The Bullerin of the Internationol Lineor Algebrc Society frL lLl Lls rsJ Servingthe Internotionol Lineor Algebro Community
IssueNumber 24rpp. L-32, April 2000
Editor-in-Chief; George P. H. SrYAN styan@ total. net, styan@ together. net Dept. of Mathematics& Statistics,McGill University 805 ouest, rue SherlcrookeStreet West Editorial Assistants: Montr6al,Qu6bec, Canada H3A zKG MillieMalde & EvelynM. Styan
SeniorAssociate Editors: Steven J. Leoru,Chi-Kwong Lr, Simo PUNTANEN,Hans Joachim WeRrueR. AssociateEditors: S. W. Dnunv, StephenJ. KtRKLAND,Peter SeuRl, Fuzhen Zrarue.
PreviousEditors-in-Chief; Robert G. Tnoupsoru (1988-1989); StevenJ. LEoN & RobertC. THoMPSoN(1989-1993) StevenJ. LEoN (1993-1994);Steven J. LEoN & George P H. Srynn (1994-1997).
ILASPresidenWice-President's Annual Report: April 2000 (Richard A. Brualdi& DanielHershkowitz) ...... 2 Tributeto JimWeaver (Richard A. Brualdi) ...... 3 Treasurer'sReport: 29 February2000 (James R. Weaver) ...... 2O ELA:The ElectronicJournal of LinearAlgebra Volumes 1-4 (JamesR, Weaver) ...... 23 Thelnfusion of Matricesinto Statistics (Shayle R. Searle) ...... 25 MRLookup-A Reference Tool for Linking (Annette Emerson) ...... 32 Eigenvaluesand Less Horrible Words (Hans Schneider) ... .32 Letus Rootfor Roof (Shayle R. Searle) ...... 32
IMAGEProblem Corrrer ProblemsandSolutions...... 5 NewProblems 24-1:Rankof aSkew-SymmetricMatrix(S.W Drury) ...... 16 24-2:Nonnegative Matrixwith All Entries Summing to One (S.W. Drury) ...... 16 24-3:A PossibleGeneralization of Drury's Determinantal Equality (George P. H. Styan) ...... 16 24-4:fhe PositiveDefiniteness of a Matrix(YonggeTian) ...... 16 24-5:Two Products lnvolving ldempotent Matrices (Yongge Tian) ...... 16 24-6:Rank of a PrincipalSubmatrix (Fuzhen Zhang) ...... 17 24-7:Partitioned Positive Semidefinite Matrices (Fuzhen Zhang) ...... 17 24-8:An Inequalitylnvolving Hadamard Products (Fuzhen Zhang) ...... 17 24-9:Decomposition of SymmetricMatrices (Roman ZmySlony & GotzTrenkler) ...... 17 Linear Algebn Evenb May2*27,2000:Winnipeg, Manitoba ...... 18 October2Y26,2000: Raleigh, North Carolina .. ... 18 Decemberf13, 2000:Hyderabad, India ...... 18
Printed in Canada by the McGill UniversityPrinting Service, Montnial,Qu6bec page 2 fMAGE 24: April2000
ItAS President/Vice-President'sAnnual Report April 2000
L. The following have been elected to ILAS offices with terms o The 10th ILAS Conference, 'Challenges in Matrix The- that beganon March 1, 2000: ory," Auburn University, Auburn, USA, June 10-13, 2002. Secretary/Treasurer : Jeff Stuan (three-year term ending February 28,2003). o The llth ILAS Conference, Lisbon, Portugal, Summer 2004. Board of Directors: Harm Bart and Steve Kirkland (three-year terms ending February 28,2003). 8. The seventhSIAM Conference on Applied Linear Algebra The following continue in their offrces to which they were pre- will take place October 23-26, 2000, at North Carolina State viously elected: University in Raleigh, North Carolina, USA. This conferenceis being held in cooperation with ILAS. There will be two ILAS Vce President: Daniel Hershkowitz sponsoredinvited speakersat the conference: Eduardo Marques (term ends February 28,2001) de Sa' and Hugo J. Woerdeman. The 1999 ILAS Board acted as the selection committee. Under the terms of the agreement Board of Directors: with SIAM, ILAS members who are not already a member of Jose Dias da Silva (term ends February 28,2OOl), the SIAM Activity Group on Linear Algebra (SIAG/LA) will Roger Horn (term ends February 28,2OOl), have the same reduced registration fee that it offers SIAG/LA Nicholas Higham (term ends February 28,2002), members. Pauline van den Driessche (term ends February 28,2002). 9. Steve Kirkland has been selected as the Olga Tauskky 2. The President's Advisory Committee consists of Chi- Todd/John TloddLecturer at the 9th ILAS conferencein Haifa in (chair), Kwong Li Shmuel Friedland, Raphi Loewy, Errd Frank 2001_The selection committee consisted of Roger Horn (chair), tlhlig. Miki Neumann,Andre Ran, md Bit-Shun Tam. 3. This fall there will be elections for Vice-President (the 1.0.Nick Trefethen has been selected as fts LA,r{ Lecturer at term of Danny Hershkowitz ends on February 28, 2001) and the 9th ILAS conferencein Haifa in 2001. This lecture is spon- two members of the Board of Directors (the terms of Jose Dias sored by Elsevier Science Inc., publishers of the journal "Lin- da Silva and Roger Horn end on February 28,2001). The ILAS ear Algebra and its Applications." fiie selection committee con- 2000 Nominating Committee has been appointed and it consists sisted of Moshe Goldberg, Volker Mehrmann (chair), George of Wayne Barrett, Jane Day (chair), Nick Higham, Chi-Kwong Styan, and Hugo Woerdeman. Li, and Michael Tsatsomeros. LL. The next Hans SchneiderPrize in Linear Algebra will be 4. An ad hoc committee has been appointed and charged to awarded at the 10th ILAS conference in Auburn in20O2. considerchanges in the ILAS Bylaws and to possibly make rec- I2.ILAS is continuing to consider requestsfor the sponsor- ommendations for change to the full ILAS membership for its ship of an ILAS Ircturer at a conference which is of substantial approval. The committee consists of Jose Dias da Silva, Raphi interest to ILAS members. Each yetr, US$1,000 is set aside Loewy, Hans Schneider (chair)o srrd Frank Uhlig. Any ILAS to support such conferences, with a morimum amount of member who has concerns about our Bylaws should convey $500 available for any one conference. The guidelines are: (i) the them to a member of this committee. conference must be of interest to a substantial number of ILAS 5. The appointment of George P. H. Styan as an editor-in- members; (ii) the same "organization" is not eligible for sup- chief of IMAGEhas been extendedto May 31,2003. In addition, port more than once every three yqrs; (iii) geographically, the "Jochen" Hans Joachim Werner has been appointed as an editor- support should be distributed widely. in-chief of IMAGE for a six year term beginning June 1, 2000 The fulI ILAS Board (three executives and six other mem- (andso endingon May 3I,2006). Thus beginningJune 1, 2000, bers) reviews proposals and decides on which, if any, will re- George and Jochen will be co-editors-in-chief of IMAGE. ceive support, ond how much that support will be. Next year 6. The 8th ILAS Conference was held at the Universitat we will review these guidelines and this may result in some Politbnica de Catelunya in Barcelona, Spain, on July 19-22, changes. 1999. A report on this conference can be found in IMAGE 23 We are now accepting requests for conferences to be held (October1999, p. 13). in 2001. Such requests should be submitted by September t, 7. The following ILAS conferencesare planned for the neir 2OOO.Electronic requests are preferred and should be sent to future: [email protected] requestshould include the follow- o The 9th ILAS Conference, Tiechnion, Haifa, Israel, June ing: 25-29,2001. IMAGE April 2000 page 3
(1) date and place of conference 16. The ILAS INFORMATION CENTER (IIC) has a daily (2) sponsoring "organization" average of 300 information requests (not counting FTP opera- tions). IIC's primary site is at the Technion. Mirror sites are 1o- (3) organinngcommittee cated in Temple University, in the University of Chemnitz, and (4) purpose of conference in the University of Lisbon. (5) invited speakers,to the extent known
