials (J. M. Gracia) o MusroNEN, Seppo, Univ. of Helsinki: "Matrix o Total Positiviry $.Ando, J. Garloffl compu- tations in Survo" o Parallel ,AsynchronousMethods (A. Frommer, D. Szyld) o ParannetrizationProblems in Linear Algebra and o Purr,lsHEIM, Friedrich, Universitiit Augsburg, Ger- SystemsTheory (P. Fuhrmann, {J. Helmke) many: "Kiefer ordering of simplex designs for second- t Combinatorial MatrixTheory (H. Schneider,B. Shader) degree mixture models with four or more ingredients" o Special Sessionin Memory of Robert C. Thompson (1. Day). (with Norman R. Draper and Berthold Heiligers) o Scorr, Alastair J., University of Auckland, New A special issue of Linear Algebra and its Applications 'An Tnaland: extension of Richards's Lrrnma, with ap- (I-AA) will publish the Conference Proceedings.This issue will plications to a class of profile likelihood problems" contain those papers which meet the standards of LAA and are approved through the nonnal refereeing processes. Submission o Sr,enlE, ShayleR., Cornell University, Ithaca, NIY,IJSA: 'oThe deadline for papersis November 30,1999. infusion of matrices into statistics" The registration fonn and further information are available o StNnn, Bimal K., Univ. of Maryland-Baltimore County, in the Conference Web page: USA: "Nonnegative estimation of multivmiate variance http://www-ma 1. upc.esli lasggi8th I LAS 99 .htm I components in rurbalanced mixed models"

by e-mail from: [email protected] send the Reg- o SryeN, George P. H., McGiIl University, Montreal, isrrarion Form by FAX to 34-93-4011713, by e-mail to: Canada: "Revisiting Hua's matrix equality and related [email protected], or by regularmail to the 8th ILAS'99 Con- inequalities, Sc,hurcomplementso and Sylvester's law of ference Secretariat, Dep. de Matemdtica Aplicada I, ESTEIB- inertia" (with Christopher C. Paige, Bo-Ying Wang & UPC, Av. Dagonal ffi7,E-08028 Barcelona,Spain. Fuzhen hang). page 20 IMACE22

The International Organi nng Committee for this Workshop A preliminary list of contributed papers is as follows: comprises R. William Farebrother (Victoria Univ. of Manch- o Ahmed, S. E., U Regina,Canada "Improved estimationof char- ester, England), Simo hmtanen (Univ. of Thmpere, Thm- acteristicroots in principal analysis" pere, Finland; chair), George P. H. Styan (McGill Univer- 'The t Akdeniz, Fikri, U Cukurova, Adana, Turkey estimation sity, Montr6al, Qu6bec, Canada; vice-chair), and Hans Joachim and analysisof residualsfor some biasedestimators in linear Werner (I-lniv. of Bonn, Borur, Germany). The Local Organi nng regression"(Poster) Committee at the tlniv. of Tampere comprises Riitta Jiirvinen, . Benesch,Thomas, Graz U l'echnology,Austria "Internet survey Erkki Liski, Tapio Nlummi, and simo Rmtanen (chair). by direct mailing" 'The o Borisov, lv{ikhail Vladimirovich, Novosibirsk, Russia in- Deadlines: Early registration: I June I99; Registration vestigationof the stability of trvo statisticalestimators to com- fee FIN{ 400 (- ca. tIS$ 8O)-after 1 June 1999, add FIN{ plex disturbancesin the data" 100; Full-time students: FIIv{ 50. F-or further information, con- o Chaganqv,hi. Rao, Old Dominion U, Norfolk, USA "statistical tact: The \trbrkshop Secretary-, Dept. of N{athematics, Statis- analysisof somemultivariate models using quasi-least squares" tics & Philosophy', tiniv. of Thmpere, Po Box 607, FIi\i-33 tOl . Chikuse,Yasuko, Kagawa LI, TakamatsuCity, Japan "statistical Tarrrpere, Finland; [email protected], FAX +358-3 -215-6157, infe.renceon specialmanifolds" http:/lwww. uta.fif sjp/wr-rrkshop 99 .htm l. o Drygas,Hilmar, LTKassel, Germany "simultaneousS\ID: Gril- lenberger'sapproach" A preliminary list of talks by students is as follows: . Farebrother,R. W., LI lv{anchester,LTK "Optimality resultsfor the method of averages" o Alpargu, Giilhan, McGill u, N{ontrdal, canada "Matrices in o Fellman, Johan, Swedish S of Econ & Bus Admin., I Ielsinki, statisticalprocedures for valid hypothesistesting in regression Finland "Robustestimators in linear models" models with autocorrelatederrors" o Gnot, Stanislaw,Pedagogical LT, Zielona G6ra, Poland "Estima- o Borisov, Mikhail Vladimirovich, Novosibirsk,Russia "The in- tion of variancecomponents by maximum likelihood methodin vestigationof the stability of two statisticalestimators to com- certainmixed linear models" plex disturbancesin the data" 'The o GroB, Jlirgen, LI Dortmund, Ge.rmany lrloore-Penrose.in- o Dumais, Mvlbne, N{cGill u, }vtontr6al, canada "The craig- verseof a partitionednonnegative definite matrix" Sakamototheorem" o Hauke. Jan, Adam Nlickiewicz LI, Pozna6,Poland "On some . Fugate.,Michael, lxrs Alamos, Ne.w Mexic.o, USA 'hdjusted matrix inequalitiesand their statisticalapplications" meansplots for detectinglack of fit in linear models" o Hovhannissian,Hrant, Northern Dept of NSSP,Ciyumri, Arrne- nia "Solving algorithm for the systemof linear algebraic o Grzadziel,Nfariusz, Agricultural LI Wroclaw. Poland "On non- equa- tions with five-diagronal,cyclic matrixes" negativequadratic estimation in linear models" o lshak,]vlaged, U Newcastle,NSW, Australia "Regressionmeth- o Guerra ones, valia, I of Math, cybernetics & Physics,Havana, 'A ods to model under registrationproblems in population data" Cuba techniquefor the numerical calculation of the lv{oore- "Recursive Penrosegeneralized inverse of an ill-conditioned matrix" I John, J. A., Li Waikato, Hamilton, New Zealand formulae for the averageefficiency factor in alpha designs" o Helle, Sami, U Tampere,Finland "Modeling of the fiber length . Kasala, Subramanyam,U North Carolina, Wilmington, LiSA distributionof mechanicalpulp using generalizedlinearmodels" "An exactjoint confidence.re.gion and test in multivariatecali- o Jensen, shane T., McGill u, Montr6al, canada "Some com- bration" mentsand a bibliography on the von Sz0kefalviNagy-Popoviciu o Klein, Andr6, LI Amsterdam,The Netherlands 'A generaliza- and Nair-Thomson inequalities,with extensionsand applica- tion of Whittle's formula" tions to statisticsand matrix theory" 'Approximation o Kollo, Tonu, U Tartu, Estonia of the distribu- o Klein, Thomas, LI Augsburg, Germany "optimal designs for tion of the samplecorrelation matrix" seconddegree /{-model mixture experiments" o L,€.e,Alan J., Ll Auckland, New Zealand "Gene.ratingcorre.late.d o Lu, Chang-Yu, Northeast Normal u, changchun, Jilin, China categoricalvariates via Kronecker products" "Some matrix resultsrelated to admissibility in linear models o Liu, Shuangzhe,tl Basel, Switzerland "On maximum likeli- and generalizedmatrix versionsof Wielandt inequality" hood estimationfor the VAR-VARCH model" o Luoma, Arto, U Tampere,Finland "Distance criterion in theory o Lomov, Andrei, Sobolev I of lv{athematics,Novosibirsk, Rus- of optimal design" sia "Identifiability of a matrix parameterby the stochastically perturtredright null space" o Nurhonen,Markku, LJN{innesota, Minneapolis, USA "N{atrices relatedto diagnosticmethods in classificationproblems" o Markiewicz, Augustyn, Agricultural Lt Poznari,Poland "Ro- bustlinear estimation in an incorrectl-vspecified restricted linear o Ollikaitten, Jyrki, U Tampere, Finland "Comparing haditional model" methodsto neural networks with large medicine data" 'A o lv{aroulas,John, Nat. Tech U, Athens,Greece geometricap- o Tian, Yongge,concordia Li, Montrdal, canada "when do matrix proach to the canonicalcorrelations in two- and three-waylay- equationshave block triangular solutions?" outs" o Wei, Yimin, FudanU, Shanghai,China "The modification of the o Mathew, Thomas, [J Maryland, Baltimore County, Catonsville, Drazin inverse" USA "Toleranceregions and simultaneoustolerance regions in multivariateregres sion" o Zhang, Bao-Xue, NortheastNormal LI, Changchun,Jilin, China 'lampere, "simple "Some results on estimatorof the mean vector and variance in o Merikoski, Jorma K., U Finland and good bounds for the Perron root" the general linear model and estimating problems for location (scale)models from groupedsamples under order restrictions" o Naumov, Anatoly, Inst. for Medical Statisticsand Epidemiol- ogy, lv,Iunich,Germany "New rnodels of optimal portfolio" April1 999 page 2 1