(6) expected attendance Richard A. BRUALDI,IIA,S President: [email protected] (7) proposed ILAS Lecturer, with some information about Dept. of Mathematics, University of Wisconsin-Madison the lecturer Van Vleck HaIl, 480 Lincoln Drive, Madison, WI 53706-1388,USA (8) amount requested. Daniel HnnssKowITZ, IIA,S Vce-President: hershkow@ tx.technion.ac. iI ILAS is sponsoring one lecture in 2000: Chandler Davis at the Dept. of Mathem.atics, Technion, Haifa 32000, Israel 5th Workshop on Numerical Ranges and Numerical Radii, Naf- plio, Greece,orr June 26-28,2000 [cf. IMACE 23, p. 16]. 13. ILAS has endorsedtwo conferencesto be held in 2000. Tributeto im Weaver They are: (i) the SecondConference on Numerical Analysis and f Applications, June 11-15, 2000, {.Iniversityof Rousse,Rousse, Bulgaria; (ii) the International Workshop on Parallel matrix Tiresday,29 February 2000, wils Jim Weaver'slast day as ILAS algorithms and applications August 18-20, 2000, Neuchdtel, Secretary/Treasurer. As you may recall, Jim was appointed to Switzerland-keynote speakers are Ahmed Sameh and Anna the position of Treasurerin 1989 by then presidentHans Schnei- Nagurney. ILAS has also endorsed the Rocky Mountain Math- der and was elected for two 2-year terms and one 4-year term ematics Consortium's Summer School on Matrix Theory to (0319242194,O319442196,and0319643/00). So for 11 years be held at the University of Wyoming in 2001-the principal he has served as ILAS Tieasurer, which was changed to Secre- speakerat the summer school will be Charles R. Johnson. tarylTreasureras a result of changesin our Bylaws. l4.ELN-Electronic jountal of Linear Algebra ILAS owes Jim a big debt of gratitude for his dedicated and professional service for 11 years. As ILAS has grown and as- o Volume 1, published in 1996, contained 6 papers. sumed a more visible and prominent place in our professional lives, the position of SecretarylTreasurerhas become more im- o VolumeZ, publishedin 1997,contained2 papers. portant and more complicated and has required additional ex- pertise. As ILAS members have witnesSed,Jim has responded o Volume 3-the Hans Schneiderissue, published in 1998, wonderfully to these new challenges. That ILAS is on a sound contained 13 papers. financial footing, with detailed documentation of our financial o Volume 4, publishedin 1998 as well, contained5 papers. and other records, is testimony to Jim's talents, hard work, and commitment to ILAS. o published papers. Volume 5, in 1999, contained 8 I hope that you will join me in conveying our deep appreci- o Volume 6-Proceedings of the Eleventh Haifa Matrix ation to Jim for his service to ILAS. I am happy to know that, Bylaws, remains member Theory Conference,is being published now. As of April according to our Jim a of the ILAS year (until 2000, it contains6 papers. Board for one after leaving office 28 February 2001). On 1 March 2000, Jeff Stuart began a 4-year term as ILAS o Voluma 7, is being published now. As of April 2000, it Secretary/Treasurer.I look forward to working with Jeff for the contains 3 papers. ELA s primary site is at the Technion. next two years. I very much appreciateJeff's willingness to Mirror sites are located in Temple University, in the Uni- .rssumethis important position, and I am very confident that he versity of Chemnitz, in the University of Lisbon, in EMIS will be a big assetto ILAS and the ILAS Community. V/elcome (European Mathematical Information Service) offered by aboard, Jeff! I also want to take this opportunity to thank two the European Mathematical Society, ffid in EMIS's 36 retiring members (as of 29 February 2000) of the Board, Jane Mirror Sites. Day and Volker Mehrmann, for their important service to ILAS, and to welcome two new membersof the Board, Harm Bart and Volumes 14 of ELA are now available in book form. The Steve Kirkland. list price is US$20 with a discounted price of US$16 for ILAS members (these prices include postageand handling). 15. ILAS-NET: As of April16,2000, we have circulated953 Richard A. B nuALDI, IIA,S P resident: brualdi@ math.wisc.edu Dept. of Mathematics, University of Wisconsin-Madison ILAS-NET announcements.ILAS-NET currently has 533 sub- Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388,USA scribers. Special Functions Permutation Groups George E. Andrews, Richard Askey, and Ranjan Roy Peter Camenon This treatisepresents an overviewof specialfunctions, This text summarizesthe latestdevelopments in focusingprimarily on hypergeometricfunctions and this areawith an introduction to relevantcomputer the associatedhypergeometric series, includi.g Bessel algebrasystelrrs, sketch proofs of major theorems, functionsand classicalorthogonal polynomials, and and many exarnplesof applying the Classification usingthe basicbuilding block of the gammafunction. of Finite Simple Groups.It is airnedat beginnitg In addition to relativelynew work on gammaand beta graduatestudents and expertsin other areas,and functions,such as Selberg's multidimensional integrals, grew from a short courseat the EIDMA institute many irnportantbut relativelyunknown nineteenth in Eindhoven. century resultsare included. London Mathenatical Society Student Tbxts 45 Enqrclopedia of Matbematics and its Applications 7I 1999 230pp. -55378-9 perback 2000 680pp. 0-521 Pa $2s.9s 0-521 -78988-5 Paperback about $34.95 New Perspectives in Enumerative Combinatorics Algebra ic Combinatorics Volume 1 Louis J. Billera, Anders Bjcirnen Curtis Greene, Richatd P. Stanley Rodica E. Simion, and Richard P.Stanley, Editors "...sureto becomea standardas an introd.uctory Basedon a full academic-yearprogram on combinatorics graduate text in combinatorics." at the MSRI, with specialemphasis on its connections -Bulletin of the AI{S to other branchesof mathematics,this book features work done or presentedat the program'sseminars. It "...utill engagefo* start n finish the aftention of containscontributions on matroidbundles, cornbina- any mathematicianwho will oPenit at page one." torial representationtheory, lattice points in polyhedra, -Gian-Carlo Rota bilinear forms, combinatorial differential topology Cambridge Studies in Adaanced Matbematics 49 and geometry,Macdonald polynomials and geometry, 2000 338pp. enumerationof matchings,the generalizedBaues 0-521-56351-2 Paperback $29.9s problem,and Limlewood-Richardsonsemigroups. Mathematical Sciences Research Institute Publications 38 Volume 2 1999 355pp. Richard P. Stanley 0-521-77087-4 Hardback s49.95 This secondvolume coversthe composidon of Proofs and Confirmations generatingfunctions, trees,algebraic generating noncom- The Storyof the AlternatingSign Matrix Conjecture functions, D-finite generatingfuncdons, David M. Bressoud mutative generatingfunctions, and symmetric functiorls. "...oneof the mostbrilliant examplesof mathematical expositionthat I haueenczuntered in many!€ars of read- "...tlie truaeof exerciseswith solutionswillform a ing t/tesame. Bressoud reuards...readers with a panorAmA uiml resnurce;indeed, exercise6. 19 on the Catalan of combinatoricstoday and with A renewedAwe at the numbers,in 66 (!) parts,justzfies the inuestment human ability to penetratethe deeplyhidden mysteries by itself Both uolurneshighly recommendedfor of pure mathematics." all libraries." -Herbert S. t$f/ilf, Science* -Choice Spectrttm Cambridge Studics in Adaanced Matbemartcs 62 Copublished with the Mathematical Association of America 2000 c.592pp. 1999 290 pp. 0-521-78987-7 Paperback about $34.95 0-521-65170-5 Hardback 57435 0-521-66545-5 Paperback $29.95 IMAGE 24: April 2000 page 5
IMAGEProblem Corner: Problems and Solutions
We arestill hopingto receivesolutions to Problems18-1, 19-3b,2l-2 & 23-1,which arerepeated below; we presentsolutions to Problems23-2 through23-7, which appearedin IMAGE23 (October1999), pp. 28 &27.In addition,we introduce9 newproblems @p. 16-17) and invite readers to submit solutions as well as new problemsfor publication in IMAGE.Please send all material in I4IEK - (a) embeddedas text in an e-mail to [email protected] (b) 2 copies(nicely printed please) by p-mail to GeorgeP. H. Styan,PO Box 270,Franklin, VT 054574270,USA. Please make surethat your narneas well as your e-mail and p-mail addresses(in full) are included!
Prcblem 18-1: 5 x 5 Complex HadamandMatrices Proposedby S. W. Dnunv: [email protected];McGill University,Montrial, Quibec, Canada. Show tlrat every 5 x 5 matrix t/ with complex entries uj,* of constant absolute value one that satisfies U*U = 5I canbe rcalized as the matrix (ri*)i,r where o.ris a complex primitive fifth root of unity by applying some sequenceof the following: (1) A rearrangement of the rows, (2) A rearrangement of the columns, (3) Multiplicationof a row by a complex number of absolute value one, (4) Multiplication of a column by a complex number of absolute value one.
The Editor has still not received a solution-indeed even the Pruposer has not yet found a solution!
Problem 19-3: Characterizations Associated with a Sum and Product Proposedby Robert E. HARTwIG: [email protected];North Carclina State University,Raleigh, North Carclina, USA, Peter Snunr: [email protected];University of Mariborl Maibor, Slovenia, and Hans Joachim WenNen: [email protected];Universitiit Bonn, Bonn, Germany.
(a) Characterize square matrices A ard B satisfying AB = pA * qB, where p Nd q are given scalars.
(b) More generally, characterize linear operators A and B acting on a vector space .t satisfying ABx e span(Ar, Bc) for every x€X.
TheEditor has still not rcceiveda solutionto PrcblemI9-3b. Thesolution by the Prcposersto Prcblem19-3a appeared iz IMAGE 22 (;Apil1999),p.25. Welookforvardto receivinga solutiontoProblem 19-3b.
Prcbfem 21-22The Diagonal of an lnverse ProposedbyBeresfordPentEtt:[email protected];Universityof Califurnia,Berkcley,Califurnia,USA, via Roy Mernres: [email protected];College of WilliamandMary, Williamsburg,Vrginia,USA. l'et J be an invertible tridiagonal n x n matrix that permits triangularfactorization in both increasingand decreasingorder of rows:
J = L+D+U+ and J =U-D-L-.
(Here the -t's are lower triangular,the t/'s areupper triangular,and the D's ue diagonal.).Show that
(J-t)**_ [(D+)rr * @-)** - J**]-t
The Editor has still not rcceived a solution!
Problem 23-1: The Expectation of the Determinant of a Random Matrix Proposedby Moo Kyung Cnuxc: [email protected];McGilI University,Montrdal, Qudbec,Canada. 'vec' Irt the n'r x n random matrix X be suchthat vec(X) is distributed as multivariate normal N(0,,4 I 1,), whetre indicates the vectorization operator for a matrix, the m x rzl matrix A is symmetric non-negative definite, E stands for the Kronecker product, ffi) n,and In isthen x n identitymatrix. Foragiven m x m symmetricmatrixC, findEdet(X'CX) inaclosedforminvolving only C and A. Is this possible? (Finite summation would also be fine.)
Welookforward to rcceiving a solutionto this ptoblem! page6 fMAGE24: April2000
Problem 23-2: The Equality of Two 4 x 4 Determinants Proposedby S.W. Dnunv: [email protected];McGill University,Montrdal, QuCbec, Canada. Showthat
a1 A2 ag A4 A1 a2 A6 a4
b1 bz bs ba bz b1 ba be
arbr azbz asbs a+b+ atbz azbt asb+ a+bs
The solutionby the proposer,who hasno imagination,is to expandeach determinant and to matchthe resultingexpansions term for term. The proposerhas "first dibs" on this solution. Respondentsare therefore asked to provide a more elegantsolution.
Solution 23-2.1by Chi-KwongLr: [email protected]; The College of Williamand Mary, Wlliamsburg,Virginia, USA.
Let lt r\ (t r\ lu, ar\ (0" an\ o=l l,B=l l,c=l I D- | l, \ or az/ \ o, on) \ orr, orb,I \ orau onbnI *) ,=(u' "),u=(u^ \ orb, orb,) \ orbn onb"I
If a2 I a1 and cta* ot, then A B\ det - det(A) det (D - C A-r B) and det - det(A) det(D - C,q-r B). (: ) Cr)
Let F- bn-1 - cA-I - B E1B- 1 - AEzA where (un o\ Et tl and Ez = \ o u'l (T; Suppose ( o 1\ R- tl \-r o) One readily checks that RFt R-L - B@EtR-r)n' - A(RErR-r)A-t _ DB-L - CA-',
and hence - dettb - C e-r B) = det(F) det(B) = det(RFt R-r) d*t (B) = det(D CA-r B).
Thus det(A)det(D - Ce-'B) = det(-A)det(D - e A-L B)
underthe assumptionthat o1 f a2 md as # a+.By continuity,we seethat the two determinants,as polynomials of the variables art...,e4,br,...,b4, areidentical. Hence, theresultistrue formatrices over any commutingring. a IMAGE 24: April2000 pageT
. -tution 23-2.2 by Hans Joachim WenNBn: [email protected];IJniversitiit Bonn, Bonn, Germany. .{r4 Our elementary solution to this nice problem will show that the determinant in question is even invariant to several fixther pairwise permutations in the underlying matrix. In what follows, let aa andb6 (i = 1, 2,3,4) be given real numbers.For eachpair (i, j), *itlt i, j e {I,2, 3,4},we define
A,j:- anc, (; ;) In addition, we put / An As+ \ M :- andrnd 1v :- [,, n,tz Ae+u"-) l, nEzt As+En/
We notice that det(M) - det(t/) is the claim of Problem 23-2. ln order to prove this identiry we begin by deriving an explicit closed fonn expression for the determinant of ,4,1.
THEoREM l. Izt M be definedas before. Then
det(M) = (az - at)(a+- as)lbsba+b1b2l* (on - ar)(ae- a2)lbfta*b2bsl- (o"- at)(a+- a2)1bft3*b2bal. (1)
Proof: Ac,cording to the well-known Laplace expansion theorem (see, e.g., $14.2 in [HJK, p. 92]), we obtain
det(M) = det(A12)det(A3a)det(-B3a)- det(,413)det(A2a) det(B2a) * det(,41a)det(A23)det(.O23)
* det(,423)det(,41a) det(E1a) - det(A2a)det(,413) det(813) * det(,43a)det(,412) det(E12)
= det(A12)det(:{3a)[det (Ezs) + det(E'12)]* det(A1a)det(A23)[det(Er+)+ det(E'23)] - det(,413) det (A2a) [det (86) + det(-E2a)] byexpandingdet(M) alongthefirsttworowsof M. Since,inviewof det(Aij) = aj - o; and det(E;1)=b;bj,thisisidenticalto (1), the proof of Theorem L is complete. D
'symmetr5l' The following corollary is an immediate consequenceof the in our expression(1) in Theorem 1.
Conollanv l.l: Let M be as before. Consider thefollowing six gtoups of replacements
(R1): bt # bz, bs e ba (R2): br $ bs, b2 e ba
(R3): b1 e ba, b2 e bs (R4): a1 # a2, as # aa
(R5): dy Q as, a2 I a4 G6): a1 # tr4, a2 Q as, and,foreachi€ {1,2,...,6},letM;denotethematrixwhichisobtainedfumM accordingto(Ri).Wethenhave
det(M,) = det(M), irrespective ofthe choice of i.