Nechval,Nicholas A., Aviation [J Riga, Latvia "Estimatingthe to attend the workshop or if you want to receive more informa- true order of a multivariate ARMA model via matrix determi- tion, pleasecontact the organizer : John ManoIJLAS, Dept. of nant minimization" Mathematic.s, Nati onal Technical LTniversity, Zo gr af ou L-ampus, Nechval, Konstantin N., Aviation U Riga, l^atvia "Uniformly undominatedestimation of a multivariate normal mean via in- Athens 15781,Greece; [email protected]. variant embeddingtechnique" Deadlines: To confinn your participation at the workshop : Paramonov,Yuri M., Aviation ij Riga, l,atvia "Order statistics 1 January 2000. To reserve your acpomodation : I lvfarch 2000. covariancematrix use for maximum expectation of P-bound for To submit the title and absract of your talk : I April 2000. random variable" Polasek,Wolfgang, lJ Basel, Switzerland "Outliers and break points in multivariatetime series" TheThird lnternationalConference on hrntanen, Simo, U Tampere,Finland "More on partitionedlin- ear models" Matrix-AnalyticMethods in StochasticModels . Shakil, Iv{ohammad,Pembroke Pines, Florida, USA "On com- plexity boundsfor some optimization problems" Leuven,Belgium: uly 12-14,2000 o Stgpniak,Czeslaw, Agricultural U Lublin, Poland "Information f channelsand comparisonof linear experiments'n This conference will provide an international fonrm for the . Tiit, Ene-Margit, U Tartu, Estonia "Some extremal properties presentation of recent results on matrix-analytic methods in of correlationmatrices" stochastic models. Its scope includes development of the o Todoran, Dorin, U North Baia Mare, Romania "Mathematical methodology as well as the related algorithmic implementations modelling and robot precision" and applications in comnrunications, production and manufac- o Trenkler, GOtz, U Dortmund, Germany "Powers and roots of turing engineering; it also includes computer expriments in the quaternions" investigation of specific probability models. . Ttoschke, Sven-Oliver, U Dortmund, Germany "Quaternions: A rnatrix oriented approach" The topiqs of interest include but are not limited to: method- . Vichi, h{aurizio, Societa Italiana di Statistica, Roma, Italy ology, general theory, computational methods, computer exper- "[.east squaresapproximation of non positive (semi-) definite imentation. queueing models, telecommunications modeling, matricesof correlationcoefficients" spatial processes,reliability problems, risk analysis, and pro- o Werner,Hans Joachim, u Bonn, Germany "More on the linear duction and inventory models. aggregationproblem" The organizers wish to encourage students to attend the con- Zalhraiev, Oleksandr, N. Copernicus I-I, Torufl, Poland "Dis- tance optimality design criterion and stochasticmajonzation" ference. To that effect, financial assistancewill be made avail- 'Testing able on a limited basis and a streamlined submission procedure Zmy(lony, Roman, Tech U Zielona G6ra, Poland hy- pothesesfor parametersin mixed linear models" will be implemented. Details will be published on the confer- Zontek, Stefan,Tech Ll Zielona G6ra, Poland "On construction ence \\reb page: http://www.eco n. .a c .be/ma m3 . @eries of a basis of Jordan algebra of matrices with applicationsto a should be addressedto fvtAtvt3 @eco n .ku leuve n. a c .be . speciallinear mode.l." TheNinth lnternationalWorkshop The Fifth Workshopon on Matricesand statistics, NumericalRanges and NumericalRadii in Cefebrationof C, R.Rao's B0th Birthday