Since modifying M according to (R1) results in -ly',our corollary, in particular, shows that det(M) = det(N) is, as claimed, indeed correct. We continue with considering some special cases of the matrix M. page8 |MAGE24: April2000
Conollanyl,.2:LetMbeasbefore.Inaddition, leti,j,r,sbesuchthati< j,r<-s,and{i,j,r,s}- {I,2,3,4}.
(i) If aa= aj, thm (1) rcducesto
det(M) - (-1)i+i fuo- o,)(ai - o,)(bi - b;)(b, - b,).
(ii) If bi : bj, then (l) reducesto
Corollary 1.2 follows directly from Theorem 1. We conclude our discussion of Problem 23-2by extending our results to a slightly more general class of matrices. The respective proofs follow along similar lines and are hence omitted.
THEoREM 2. For i = 1,2,3,4: Iet ai, b,;,c,; be given rcal numbers.If
c1 c2 cg c4
a1 a2 a3 a4
crbt czbz csbs c+b+
atht azbz asbz a+bq
= then- det(IU) Q)
("fl2-c2a1)(caaa-caas)(fuba+bft2)*(c1aa-caa1)(c2as-csa2)(bfta+bz|')-("to"-csa1)(c2aa-caa2){bfts*b2ba).
Conor.r,nny 2.1: Let W be defuedas inTheorcm2. Considerthefollowing sixgroups of replacements
(Rl): br e bz, bs ++ ba
(R2): b1 Q bs, b2 <+ ba
(R3): b1 H ba, b2 ++ bs
(R4): a1 Q a2, as # a4, cy H c2, cs Q c4
(R5): 01 S o3, a2 Q a4, cr 1) cz, c2 # c4
(R6): ar + ct4, d2 Q as, c1 1) c4, c2 e ca,
and,for i € {1, 2,. . . ,6},letW6 denotethe matrixwhich is obtainedftomW according to (N). Then
det(W;) - det(W),
inespective of the choice of i.
COROLLARY2.2:I*tWbeasabove.Inaddition, leti,j,r,sbesuchthati< j,r Reference [I{IK] H.-J.Kowalsky (1967). Lineare Algebra.'l,lalter de Gruyter,Berlin. IMAGE24: April2000 page I Prcblem23'3: An lnequalityInvolving a SpecialHadamard Product ProposedbyShuangzheLru:[email protected],Canbena,Australia. LetA> 0beann x npositivedefiniteHermitianmatrixwitheigenvalues)1)....) 1,, AT bethetransposeof A,A-T bethe inverseof AT , /, be an n x n identity matrix and O denotethe Hadamardproduct. Showthat then,in the liiwner ordering, AoA-r- sX#," )?+'2 Solution 23-3.1by S.W. DRURY:[email protected]; McGill University,Montrdal, Qadbec, Canada. I'etH = AO A-TandletthespectraldecompositionofAbeA= t/*diag(,\1,...,\n)U witht/ aunitarymatrix.Then,a76 = - Dt@\rutx. Also,A-r t/*diag(.\!1...,\;L1Lr andif we denoteB = A-r = [i"diag(li1..., ^;t)7, wehavebi* = D^u*j\itu*n. We find that hi* = ainbi* = Dz,^ufi)aus1,u^i\;tu**. We needto boundthe largest eigenvalue p of the 1 /, \ \ hermitianmatrixffUV * ( 3 + p ). No*pr isthemaximumvalueofz*Ilz aszrunsoverallunitvectors. Wefindthat f\4" At/ -\- z*H:- I ,ihik:k -.L lrn*l'AnA;t i,k j ,ls,L,m t,ffi where N is the norrral matrix given by ntm = l^ us1"z1ru^1,.By comparing the diagonal entries of i[,]/* and I[*1/ we find that for eash I I lrn^l'=I lr,nrl' Furthermore, we have r uAiuapu*purnj zi zn -.L-v I lnn*l'=)_ lril' t,ffi i ,k ,L,nt, j It remains to apply the followingpropositionto gt,m : \ry,^l'. PRoPosITIoN. Let Q = km) be an n x n nonnegative matrixwith the sum of all entries equal to 1. If further the tth ,ow sum equals the lth cohnnn swnfor all !., thenfor any ,\1 ) ,\2 ),, QL,m,. S I Avn *(*.*) t,rn' Proof. Let pr. S p forall t and m. Then we must show that p P'n sinh ' QtmrPt- ( coshp. Towardsthis, we define the function g@) - et - r. It is clear that p(+p) I p t,ffi sincegisconvex,thatp(n) ( coshpfor-p 1r 1p. Therefore,since the sum of all entries of Q is unity, \.- - ) qe*P(q P^) l coshP L,m which is same qtm(k - g^) = the as the desired conclusion since I 0, by the condition on the row and column sums. I,m page10 fMAGE24: April2000 Problem23-4: Traceand A Partitioned Matrix Proposedby HeinzNeuoecxen: [email protected];Cesarc, Schagen, The Netherlands. Considerthepx(f+t;matrixX'=(r: Y'),whereYispxpnonsingularandrlQlflYteowitheothepxlvectorwith eachelement equal to 1..Let Mp := Ip - (l I plere'o denotethe p x p centeringmatrix. Provethen that the trace - ft {(xt Me+rx)-rY'MpY) = p t. (3) Solution 23-4.1by GiilhanALIARGU: [email protected]; McGill University,Montrdal, Qudbec, Canada, andHans Joachim WBwnn: [email protected];UniversitiitBonn, Bonn, Gennmy. We note first that there are situationsunder which the equality (3) doesnor hold! We derive, therefore,a condition which is both necessaryand sufficientfor (3) to hold. I.et the p x p matrix Y be nonsingulr,let r I $ lflY' eo,and let X' = ( r, Y/ ). Then, by someeasy algebraic transformations, we obtain X'Il{oa1X = !v'',)('- +Y'Mo' (4) h("- ;t',0) This shows thal Xt MpgX can be written as the sum of two nonnegative definite matrices. Since c I $ I flY' eo, the first summand has rank 1. Since Y is nonsingular and M, is an orthogonal projector with rank p - 1, it is also clear that the second summand has rank p - 1. So one could be tempted to believe that Xt MpgX is always nonsingular. But this, however, is not always the case. More precisely, we see from our decomposition (4) that X' Mp1X is nonsingular if and only if r is such that 1 x - :Y'ep eR(Y'Mp) (s) twith A(.) denotingthe range (column space) of the matrix (.)l or, equivalently,if and only if r'Y-reo f L (6) In other words, when going ftomY'MoY to X'MoqtX, then the rank does not always go up by l. The rank goes up if and only if condition (5) is satisfied.This shows that (3) can hold only if this condition (5) is satisfied. Let us, therefore, now assume that (5), or equivalently, that (6) holds. Then it is clear that the row spaces and the column spaces of the two summand matrices in decomposition (4) do have only the respective origin in common. Such matrices are said to be weakly bicotnplernentary to each other; see fWGIl. (A pair of weakly bicomplementary matrices is also often said to be a pair of disjoint matrices; cf. [MFR].) lf A, B is such a pair of weakly bicomplementary maffices, then each g-inverse of .4 * B is also a g-inverse of A as well as a g-inverse of B; see [JMW Th. 2.3]. Therefore, in particular, Yt MpY (x' Mparx)-rY' MoY = Y' MpY, which in tunr implies that the matrix \'t MpY (Xt Mp+tX)-t is a projector onto R(Y'MrY). The rank of this projector thus coincides with the rank of the matrix Y I MpY which is p - 1. Since for each projector its rank coincides with its trace, our proof is complete. Remark: We note that the concept of weak bicomplementarity was used in Solution 22-3.4 (IMAGE 23, p. 23). References UMW S. K. Jain, S. K. Mitra & H. J. Wemer (1996). Extensionsof 9-based matrix partial orders. Str{M foumal on Matrix Analysisand Applicartons, 1 7, 834-850. IMFRI S.K. Mitra (1972).Fixed rank solutionsof linearmatrix equations.Sankhyd Series A,34,387-392. IWGII H. J. Werner(1986). General2edinversion and weak bi-complementarity.Linear and Multilinear Algebra, 19,357-372. A solutionwas alsorcceivedfrcmJos M. F. TEN BERcE: [email protected];Rijl Problem23-5: An Inequalitylnvolving Diagonal Elements and Eigenvalues Proposedby Alicja Srraortuuowrcz: [email protected];Wqrsaw University of kchnology,Warsaw Potand. Provethateach eigenvalue,\ of A € Cnxnsuch that \ * at.t forall i - 1, . .., n, satisfies n r on,i r 1- I It k=I j =L,j #k Solution 23-5.1 by Giilhan Alpencu: [email protected];McGilt University,Montrdal, Quibec, Canada. IJtlbeaneigenvalueof A satisfying\* on* forall& - 1,...,n. ByGerschgorin'sTheoremwethenhave,foreach k, that ltr-o*,*l ld,xor,equivalently, ll-o*,*12 ldfl,wheredo=Di=r,is*lo*,il.FromtheCauchy-Schwarzlnequalitywethen obtaind! < (n - 1)rl. Combiningourobservationsnow,foreachk = I-,2,...,fl,resultsin 1 -/ r2k r7 - 1 Since this in turn implies n .+ rzk ?r-I - ak,klz' 3ll- our proof is complete. tr Solution 23-5.2 by R. B. Baper: [email protected] Statistical Institute-Delhi Centre,New Delhi, India. I€tAbeaneigenvalueofAsuchthat).lai,;forallf = l,...,nandletcbeaneigenvectorcorrespondingto,\withDi=rlrol'= 1. Then n, (A- an,n)rn T L j =r,j #k It follows, by the Cauchy-SchwaruIne,quality,that - L-1 ll ak,kl'l* ol' , tr, Ir Therefore l**l' - + l'rl' I r - - (7) k-r k=I t;= L,j#kl*il' ?'1 I'tlt' where rf - D?=r,i**lor,il'. (Since,\f a;6fori = I,...,tu,o hasatleasttwononzerocoordinatesandhence1- lr7"l2 is nonzero,k : I, . . . , n.) By the weightedarithmetic mean - harmonicmean inequality, + lr,ul' (8) ^:L-ffi D?_rlrnl'(1- l*nl') 1- D7_rlrkl4' Since|,T=L lr kl2 1 l* *ln r n k=l and hence n (e) - 1 1 D;= Llnkl4 n- L The result then follows from (7)-(9). page 12 IMAGE24: April2000 Solution 23-5.3 by Lajos L(szt-6: [email protected];Eiitviis llrdnd University, Budapest,Hwrgary. lrt I bean eigenvalueofthen x n matrixA satisfying\ * ou,i = L..n, andr betheassociatedeigenvector.The /cthcomponent of Ar -.\r = 0 gives l, l' - ar+r).ra- akjrjl < q'*, l() l2 lf v*n I "* wirh rtz /n' \r/2 / n \ r1x: and = t t lou,l' I Qr* (r wiP) \rfe I \r llc / - - - Here,qp + 0,for qp 0 wouldimplyra 0 andhencer 0, whichcontradicts the definition of eigenvector. Thus we have rZ w.7, irrd, by summing up all the n tenns, +F4*-T# =r@)'# The last inequalityfollows from a resultin Chapter3, $D.6of Marshall& Olkin [MOI]. Remark Of several possible generalizations using Hiilder's inequality, one is especially interesting because of the same minimum value. To see this write - ( lQ, app)rpl maxry*l"ril Dl*xl: RnQx, i+k whereQe * ibythe samereasoning as above,and obtain ftn n.. I-,. ,tf l{4=F(xl>-- \*/ - ?l^-or*l- ? Qo n-I' sincethe minimum valuesfor the functions f andF obviouslycoincide. Refercnce tMOfJ A. W. Marshall & t. Qlkin (1979). Inequalities: Thcory of Marjorization and its Applications, Academic Press, New York. Solution 23-5.4 by thehoposer Alicja SIuorruNowlcz: [email protected];Warsaw University ofTbchnology, Warsaw, Poland. We prove that this inequalityis essentiallybased on the inequalitybetween the harrronic and arithmeticme,an: h*...*c", n , = (10) AT"'T r* where c6 Let.\ be an eigenvalue of A, and suppose Ar - ),n, r + 0. This means that - ak,Irt* ak,znz*. .. + alr,nffn \rn, k : 1, . . . )n which is equivalent to t1, (^ - ak,k)*n = T /-.r j =r,j #k We obtain |)-&k1 ll** t< r Iak,ill *i I j =I,j #k fMAGE 24: April2000 page 13 forall k - 1,.. .,n BytheCauchy-Schwarzinequalitywehave q q +, T on,i Ir ni L, lon,ill*i lS L I l' 2 | l' --t j ,i *k j =t,j *k j =r,j #k Thus q ll-o*,rl'l**l' To simplify the notation, let n b*= t l*ilr. j-1,i*k The elementswe havejust defined satisfy b1, and 1) . 7u- ll" llr' k=I Thus, we can rewrite (11) as follows I ) - ak,kl'l ** It S rkT bx, k = 1,.. .)n. Notice that by our assumptions,all bk and I n1 2 -\rkTnt12 tt: -n. (12) ?bn rffi>t+ K=I By ao application of the harmonic-arithmetic mean inequality (10), with c* = I lbn, wo have n, .l n2 ll"llr'11*.- n,-1 This result and (12) give the desired inequality. Problem 23-5: Linear Combinations and Eigenvalues Proposedby Jos M. F. TEN Bnncr: [email protected]; Rijksuniversiteit Gruningen, Gruningen, The Netherlands. Suppose we have two real matrices of order p x p, with p even, and with all eigenvalues imaginary. Is it possible to find p linear combinations of the matrices that have at least one real-valued eigenvalue? I expect that this is not always possible and that the set of mafrix pairs that does allow real eigenvalues for linear combinations has positive measure. Is there a place in the literature where I can find such things? Sofution 23-6.1 by Hans Joachim WenNen: [email protected];Universitiit Bonn, Bonn, Gertnany. We begin with considering only 2 x 2 real matrices. From the literature, cf. e.g., [PLI, p. 55], we know that the characteristic polynomial of a2 x 2 matix A, say, can be written in the form cA(r) - .\2 - .\tr(A) * det(A); (13) here tr(.) and det(.), respectively, denote, as usual, the trace and the determinantof the matrix ('). SV eig(,a) we will denote the - setof alleigenvaluesof A,i.e.,thesetof solutionstothecharacteristicequation",q()) 0.Inviewof (13),wehavethefollowing characterization. page 14 IMAGE 24: April2000 TgeonEM I: Let Im denotethe set of all thosereal 2 x 2 mntrices with purely imnginary eigewalues. Then A e Im if and only if tr(A) - 0 and det(A) > 0. In whatfollows, let A, B e Im. Then,by Theoretrl1, /ot as \ A-t and B- I j,) -ot \az / fi where-ai - azas > 0 and -b? - b2ba> 0. We notice that @\ -0andtr(B) For fixed real numbersp and q, what canwe sayabout the eigenvaluesof the matrix C := pA+qB? Sincetr(A) =0, clearlYtr(c)=oandso (, z-----;-l eig(C)= t+./-det(C)j. Hence,if det(C) > 0 or, equivalently,if -p" ("? + azaz)- s'Q? + b2h) > pq(2a1b1+ azbl* a*z), thenC e Im. Otherwise,i.e., if det(C) < 0 or, equivalently,if -p2 fu?+ azas)- q' (b?+ bzrt.){ pq(2oth * azbe* asbz), (14) then C has, in contrast to A and B, two purely real eigenvalues, and we indicate this by writing C e Re. The following example illustrates that both situations can indeed occur for different values ofp and q. ExeuPLE 1: Consider -'\ n) o,=(t and \z -t) ",=('\-s -z) Then,eig(A) {+;.'/s},eig(:{ + a1= {*rt}, eig(,A+28\ = {+1},andeig(,4 + 38) = {+t'/zt\. Moreover, as the reader may verify, A+qB€ Im <=+ q2 Our example,in particular,shows that theredo exist matricesA, B e Im with (i) pA + qB e Im for infinite many pairs (p, g), and, at the sametime, with (11)pA + qB e .Re for infinite many other paits (p, q). But doesthis also meanthat both casesare possiblefor any pair A, B € Im? \\x the answeris negativeis illustratedby our next example. ExanaPLE2: Consider 0100 -1 0 0 0 and B- 0001 0 0 -1 0 Then, for each pair of real numbers p and g eig(p.4*qB) - p2+q2 {*o IMAGE 24: April2000 page 15 Weuotice,inpassing,thatthecharacteristicpolynomialof pA+qBiscpaaqn(.\) =,\a *2(p'+q')\'+ (p'+ q2)2.Moreover, - -i,+i,-i} eig(A) {+i, -i,+i,-i} , eig(B)= {+i, . Hence,wedo notonlyhave A e Im and B € frn butalsopA + qB € Im,for eachpair (p,q) * 0. tr We now let A, B e Imbe again given 2 x 2 matrices and assume that for some given real numbers p nd Q, the inequality (14) strictly holds. Then, by continuity, (14) also holds for all other maEix pairs lying in the intersection of Im and a sufficient$ small neighborhoodof(A,B). Needlesstoemphasizeoncemore,C e Im onlyif tr(C)= 0. Refercnce IPLII P.Lancaster (1969). Theoryof Matrices. AeadenrcPress, New York. Problem 23-72 An Inequality lnvolving Rank and Matrix Fowers Proposedby Yongge TteN: [email protected];Queen's University, Kingston, Ontario, Canada. lI Let A be a square matrix of size n x n. Show that there are n vectors r0, t7,...tfin-r such that the square matrix . . ., (r g, Ar 1, 42 *2, An-' * n -r ) is nonsingularif and only if min{1* rankA,2+ rankA', ..., n - I+ rankA'-1} > n. [email protected]; Solution 23-7.1 by John AsroN: McGitl (Jniversity,Mont6ql, eu6bec, Canada. Below we make use of the fact that for each nonnegativeinteger f, we have R(Ao) g R(An-t), with 7?