Nafplio, Creece:fune 26-28, 2000 Hyderabad,India: December9-13t 2000 The Sth Workshnp on "Num€rical Rangesard Numerical Radii" The Ninth International Workshop on Matrices and Statistics, will be held in the historical and enjoyable town of Nafplio in Celebration of C. R. Rao's 80th Birthday, will be held in the (Nauplia) in the Peloponnese, Greece, from Monday, June 26 historic walled city of Hyderabad, in Andra Pradesh, India, on till Wednesday,June 28,2M, starting with a reception on Sun- December 9-13,2000. The progfam will start with a two-day day evening, June 25. The purpose of the workshop is to stim- course on recent advancesin Matrix Theory with Special Refer- ulate research and to foster the interaction of researchers on the ence to Applications in Statistics, on Saturday,December 9, and subject. The informal workshop atmospherewill guffantee the Sunday,December 10, 2000. This will be followed by presenta- excharrge of ideas from different research areas and in partieu- tionof researchpapers, which will be publishedin aprofessional lar, the researchersmay he better inforrned on the tatest devel- journal after refereeing. opments and the newest techniques. The Intemational Organinng Cornmittee for this TUorkshop There will be no registration fees and it is not possible to comprises R. W. Farebrother (Manchester), S. Puntanen (Tam- provide financial support to the participants. All participiants pere; vice-chair), G. P. H. Styan (McGill), and H. J. Werner though will have the chance, if they wish, to atteRd the social (Bonn; chair). For furtherinformatione-mail the l,ocal Organiz- events during the work^shopwithout charge. lnformation about ing Committeein India: K.VtswaNATH, [email protected] a@omodation will be available by December 1999. If you plan R. J. R. SweMY, [email protected]. page?2 IMACE22

ILASTreasurer's Report: March 1, 1998-February 28, 1999 by famesR. Weaver, Universityof West Florida, Pensacola

Balance on hand Uarch L_,-1998 Certificate of Deposits ( CD) t9 ,000. 00 Vanguard 8,882.L5 Checkinq Account 27 ,702.7L 55-584-t6 * ** * * **** * ** ** ***** *** ***** rt**** ***** * *** ***** * * ****** ** ************* * Checkinq Account Balance on March L- 1998 27 ,702,7L

March 1998 I ncome : Dues 844.00 Interest ( Gen) 134.'19 Contr ibut ions General Fund 40 .00 Hans Schneider Prize 50.00 Service Charge Refund 10.79 1,079.58 Expenses : Sec. of State 70 .00 Judy K. Weaver( IIAS Files ) L27.7 5 t97.75 881 . B3 April 1998 I ncome : 7th ILAS Conference 4 ,o60.0o Interest ( Gen) L7 .94 4,077 .94 Expenses: Service Charge 5.05 5.05 4 ,072 .89 Mav 1998 f ncome: Dues 640.00 7th ItAS Conference 4 ,304 . oo Interest ( Gen) 19. 87 Interest on CD (HS) 74,56 Interest on CD ( OT/ n 7 29 .00 Interest on CD ( FU) 23.L2 ILAS/Lee Lecturer 1 , o0o. 00 Contr ibut ions General Fund 25.00 Hans Schneider Prtze 25.00 6 ,L40. 55 Expenses: Judy K. Weaver ( IIAS Files ) 35.00 Producing & Mailing "Image" George P, H. Styan 1,315.10 7th ItAS Conference 6 ,450 . 0o Service Charge 5.35 7 ,805.45 ( 1 .664. 90 ) June 1998 I ncome : Dues 570.00 Interest ( Gen) 20 ,46 7th ILAS Conference-Cash 340.06 7th ILAS Conference-Checks L ,407 . o0 Cont r ibut ions General Fund L2 .00 2 ,349.52 Expences: 7th ILAS Conf.- Supplies 2,L00.00 7th ILAS Conference- Food/ ne f re shment s L ,LB9. oo 7th IIAS Conference-Refund 438.00 Judy K. Weaver ( ILAS Files ) B4.00 Service Charge 8.80 I .819.80 (t,470.28\ April1999 page

Julv 1998 f ncome : Dues 60.00 Interest ( Gen) 18.14 Interest on CD (HS) 75.39 Interest on CD (OT/lfl 29.32 Interest on CD ( FU1 23.63 7th ILAS Conference-Supplies 25.00 7th IIAS Conference-Reg. 90.00 311. 47 Expenses: Postmaster ( Stamps & Mailing ) 226.96 Judy K. Weaver ( Dues Notice ) 84.00 Service Charge 7.95 319.91 ( 7 . 34 ) Auqust 1998 I ncome : Dues 2 ,O36.51 7th ILAS Conference t ,921 . 00 Interest ( Gen) 16.16 Contr ibut ions General Fund 191.00 Hans Schneider Prtze 191.00 OT/JT Lecture Fund 181.00 F. Uhlig Fund 71.00 Conference Fund 51.00 4 ,659.67 Expenses : Judy K. Weaver ( IIAS Files ) 59 . 50 Service Fee Z ,59 62.09 4,596. 5g September 1998 fncome: Dues 280.00 Interest ( Gen) L2 ,96 Contr ibutions General Fund 45.00 Hans Schneider Prize 80.00 OT/JT Lecture Fund 30.00 F. Uhlig Fund 5.00 Conference Fund 25.00 477.86 Expenses: Office Depot ( Supplies ) 61. 54 Postmaster 29t.04 Service Charge 16.50 369.08 108. 78 October L997 I ncome : Dues 450.00 Interest (Gen) t2 ,95 Interest on CD(HS) 76.23 I nterest on CD( OTI JT ) 29.64 Interest on CD(FU ) 21. 55 Collection Item 20.00 Contr ibut ions General Fund 20.00 Hans Schneider Prize 50.00 Conference Fund 10.00 700.3'l Expenses: Judy K . Weaver ( ILAS Files & Supplies ) 170.94 Service Charge 2 .3L L73 ,25 527. L2 November 1998 I ncome : Dues 640.00 Interest ( Gen) L3.24 Interest on CD (HS) 25.68 I nterest on CD ( OT/ Jf 7 9.99 Contr ibut ions page24 IMACE

General Fund 31. 00 Hans Schneider Prize 40.00 OT/JT Lecture Fund 40.00 F. Uhlig Fund 30.00 Conference Fund 30.00 g5g.g1 Expenses : Producing & Mailing I'IMAGE" George P. H. Styan 1 , 325 . 30 Service Charge Z .96 L ,329.26 ( 468.3f) December 1998 I ncome : Dues 940. 00 Interest ( Gen) 13.39 Interest on CD(FU) 2L,32 Contr ibut ions General Fund 180. 00 Hans Schneider Prize 220.00 OTIJT Lecture Fund 200.00 Conference Fund 35.00 L,609.70 Expenses: Postmaster 17Z,BO Judy K. Weaver ( Dues Notice & ILAS files ) LLZ.00 Service Charge Z.S3 287.33 L ,322.37 Januarv 19 9 9- I ncome : Interest ( Gen) L2.96 L2.96 Expenses : ItAS Lecturer 500.00 Service Charge 6,26 505.26 ( 493.30l Februarv 1999 f ncome : Dues 1,360.00 Interest ( Gen) Lt.t7 Interest on CD(HS) 70.78 Interest on CD(OTI JT ) 27.53 Contr ibut ions General Fund L52.00 Hans Schneider Prize 60.00 OT/JT Lecture Fund 2.00 F. Uhlig Fund L2.OO Conference Fund 25.00 .L,720 .49 Expenses : 00,00 00.00 t ,'120. 49 ********************************************************************** Februa

Account Balance Checking Account 36, g2g.59 Certificate of Deposit (Fu) 1,500.00 Certificate of Deposit ( Gen. ) 10,000.00 Certificate of Deposit (72t HS & 28t OTlJT ) 7 ,500.00 Vanguard (72t HS & 28r OrlJr) 9,523.65 65,352.24

General Fund 26,378 ,70 Frank Uhlig Educational Fund 2 ,906 .62 Hans Schneider Prize L6 ,235 . g',l Olga Taussky Todd/John Todd Fund 8,533. 11 Conference Fund 9 1707,94 ILAS/LAAFund 1,590.00 ***********************************************************************65,352.24 - - ,a. t t aF/i;dL V,;;- Z/it / April1 999 page25

IMAGEProblem Corner: Problems and Solutions

We look forward to receiving solutions to Problems 1&1, 19-3b, l9-4 & 2l-2, which we repeat below; we also present some further comments conceming our solutions to hoblem l9-5 in IMAGE 2O(April 1998), pp. 28-31, and the comments on these solutions on page 22 in IMAGE 21 (October 1998). We present solutions to Problems 2l-1, 213, 2l-4 & 21-5, which appearedin IMAGE 2l (October 1998), p. 28. In addition, we introduce seven new problems (p, 32) and invite readers to submit solutions, as well as new problems, for publication in IMAGE. Please send all material both in l4tg[. embedded as text in an e-mail to [email protected] and by pmail to G. P H. Styan, Dept. of Mathematics & Statistics,McGill University, 805 SherbrookeSheet West, Montrdal, Qudbec,Canada [f]A 2K6.

Problem 18-1: 5 x 5 Complex Hadamad Matrices Proposedby S. W. Dnunv, McGill University,Montdal, Qudbec,Canada. Show that every 5 x 5 matrix U with complexentries ui,* of constantabsolute value one that satisfiesU*U = 51 cm be realizedas the matrix (rjo )i,* wherec.r is a complexprimitive fifth root of unity by applyingsome sequ€nce of the following: (1) A rearrangernentof the rows, (2) A rearrangeme,rt of the columns,(3) Multiplication of a row by a complexnumber of absolutevalue one, (4) Multiplication of a columnby a complexnumber of absolutevalue one.

The&litor has not yet receiveda solution to this problem-indeed eventhe Proposerhas not yetfound a solution!

Problem 19-3: CharacterizationsAssociated with a Sum and Product Propo,sedby RobertE Hanrwtc , North Carolina State University,Raleigh, North Cmolina, USA, PeterSEMRL, University of MaribonMaribon Slovmia& HansJoachim WERNER, Universitiit Bonn, Bonn, Germany.

(a) Chracterize squarcmatrices A nd B satisfyrngAB = pA * gB, wherep md g arc grvenscalrs. (b) More generally,characterize linear operators,4. and B acting on a vector spaceX satisfyingABo € Span(Ac, Bc) foreveryo e N.

Solution 19-3a.1by the Proposers:Robert E. HaRrwtc, PeterSruu & HansJoachim WsnNsn. Clearly,AB=pA*qBif andonlyifA(B-pI)=qBo\equivalentlsqpl=(A-cI\(B -pI).HenceAB=pA*qB if andonlyifqB(B -pI)-(B -pI) = qB nd,forsomematrixZ,A- qB(B -pI)- + ZII -@ -et)(B -pl)-1. Here (.)- ref€rs to an arbitrary generalizedinverse. In paticular, wheneverpC + O,then AB = pA + qB if nd only if (B - pl)-l existsand A - qB(B - pI)-L

Asolutionto Part(a\,whenpqf 0,wasalsoreceivedJromChristopherC. PAIGE, Mccill University,Monb6al,QC,Canada. Welookforward to receivinga solutinn to Part (b) of this problem.

Problem 194: Eigenvaluesof Non-neg3tiveDefinite Matrices Proposedby Fuz.henZsauc, Nova SouthcasternUniversity, Fort Lalderdale, Florida, USA. ShowthatthereareconstantsT€(0,|]ande€(0,+)suchthat,ifA1,...,AnTenon-negativedefiniterxrmatricesof rmk one satisfying

(a) Ar + "'+ An = I,

(b) trace(.A;)< 7 foreachf=1,...,n,

thenthereisasubsetoof{1,2,...,n} suchthattheeigenvaluesofAo=DnroA;alllieintheinterval(e,1-e). Welookforwqrd to receivinga solutionto thisproblem! page26 IMACE22

Problem1 9-5: SymmetrizedProduct Definiteness? Proposedby IngramOLKIN, Stanford University, StanJord, Califurnia, USA. lrt A md B be real symmetric rnatrices. Their prove or disprove:

(a) If A and B are both positive definite (non-negative definite), then AB 1. B A is positive definite (non-negative definite).

(b) If A arld AB * BA are both positive definite (non-negative definite), then B is positive definite (non-negative definite).

Solutions 19-5.1-19-5.5 respectivelyby David C,q,lI-aN,Shuangzhe Ltu, David LoNooN, HansJoachim WEnNpn and Henry Wolxowlcz, appearedin IMAGE 20 (April 1998),pp. 28-3I; for commentson thesesolutions by Karl E. GustarsoN seeIMAGE 21 (October198),p.22.