(.) indicating the range (columnspace)of (.). ObservethatweputA0:= 1,,where1, standsforthen x nidentitymatrix. Forprovingnecessiryletthere existn vectorsfio, xr,- -.tfin-7 suchthatthematrix ( ro Art A"-r rn-t) is nonsingular. Observe that in view of our above inclusion this can happen only if nnk(A, ) > n - i is satisfied for each f = I ,2, . . . , n - I. Since the latter is equivalentto min{l * rank,4,2 + rankA', ..., n - I+ rank,4"-1} > n, proof i) the of necessityis complete. To prove sufficiency, let rank(, ) n - f be satisfiedfor each i = I.2, In virtue of our above-mentioned inclusion, we can then, of course, choose in backward order the vectors r; (i = n - l. 1, 0) such that the matrices M; :- ( Ao*t Ai+t *o+o An-trn-1), i-n-1,fr-2, are all of full column rank. This completes the proof. A solutionwas alsoreceivedfrcmthe Pruposer Yongge TnN: [email protected];Queen's University,Kingston, Ontario. page 16 fMAGE24: April2000 IMAGEProblem Corner: New Problems V"tt' ?"426' f 4'v' P'z'7 probfem24-1: Rankof askew-symmetricMatrix Proposedby S.W Dnunv: [email protected];McGill University,MontrCal, Qudbec, Canada. l*t A be a complex skew-symmetricmatrix. Show that the non-zerosingular valuesof A are equal in pairs and deducethat A necessarilyhas even rank. Problem24-2: NonnegativeMatrix with All EntriesSumming to One Proposedby S. W. Dnuny: [email protected];McGill University,Montrial, Qu6bec,Canada. jth I,et Q bea real n x n elementwisenonnegative matrix with the sum of all entriescqual to 1 and,for eachi, the row sum equals the jth column sum. Showttrat I is a convexcombination of matricesR(X, o) describedas follows. I,rutX be a nonemptysubset - = a) is givenby of In 7I,2, .. ., n) with & elementsand let o be a cyclic permutationof X. Thenthe matrix R R(X, 1 ." . ,ri=E itieXandj=o(i) and ,oi--0 otherwise. Remark This Problemis relatedto Solution23-3.1 in this IMAGE,page 9. Problem 24-3: A PossibleGeneralization of Drury's Determinantal Equality hoposed by GeorgeP. H. Srylr.l: [email protected] et; McGiIl lJniversity,Montrdal, Qudbec,Canad'a. I.r:tA, B, C, D, E, andF be n x n complexmatrices with B\ lA B\ (A R=f I and ^e=l I \c D) \r F) = det(S). anddet(C)= det(E)anddet(D)= der(F).Findturthernecessaryandsufficientconditions(if any)sothatdet(.R) Remark This problemis inspiredby Problem23'2by S. W. Drury-see this IMAGE24,pp' G8' of a Matrix frcblem 24-4: The PositiveDefiniteness Proposedby YonggeTHN: [email protected];Queen's University, Kingston, Ontario, Canada' roots and that A is any Supposethat p()) and q(I) are any two nonzeropolynomials with real coefficientsand without oornmon matrix p(A)p(A.) -l q(A:)q(A\ is always positive squarematrix of order rn. Show that the Herrnitian |efintlet "o-pl"* problem 24-5: Two Productslnvolving ldempotent Matrices WnYnrcl"*"?::'.;';A^a X'c'if* kdt4- /ter 'l&ol^na'!pla+J. Canada.ryada. ,iat hoposed by yongge TIeu: ytian@ mast.queens u.cai Quem's University,Kingston, Ontario, "il.^'-ler& rttab& rrst w4 (a)SupposethatAandBaretwocomp1exHermitianidempotentmatricesofthesamesize.Showthatffi"m.* - -- ,uf0d.q lJ.-4X ^l Ae d{A),* Prcblem 24-5: Rank of a Principal Submatrix Proposedby Fuzhen ZnaNc: zhang@ nova .edui Nova SoutheasternUniversity, Fort Lauderdale, Florida. I.et Abe a (complex) positive semidefinitematrix and write [A]" for theprincipal submatrix of A consisting of rows o and columns a. Show that for any positive integer & rank([A].) = rank([Ak]") = rank({[A],]k). Does this generalizn to real numbers fr or to Hermitian A? Probfem 24-72 Partitioned Fositive Semidefinite Matrices Proposedby Fudren ZnnNc: zhang @nova.edui Nova SoutheasternUniversity, Fort Lauderdale, Florida. l,et Abe a positive semidefinite matrix. With t for Moore-Penrose inverse, o for Hadamard product, and ) for L0wner ordering, show that Ao AI > At Ao AAt and A+ AI > At A+ AAt. In particular, it A is invertible, then AoA-t)I and A+A-'>21. Problem 24-8: An lnequality Involving Hadamard Products Proposedby Fuzhen ZH^e,Nc:zhang @ nova.edu; Nova SoutheasternUniversity, Fort lauderdale, Florida. I.et A and B be complex positive semidefinite matrices of the same size. If B is nonsingular, with o for Hadamard product and ) for l,iiwner ordering, show that A2 o B-r > (diag,4)2(diagB)-1. Inparticular,takingB: land A= I,respectively,itfollowsatoncethatdiag(Az)> (diag,4)2anddiag8-l > (diag8)-1. Problem 24-9: Decomposition of Symmetric Matrices ProposedbyRomanZnavSLoNv: [email protected],Poland, and Giitz TneNrl-rn: [email protected];Universitiit Dortmund, Dortmund, Germany. Itiswell-knownthatareal synrmetricn x n matrixA canbeuniquelywrittenas,4 = A+- A-,where,44 andA- arenonnegative definite matrices such that A+A- = 0. Find a procedure to calculate the pair (A+, A-) withoutusing the eigenvalues and eigenvectors of the matrix A. Assume n ) 3. Remark: This decomposition of a real symmetric matrix plays a crucial role in the estimation and testing of variance components in linear models. Pleasesubmit solutions,as well as new problems,in 14$ - (a) embeddedas t€xt in an e-mail to [email protected] (b) 2 copies(nicely printed please)by p-mail to GeorgeP. H. Styan,PO Box 270, Franklin, W 05457-0270,USA. Pleasemake surethat your name,as well as your e-mail andp-mail addressesare included! We look forwardparticulady to receivingsolutions to Problemsl8-1, 19-3b'2l-2 &23-ll page18 IMAGE24: April2000 SelectedForthcoming Linear Algebra Events Western Canada Linear Algebra Meeting cafeteria during weekdays. An information sheet and ac- commodation reservation form must be used for reserva- Winnipeg,Manitoba: May 26-27, 2000 tions. E-mail Michael Tsatsomeros(tsat@ math.uregina.ca) or Rob Craigen(craigen @ server.maths.umanitoba.ca) to ob- tain theseforms. Rates: single C$34.77, double C$25.65per The Western CanadaLinear Algebra Meeting (W-CLAM) pro- person,ffid there is a one-time adminisffation fee of C$5. vides an opportunity for mathematicians in western Canada working in linear algebra and related fields to meet, present ac- countsof their recentresearch, and to have informal discussions. While the meeting has a regional base, it also attracts people The SeventhSIAM Conference on from outside the geographical area. Participation is open to any- Applied Linear Algebra one who is interested in attending or speaking at the meeting. Previous W-CLAMs were held in Regina (1993), Lethbridge (1995),Kananaskis (1996), and Victoria (1998). Raleigh,North Carolina:October 23-26,2000 '00 The W-CLAM will be held at the University of Mani- The SeventhSIAM Conferenceon Applied Linear Algebra will toba, Winnipeg, May 26-27, 2000, and is supportedby the Na- be held, in cooperation with the International Linear Algebra tional Program Committee of the Institutes CRM, Fields, PIms Society, in the