Additional Comment by Karl E. GustersoN, Llniversityof Colorado,Boulden Colorado, USA. In my ealier commeNrtsI pointed out that the cormter-examplesto (a) could all be seenin termsof the sufficientcondition: sin B < cosA, cf. Gustafson(1968). Without going on ad infiniturn,I wouldlike to addtwo obseryations: (1) The pagenumbers in my citedreferences [1,2,3] in IMAGE 21,p. 22,shouldbe p. I26,p. 334,p. 62, respectively. (2) As observedin IMAGE 20 (April 1998),p. 28, this problemis discussedon pp. 120-121in l-ax (1997\. I-ax (cf. pp. 12O-12Land 137-138) gave two proofs of (b) in order to exposenice usesof homotopyand squareroot arguments, respectively. Solutions 19-5.1-19.5.5to (b) were (naturally, since the questionwas statedfor real matrices)of the matrix variety,whereas lax was treatingarbitrary selfadjointoperators A and B. Here I would like to provide an altenrateproof to (b) which, moreover,bers on (a). This alternateapproach is basedupon the usefirl result of Willims (1967): If zero is not in the closureof the numericalrange of an operatorA in B(X), i.e., 0 f W@, thenfor my operatorB in B(X) one has

c(A-la\ cW6 /W6.

Williams's result has the additionalmerit of needing(in Hilbert space)only a few-linesproof dependingessentially only on the fact that the spectrumof a boundedoperator is containedwithin the closureof its numericalrange. With more work, the result holds as well in Banachspace. Wielmdt had someearlier resultsfor matricesalthough it is unclearwhether Williams wasaware of those.See Gustafson & Rao.(1997, pp. 34-36) for more details. The proof of (b), for the postivedefinite cas€,now goesas follows. Since,4,B + BA - 2FieBA = 2PieAB ) 0 md A > 0, we write B - A-r (AB), from which

a(B)cw64lW The result (b) follows immediatelybecause W@ = convexhull{o(B)} = a real interval when B is selfadjoint. Moreover we may also immediatelyexhact the affirmative spectralcontent of (a). Given A andB positive definite, we writeA = (A-t;-t, from which o(AB)cW@ /WWl Thus, eventhough W (AB) neednot be positive,o(AB) alwaysis. To getW (AB) positive,I introducedmy operator higonometricconditions (cf. Gustafson,1%8).

References [1] Karl E. GUSTAFSoN(1968). The angle of an operator and positive operator products. Bulletin oJ the Americot Mdhematical Societl 74,&492. [2] Karl E. GUSTAFSoN & Duggirala K. M. RAo (1997). Numerical Rorye: Tttc Field oJ Valaes oJ Linear Operaton od Morices. Springer-Verlag, New York [3] Peter LAx (1997). Lineat Algebra Wiley, New York. [4] J. P WILLIAMS (1!X7). Spectra of products and numerical ranges. Joumal of Matlumatical Analysis otd Applicatbns 17, 2l+22O. April1 999 page27

Problem21-1: Square Complex Matrix of Odd Order Proposedby LudwigEI-sNpR, Universitiit BieleJeld, Bielefeld, Germany. Irt ,4 denote trt n x n complex matrix, where n is odd. Show that there exist mrmbers F;, i = I,2 on the unit circle and nonzero vectors rr, tz e C" satisfying - AH *, thArt, Ar, = FzArz.

Solution 21-1.1 by RogerA. HonN, IJniversityof Utah,Salt lnl

First supposethat ,4 is nonsingular,so we are to show that B = A-r AH and C = A-r A eachhas at least one eigenvalue withunitmodulus.Then C-l - CandB-1 = (e-r;aAis similartoA(l-1n = BH,whichis similartoB.Consider the Jordanstructure of a nonsingularmatrix D whoseinverse is equalto or simila to its conjugate:D- I and D havethe same JordanCanonical Form. If ,\ is an eigeirvalueof D that doesnot haveunit modulus,then ,\-l I \ nd,every Jordanblock of D with eigenvalue) is pairedwith mother block of the samesize with eigenvalue). Thus, the sum of the sizesof all the Jordanblocks of D with ooo-unit eigenvaluesis even. If D has odd order,it must thereforehave at least one eigenvalueof unitmodulus. If .4 is singular,then .4. = A * e1 is nonsingularfor all sufficiently s-all positivee . I-et )" and r. be a rmit eigenvalue andunit eigenvector,respectively, of B, : A;t A! , so Brc, = \ere andA! r, = \rArxr. By compactnessof the unit circle and the rmit ball of lC" thereis a sequ€noe€3 -+ 0, a 'nit scalar,\, anda unit vector r suchthat )ro 4 ,\ md_cr. -f c, soAHx: lim*-+- A!*rrr = lim*-+oo\r*Ar*c"r = ),Ao.Thesameargum€ntgivestheassertionaboutA aordA. tr

Sofution 21-1.2 by FuzhenZnaltc, NovaSoutheastern University, Fort Lauderdale,Florida, USA.

We first establisha lernma,which is of interestin its own right. lcmma.LetXandYlorcnxnHermitianmdskew-Hermitimmatrices,respectively.ThenforanyeigenvaluelofXY, either.\ € ilR, or ,\ and -.\ occur in pairs as the eigenvaluesof XY with the samealgebraic multiplicity. The assertionalso holdsformatricesX andY satisfyingX = X andY = -Y. Proof. For any tr € R, we have

p(l) :: det(f.\I - XY) = det(-iM - YH xH) = d"tH)I+ vX) = detll.\r + XD = (-l)"detft)/- x7) = (-l)"t(.\t.