McKimmon Conference Center, North Carolina and the University of Manitoba. StateUniversity, Raleigh, North Carolina, October 23-26,2000. We will have three distinguished invited speakers: The Web site addressis o Hans Schneider(University of Wisconsin) http://www. s ia m.o rglmeetin gs/l a00/ o Bryan Shader (University of Wyo-irg) o Henry Wolkowrcz (University of Waterloo). Plenary Speakcrs.'Gene Golub, Tom Kailath, Eduardo Mar- ques de Sa GLAS), Gil Strang, Charlie Van Loan, Richard The Web site addressis Varga, Hugo Woerdeman (ILAS), Margaret Wright. Irwited Concurrent Speal The program will start with a two-day oourse on recent ad- abstract, plus the names, e-mail addressesand affiliations of all vances in Matrix Theory with Special Reference to Applica- authors by e-mail to Hans Joachim Werner at tions in Statistics on Saturday, December 9, and Sunday, De- [email protected] This will followed by the presentation of cember 10, 2000. be and a copy by regular mail (p_mail) or FAX to both research papers in the Workshop proper on Monday, December ll-Wednesday, December 13, 2000; it is expected that many of Hans Joachim Werner, Institut fiir Okonometrie & Opera- these papers will be published, after refereeing, in a Special Is- tions Research, Okonometrische Abteil'ng, Adenauerallee 24- sue on Linear Algebra and Statistics of Linear Algebra and lts 42, Rheinische Friedrich-Wilhelms-Universit[t Bonn, D-53113 Applications. Bonn, Germany;FAX (49-228) 73-9189, and to The International Organizing Committee for this Work- shop comprises R. William Farebrother (Victoria University of P. Bhimasankaram, Indian Statistical Institute, Street No. 8, Manchester, Manchester, England, UK), Simo Puntanen (Uni- Habsiguda,Hyderabad-50O 007, India; FAX (91-40) 7I7 -3602. versity of Thmpere, Tampere, Finland), George P. H. Styan The for receipt of abstractsof contributed papers is (McGill University, Montreal, Qu6bec, Canada; vice-chair), deadline and Hans Joachim Werner (University of Bonn, Bonn, Ger- September30, 2000. The registration fees will be as follows: -*y; chair). The Local Organizing Committee in India in- o For payment received by September30, 2000: cludes Rajendra Bhatia (Indian Statistical Institute-Delhi Cen- Short Course: US$30/C$45 (Rs. 25Ol-Indian residents), (Indian tre), P. Bhimasankaram Statistical Institute, Hyderabad), Workshop: US$60/C$90 (Rs. 500/- Indian residents). V. Narayarua(Indian Statistical Institute, Hyderabad), M. S. Rao (Osmania University, Hyderabad), B. Sidharth (Birla Science o For payment received after September30, 2000: Centre, Hyderabad), LJ.SuryaPrakesh (Osmania University, Hy- Short Course: US$3 6lC$54 (Rs. 3001-Indian residents), derabad), R. J. R. Swamy (Osmania University, Hyderabad),P. Workshop: US$72IC$108 (Rs. 600/- Indian residents). Udayasree (University of Hyderabad), K. Viswanath (University There will be no registration fees for students for the Short of Hyderabad). Course. For students and retired persons the regisffation fee for This Workshop is being organuzedin Hyderabad by the In- the Workshop will be US$30/C$45 (Rs. 2501-Indian residents). dian Statistical Institute and the Society for Development of There will be no registration fees for accompanying persons. Statistics, in collaboration with the Birla Science Centre, Os- The regisffation fees cover all conference materials and hand- mania University and the University of Hyderabad. outs, lunches, and tea and coffee. We expect that lodging will The Short Course will cover generalizedmatrix inverses(in- be available at low cost: fulI details will be given on the Work- cluding Moore-Penrose,Drazin and group inverses),matrix par- shopWeb site: http://eos.ect.uni-bonn.de/HYD2000.htm,which tial orderings (including minus and sharp orders), matrix in- will be updated regularly, and in the Third Announcement. The equalities (including inequalities for determinants, traces,eigen- Workshop banquet (US$10/C$15; Rs. 150/- Indian residents) values and singuar values), togetherwith statistical applications. will be on Monday, December 11, 2000. In addition there will be a sessionon statistical proofs of maffix 'We Indian residents should pay the registration fees through results. expect that the participants will be researchersin Crossed Demand Draft on any nationahzed bank, payable at Mathematics, Statistics, Computer Science,Engineering, Physi- Hyderabad, in favour of R. J. R. SwamyA.{TWMS-2000.Partic- cal and Earth Sciencesand related areas. It may be assumedthat ipants not resident in India may pay the regisffation fees in ad- all the participants have a knowledge of Linear Algebra at the vance by personal check in either US or Canadian dollars drawn level of Hofftnan & Kunze lLinear Algebra, 2nd od., Prentice on a US or Canadian bank and made payable to George P. H. Hall L9711or Noble & Daniel lApplied Linear Algebra, 3rd ed., Styan, and sent to George P. H. Styan, PO Box 270, Franklin, Prentice Hall 1988.1 A set of lecture notes will be provided for vT 05457-0270,USA. each participant. We regret that we are not able to accept payment of regis- The Short Course will be followed by the Workshop proper, tration fees by credit card. All correspondenceregarding regis- presentation which will include the of both invited and con- tration fees from participants resident in Europe should be di- papers 'Werner: tributed on matrices and statistics. There will also be rectedto H. J. [email protected]; FAX a special sessionfor paperspresented by gfaduate students.We (49-228) 73-9189, and from participants not resident in In- expect that the topics to be coveredwill be similar to thosecov- dia or in Europe should be directed to George P. H. Styan: ered by the severalSpecial Issueson Linear Algebra and Statis- styan@ total. net, EAX (1-5 I 4) 398-3899 (e-mail preferred). tics of Linear Algebra and lts Applications. The Web site ad- Alt conespondence regarding lodging in Hyderabad for dress is this Workshop should be directed to R. J. R. Swamy, Dept. http://eos. ect. u n i-bon n . de/HYD2000. htm of Statistics, Osmania University, Hyderabad-SO0007, India; rjrs @ cool mai l. com, FAX (9 1-40) 7 17 -3602. 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Weaver,Dept. of Mathematics& Statistics,University of West Florida I1000 UniversityParlclvay, Pensacol4 FL3251+5751, USA Orderswill be filled on a fint-come, fint-served basis ILAS plans to produceprint versionsof future volurnesof ELA, so start your collection now. ls your linearalgebra textbook older than the World Wide Web? Doesit predatethe useof matrixdecompositions to build internetsearch engines? lf so,you shouldconsider this excitingnew textbookthat bringsmodern linear algebra to life! MnTRX ANATYSISAND AppLIEDLINEAR AIcEBRA By Carl D. Meyer Approx.700pages Hardcover with so many linear algebra textbooks available already, ISBN0-89871-454-0 why should you choose this one? ListPrice $75 This is an honest math text that circumventsthe traditional SIAMMember Price definition-theorem-proof format that has bored students in $eO the past. 