Thus for all real .\, p(f) = p(I) if n is even, anrdp()) = -p(I) if n is odd. Accordingly, we nay assumethatp(,\) = f(,\) if n is even and p(,\) = i/(,\) if n is odd, where f(,\) is some real coefficient polynomial. Further if C(,\) is the characteristic pollnomial of XY, then C(.\) = p(-i)) = f(-i.\) o12f(-lD Now if .\ is an eigenvalueof XY, then C(.\) = 0 or f (-il) = 0. Thus -t.\ € lR, i.e. .\ € rlR, or -f.\ and -fI = r.\ occur in conjugate pairs (as zeros of /(,\) with the same multiplicity). Thus C(-l) = f (-t(-l)) = t(il) = 0, and -l is an eigenvalueof XY . The le,mmais proved. n

It follows from the lemma that if X and Y are further nonsingular, then XY has at least one eigenvalue in the form ir for somereal r, sincedet(XY) = det(X)det(y) € iR. Now we are rcady to solve hoblem 21-1. Notice that the assertation is equivalent to sayrng that AH - hA nd A - pzA are singular for some p1 nd pz ol1 1frs lnit qircle. We show the case for A' - ,rA. The other one is similar. tf AH + A or AH - A is singula, then the assertion is true. Suppose otherwise they are both nonsingulr. Then by the above qgum€trt, -A)hoatleastoneeigenvalueine'R,i.e.,det -A)] thematrix(AH + A)-t14n [itl +(A'+A)-r1g = 0forsome , € R. Thus / detII t' - l.*r) --''--et(AH +' A)det[itl--/---L- + (AH + A;-r (A' -A)] = o. \ | + it" ) (l +;t;" ThereforeA* - rrAis singular,wherepl= (1- it)/(L+ it)isontheunitcircle. tr We note that the conclusiondoes not hold in generalwhen n is even. page28 IMACE22

Solution 21-1 .3 by the Proposer:L,udwig ELS NER, Universitiit Bielefeld, Bielefeld, Germany. 'We assnmefrst that A is nonsingulr. In this ciasewe haveto show that the matricesBr = A-t AH andBz - A-L Ahave eigenvalues on_thermitcircle.From B!fr - AH A-r we havethat Br is similarto BIH ,md hencefor any ,\ in the Ft, Fz - spectnrmof .B1also (.\) 1 is in this spectnrm.As thereis an odd numberof eigeirvalues,in this matchingone eigenvaluep1 is matchedwith itself, henceit is on the rrnit circle. In the secondcase we set ,4 = T + iS with ,S,? real. We assumealso that ? is nonsingula. C = T-r S hasa real eigenvaluea. It follows from

82=(T +ifl-1("-rS) = (r+ic)-Lg-ic)

thatBz hasaneigenvalue p2 = (I-io)l$*io), he,ncelpzl = l. Thegeneral case follorrsby theusualcontinuityargume,nt. tr

Prcblem 21-2: The Diagonalof an Inverce Proposedby BeresfordPARLETT, University of Califurnia, Berlrelq, California, USA, via Roy MATHIAS,College of Williamand Mary, Williansburg,Vrginia,USA. kt J be m invertible tridiagonaln x n matrix that pemits rimgulr factoriziationin both increasingmd decreasingorder of rows: J = L+D+U+ and J =U-D-L-. (Herethe ["ls re lower trimgulr, the U's re uppertimgulr, md the D's are diagonal.).Show that

("I-1)* = [(D+)r* * (D-)g - Jxx]-'.

Welookforward to receivinga solutionto thisproblem!

'li^o Problem 21-3: Eigenvaluesand Eigenvertorsof SymmetricMatrices Proposedby Willim F. Tnsxcu, Trinity University(Emeritus), San Antonio, Texas& Divide, Colorafu, IISA. Find the eigeirvaluesmd eigenvectorsof the n x n symmetricmatrices

,4 = {min(i,i)}T,i=r md B = {min(2i - I,zj - 1)}ii=r.

Solution 21-3.1 by the Proposer:Wliam F. TnsNcs. For the momentconsider the matrix - C {min(ci - b,aj - 6)}ii=, wherea ) 0 md a t' b. ltis straighdorwrd to verify that

Za-.b _14 a-b 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 -: C-1 ::::::: 0 0 0 0 -1 2 -1 ; 0 0 0 0 ... 0 -1 2 -1 0 0 0 0 0 0 -1 1

If rs , r! , xn+1 satisfy the difference equation

rr-1 - (2-acr)r'r*r.r*1 =0, 1< r1n, (l)

and the boundary conditions (a - b)ro * br1- 0 and tn - rn*1 : 0r (2) April1 999 page29

- then c lrrr, ... rnfT satisfies C-rr = ac; therefore, a is m eigenvalue of C-L if md only if (1) has a nontrivial solution satisfyrng (2), in which case r is an o-eigenvector of C- 1. The solutions of (1) are given by - nr c1(' + czC-' , (3)

where( is a zeroof the reciprocalpolynomial

p(r) = ,2 - (2- oo)+ 1.

Sincep(z)=(r-OQ-Il0=r'-(C+llQz+l,itfollowsthatif(3)determinesaneigenvectorof C-Lthenthe correspondingeigenvalue of A- 1 is o=l (r-r-l\ (4) o\ q)' The boundaryconditions (2) yield the system I a-b+bC a-b*bl|.1 f"'l: fol rlc"-rlc+l - Lfl-c"*' J L", J Lo I with deterninmt (-"- 1(( - 1)d((), where

d(()= (o- 6)(4zn+1+ 1)+ b(C2"+ 0. Therefore,(3) determinesan eigenvector of C- 1 (md of C) if andonly if d(O = g.

If a=1and6=0thenC=A,so(3)deternrinesaneigenvector of A-rifmdonlyif qzn+t=-l.Therefore

i(2k* r)r (=exP, 2"+l

for some integer /c. Since the boundary conditions (2) require that 16 = 0 and xn = tnlt, (3) implies that *,= "ro.ffff, r =!,2,...,n,

ae the cornponents of an eigenvector corresponding to (. From (4), the corresponding eige,lrvalueof A-r is ('l * t)") ,'- = ? (r- . o \- "o. 2n*r )'

and) = 1/ois aneigenvalueof A. trtting/c :0, 1,...,D - l yields acompletesetof eigenvaluesandeigenvectors of ,4.

If a:2mdD= lthenC=B,so(3)detenninesaneigenvectorof B-1 if andonlyif e2n =-l.Therefore

. i(2k * Ilr t = exp 2r, conditions (2) require that rs - -sr and r,. = tn*r, (3) implies that

+ r)(2r - r)t rr = sin ____i______::,Qk r = 1,2,...,n,

arethe components of aneigenvector by correspondingto (. From(4), thecomesponding eigenvalue of ,4- I is (, t=;\t-cos-2ntrt.._2 -^^(2k+1)r\ )' - and.\ = | is meigenvalue of B. Irtting lc = 0, I,...,fl l yields acomplete set of eige,nvaluesmd eigenvectorsof B. tr page30 IMACE22

Problem.214: SqqareComplex Matrix, its Moore-FenrcceInverce and the Ldwner Ordering Proposedby Giitz TnnNKLen, universitiit Dortrnund,Dortmund, Germarry. (a)Irt A&asquarematrixwithcomplexentriesmdMoore-PenroseinverseB = A+. ShowthatAA* < A*Aif md only if B B* < B* B, where I de,notesthe Ltiwner-orderingof matrices. (b) Doesthis result generalizeto operatorson a Hilbert space?

Solntion 214.1 by FuzhenZHANG, Nova Soutlrcastern{Jniversity, Fort Intdcrdale, Ftorida, USA.

Part (a) follows immediatelyfrom the observationthat if X is a fini1scorrplex matrix, thenit follows that XX* < X* X <+ = XX* X*X (i.e.,Xisnormal),sincetrace(X*X-XX*) = 0 forall*. lNote:Thisisoneofthe30importaniequivalent conditionsfor rnatrix normality in Chapter8 of Thng (1999).1 For hrt (b) the msw€r is no: The assertionin Part (a) doesnot generalizeto operatorson Hilbert spaces.Take, e.g., cf. Wang& 7h^g (1995),the ShiftingOperator .9 : (c1,c2rcys.. .) *+ (0,eL,cz,ca,. . .) f* the 12space. Then ^9* :lS+ , ("r,"r,ca,...)r-+("2,ca,...).Itiseasytoverifythat^9+^9=,S*S=landSS*

Solution 214a.1by Rocrn A. HonN, Universityof Utah, SaltLakc City,Utah,USA.

To mswer Part (a), let A = VEW* be a singulr valuedecomposition md computeAt = WE|V*, l(At\*At)t = - - VE2V* AA',and/i(.4t)* | W('j|)2W. (,4*,4;t.WeusethefactrhatX >y > 0if andonlyifp(Xty) AA* if md only if

| > p{(A. A)t (A,4-) } = p{(AA. ) (A. e\t 1 = p{l(A\. Atlt [/t (At).] ] if mdonlyif (At).1t > Al(11).. D

Sofution 214a.2 by the Proposer:Giitz TnpwrLER, UniversrtAtDortmund, Dortmund,Germany. We presenttwo solutionsto Prt (a):

Solutionl.'SinceAA* < A*AwehaveA*A = AA* +CC* forsomemakixC. Takingtraqesweobtaintr(A*A)- ft(AA.) +tr(CC-). Theidentityt{A.A) - tr(,,4A.)theneirtailstr(CC.)= 0,i.e. C= 0. Henw,AA*= A*Ashowing A to be normal (cf. Halmos,1967, Problem 160). But it is well known that a matrix is normalif md only if its Moore-Penrose inverseis normal. tr

Solution2: By Bahsaluy,hrkelsheim md Styan(11E9, Theorem 4.3) we have(A-A)+ < (AA*1+, sincerank(AA*) = *A) - rank(, (seealsoMillikenandAkdeniz,1977). Itfollowsthat A+A*+ < A*+A+,andby A*+ A** we letBB* = B*8. tr

Comment:It wasnot possiblefor the poserof the proble,mto generalizethis result to operatorson a Hilbert space(but seethe secondpart of Solution 214.1 above). Note, however,that an invertible operator? on a Hilbert spaceis hypemormal(i. e. TT* < ?*T)if mdonlyif ?-l ishypernormal(cf. Berberim,l%|,p. 161).

References [] Jerzy K- BAKSALARY, Friedrich PUIGLSHEIM & George P H. STvAN (1989). Some properties of matrix partial orderings. Linear A lge bra od its Applic otions 1 19, 57-45. [2] S. 1( BERBERTAN(l%L). Int odnction b Hilbert Space. Oxford University Press,New York. [3] Faul R. HALMoS (l9ci7). A Hilbert Sprce hoblem Book. Yan Noshand, hinceton, New Jersey. (2nd revised & enlarged edition: Springer, 1982.). [4] George A. MLLIKEN & Fikri AKDEMZ Qqn>. A theorem on the difference of the generalized inverses of two nonnegative mahices. Communbdbw in Stotistics-A, Tlwory od Metlpds 6,73:79. April 1999 page31

Problem 21-5: Determinantsand g-lnverses Proposedby SimoPuNtl,NBN, Ilniversity of Tampere,Tampere, Finland,George P. H. SryeNr,McGill University,Montrdal, Qudbec,Canada & Hms JoachimWr,nNr,n, UniversitiitBonn, Bonn, Germany. For a real matrix B,let B- denotean arbitrary g-inverseof B satisfyingBB- B = B, and let {B- } delrotebriefly the set of all g-inversesof B. Now let ,4 € R'x" and X a pnxk be two given matricessuch that A is non-negativedefinite and symmetric,X'X = Is md R(X) g n@); hereR(.) denotesthe range(col'mn space)of (.). Definethe matrix fimction M(.) from {A- } into R'x' accordingto

M(A-) := In - XX' + A- XX' A.

(a) Does M(A-) dependin generalon the choice of ,4- € {A- }? And if it is so, characterizethe restrictivecase of invariance.

(b) DoeslM(,4- ) I, with | . I indicatingdeterminant, depend on the choiceof .4- € {A- }? (c) ForeachA- e{A- },determineexplicitly{M(A-)- }intermsof A,X,A-.

Solution 21-5.1 by the Proposers:Simo PuNr.qNEN,George P. H. SryaN & HansJoachim Wr,nxsn. Thereexistsannx("-k)matrixYsothatthematrixP-(X,Y)isorthogonal,andsoXtY=O,Y'X=0,and Y'Y = 1,-3. PutW(A-) :- P'M(A-)P. Thenclealy lW(A-)l = lM(A-)lmd x' A-xx' AY w(A-)- ( {:,+-:{'Ax \ I r'A- xx'Ax In-k + YtA- xx'AY )

Since?(X) 9R(A),weknowfrom,e.g.,Theorern2.3inWemerandYryar(1996)thatXtA-Xisinvaimtforanychoice of ,4- andR(Xt A- X) = R(X').In viewof XtX = I*, clearlyrank(X) : & andso XtA- X is nonsingular.From

/ t* O \ ( X,A_ XXI AX XI A- XXIAY w(A-)- -1 tt \ Ir'r-x (x'A-n I"-n) \ o In-n ) we then get lM(A-)l: lw(A-)l: lx'A-xl .lx'AXl (2) whichis invuiantfor anychoice of A- . Observethat the nonsingularityof Xt A- X is equivalentto lX' A- Xl I 0. SinceA is non-negativedefnite and symmet- ric, AR(X) n,l\,f(X') = {0}. This in turn implies thatXtAX is nonsingularand so lX' AXI I 0. Fromour determinmtal formula (2) it now follows that M (A- ) is, irrespectiveof the choiceof A- , nonsingulu. We leaveit to the readerto check that the matrix expressionon the right handside of the equality sign in

M (E_1-T= IN* zX(X'AX)_T1Y' A- X)_I X' - X1X'AX)_I X'A _ A_ X(XI A- X)_LX" sayG, satisfieswith M(A- ) G = In thedefiningequationforthe nonsingularinverseof M (A-). Needlessto say,M(,4- )-1 is then the only g-inverseof M (A-). In passing,we further mention thal M (A-) can also be derivedrather easily,in m alternativeconstructive matrner, from M(,4-) = PW(A-)P' by exploiting the speciallower and upper block triangular structurein the factorization(l) of W(A-). Contrry to its det€mlinant,the matrix M (A-) is ge,nerallynot invrimt for the choiceof A-. This is seenas follows. Trivially,R(x) = R(x(xt AX)-t). So M(A-)x(x'AX)-t = A- x. Sinceft.(x) g R(A), ir now followsfrom Theorem2. 1 (ii) & (iii) in Wernerand Yapar(19%) that A- X md so M (A- ) is invaiant for the choiceof ,4- if and only if Aisof fullmlumnrank. SinceAissquae,thishqpensif mdonlyif .4isnonsingulamdoursolutioniscomplete. tr

Reference [1] Hans Joachim WERI.IER& Cemil YAPAR (f 996). On inequality constrainedgeneralized least squaresselections in the generalpossibly singularGauss-Markov model: a projector theoretical approach.LircatAlgebmodia Applications 237123,8,359-3y3. page32 IMACE22

IMAGEProblem Corner: New Problems

Prcblem 22-1: hwers, g-lnvenes and Core Matrices Proposedby Giilhan AlpeRcu and GeorgeP. H. Sty.e,Nr,Mccill (Jniversity,Montrdal, Qudbec,Canada & Hms JoachimWEnuER, Universitiit Bonn, Bonn, Germany.

For a real matrix A,let A- denotean arbitraryg-inverse of .4 satisfyingAA- A = ,4, and let {A-} denotethe set of all g-inversesof A. Showthat for a real squre matrix ,4 the following conditionsare equivalent: (u) DLo At e l(I - A)-| forsomenonnegativeinteger h. (b),4" = All fot somenonnegative integer s. (c) The core part C(A) of A is idempotent. [We recall that a squarematrix ,4 has index ind(A) = /c iff rhere exist a nilpotentrnatrix Na of degreeft (i.e., NL = 0 while NI-' + 0) and a corematix Ca 9.e., ind(Ca) = 1) suchthat A = Ca + trfa, CeNe = NtC,q. = 0; cf. e.g.,pp. 11*177 in Adi Bpl-Isnepl & ThomasN. E. Gnnvtu-r,, Generalized Irwerses:Theory od Applicaions, Wiley, New York, 1974;corrected reprint edition: Krieger, Huntington,}.ry, 1930.1

Problem 22-2: SquareComplex Matrices, Linear Maps and Eigenvalues Proposedby Lndwig Et sNpn, Universitiit Bielefeld,Bietefetd, Germany. (a)Irt M' denotethesetofn x n complexmatrices.Finda linearmap S : Mn 1 Mn, withthefollowingproperty: If A 9 IUIJ,has the(notnecessarily real) eigenvalues \;,i = I,...,ft thenS(A) hastheeigenvalues A;, i = 1,..., n, aswell -f,Tr' as l,{ and for all pairs i < j. (b) Showthat thereis no liner map ^9 : Mz ---+ M2 satisfyingthe following condition: If A e Mzhas the eigenvalues ),;r theirS(A) hasthe eigeirvalues+.1/T1t.

Problenr22-3: The Rankof a Matrix Difference Proposedby YonggeTtl.N, ConcordialJniversity, Mont6at, euebec,Canodn. Irtthen x nmatrices,4andBbeidempotent.Showthatrank(A- B) = rank(-A- AB\ *rank(B - AB).

Problem 224: Determinant of a Certain FattemedMatrix Proposedby Willim F. TRENCH,Divide, Colorado, USA. I.llta,,b,carrrd,itbecomplexmrmbers.FindthedeterminantofGn=1o(i- j)*Dmin(f,j)*cmax(f, j)+dlir=r.

Problem 22-5: Eigenvaluesand Eigenvectonof a Certain Toeplitz Matrix Proposedby Willian F.TRENcH, Dividz, Colorado,USA. - Find the eigeirvaluesmd eigenvectorsof the Toeplitzmatrix ? {o cos(r - s)t f Bsin(r - s)l}L"=r, with a2 + p2 + 0, n ) 3, ndt I /ca'with ft m integer.

Problem 22-6: tdempotencyof a Certain lMatrix Quadr:aticForm Proposedby Michel vAN DE VBLosN, UniversityolAmsterdant, Amsterdam, The Netlurlands, ShumgzheLlrJ, Universiliir Basel,Basel, Switzerland & Heinz NpunEcKER,Cesarc, Schagen, The Netherlands. I-€tAbemn x n symmetricidempotentma8ixandletUbean n x k (k < n) matrixsuchthat(Jt(J = [. Characterize thosematrices U for whichUtAU is idempotent.

Problem 22-72chancterization of a square Matrix in an tnner-prcduct lnequality Proposedby FuzhenZHANG, Nova Southeastern(Jniversity, Fort Lauderdale,Florida, USA. I.et 4 bean a x n complexmatrix. If, for somereal number o ,l(AV,V)l < (y,y)'for all n-columncomplex vectors y, what can be said aboutA?

Pleasesubmit solutions both in I4f$ embeddedin an e-mailto [email protected] by pmail (nicelyprinted copy, pleasel) to G. P H. Styan,Dept of Mathematics& Satistics, McGill University,8O5 Sherbrooke Sheet West, Monh6al, QC, Canadal-ft A 2K6.