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Determinant, direct product, Doolittle, "Matrix theory was not to emerge until quite late in experimental design, generahzed inverse, multivariate statistics, the 19th century, and became prominent only from normal equations,quadratic forms, vec and vech. the 1920s, ... [The papers by] Farkas (1902), de la Vall6e Poussin (1911) and even as late as Haar 1. INTRODUCTION (1924) are typical examplesof the continuing slow- determinants Matrices were introduced to me in New Zealand in 1949 as a ness, for they explicitly worked with one-year Master's level course taught by an A. C. Aitken doc- but wrote out the matrix equations and inequalities toral graduate. Two years later at Carrbridge University's Statis- longhand." tics Laboratory, the 24-week mathematical statistics course 1930s made no use of matrices,not even in the teaching of the bivariate 3. MATRICES ENTER STATISTICSlN THE normal distribution or of multrple linear regression. This was a With Grattan-Guinness(1994) having concluded (as previously great surprise. But it is totally in line with similar comments quoted) that maffices "became prominent only from the 1920s", from others, reported by Farebrother (1997), who also quotes the year 1930 seemsa good starting point for the entry of matri- Grattan-Guinness & Ledermann (1994) as saying "the rise of ces into statistics. That was the year of volume 1 of the Annals matrix theory to staple diet has occurred only since the 1950s". of Mathematical Statistics,its very first paper,Wickselt (1930), Indeed, not even Aitken himself (much of whose research being "Remarks on regression". Today that would undoubt- centered on both statistics and matrices) made a strong pitch for edly be a welter of matrices; in 1930 the normal equations were using matrices in statistics. Neither of his two books, Determi- solved using determinants. But two yeffs later came the Turn- nants and Matrices (Aitken I939a) or Statistical Mathematics bull & Aitken (1932) book with severalapplications of matrices (Aitken 1939b), mentions the topic of the other, except for a to statistics, all of which are still important and widely used to- snippet about quadratic forms in the matrix book. For some- day: normal equationsX'Xp one having strong interests in both topics, these ar:e,surely, un- errors are assumedto be not of "equal importance and uncorre- expected omissions. They have been my motivation for trying lated"; and E(*'A*) = tr(AV) for x /\, (0, V) obtainedby to fface a little of the infusion of matrices into statistics. It all what today is considered to be a very roundabout derivation. seemsremarkably recent when viewed against the longer history Then comes the Hotelling (1933) paper on principal com- of matrices themselves. Of course, in trying to be an amateur ponents, a landmark both for its content and its marix usage, historian one immediately comes face to face with the ocean of although it doesmake referenceto Kowalewski (1909) on deter- literature available and the consequentnear-impossibility of as- minants. Cochran (1934) is a next important milestone, dealing sembling every detail and aspect of one's topic. Therefore there with the distribution of quadratic forms, and in the same year in ire assuredly gaping holes in what follows-and all that can be Bartlett (1934), we find an early use in Britain of the word vec- done is to apologize and ask for help for filling those holes. Cir- tor in a title-but there are no matrices in the paper. And the ab- cumscribed by such lacework, the paper is aranged under four senceof marices in Kolodziejczyk (1935) and in Welch (1935) main headings: origins, the 1930s, special topics, and books. is glaringly noticeable by today's standards;this is also true of And, generally speaking,these historical notes mostly go no fur- Cochran (1938). A11three of thesepapers deal with topics which ther than the mid 1980s. lend themselvesso easily to matrix representation,namely linear hypothesesand regression.But even the popular layman's book 2.THE ORIGINS OF MATRICES of that time, Mathematicsfor the Million, Hogben ( 1936),hasno In his Men of Mathematics,Bell (1937, pp. 40H03; 1953,pp. index entry for determinantsand only one reference to matrix al- 440443), attributes the invention of mafices and their algebra gebra, as being but one of severalnames used for "different ways to Cayley (1855, 1858). "Maffices" writes Bell, "grew out of the of counting and measuring"! Contrarywise,Frazer, Duncan & . . . way in which the transformations of the theory of algebraic Collar (1938) is a substantivebook on matrices with virtually invariants are combined." However, Farebrother (1997) indi- nothing in statisticsbut many applications in engineering, such cates that others in the eighteenth and nineteenth centuries may as the oscillations of a riple pendulum (p. 310) and the dis- well have made greater although indirect contributions than did turbed steady motion of an aeroplane (p. 284). So the progress Cayley, a conclusion, he suggeststhat is supported by Grattan- of matrices was slow. Infusion had begun, but the steeping was Guinness(1994,p. 67). taking a long time. Nevertheless,it was to gain speed. From thatpaper, Farebrother(1999,p. 4) gives the following Craig (1938) was, as a follow-on from Cochran (1934), the quote as evidenceof the slow adaptationof matrices to statistics. first in a long line of papers (by many authors continuing to page 26 fMAGE 24: Aprit2000 the present day) dealing with the independence of quadratic A matrix operation of particular use in multivariate statistics forms of normally distributed variables(see Section 4.1). Aitken is the vectorizingof a matrix: writing its columns one under the (1937 , 1938) continued publishing maffix results that would be other in one long vector. Its origin goes back to a Cambridge and of later value to statistics but with no explicit mention of statis- London contemporary of Cayley, namely Sylvester (1884), and tics. And the year 1939 ended with a flourish: the publication of it is now usually called vec. It, and vech, an adaptationof vec to Aitken's two books referred to at the beginning of this paper. symmetric matrices developedin Searle (1978), are considered at length in Henderson& Searle(1979,1981a). An early use of 4. SPECIAL TOPICS vech is in Aitken (1949). We now briefly describe some of the progress that maffices have Two important features of multivariate analysis are eigen- made into certain statistical topics. The choice of topics here is roots and eigenvectors (known originally as latent roots and la- undoubtedly personal and the references are certainly not com- tent vectors-see Farebrother, 1997, p. 9, for interesting com- plete. mentary). Whatever their name, they are matrix characteristics 4.7. Quadratic Forms. Ttrrnbull & Aitken (1932) show how which feature frequently in multivariate literature. Early exam- to use a canonical form of a (symmetric) matrix to reduce a ples are Hsu (1940) on analysis of variance, Anderson (1948), quadratic form to a sum of squares. Later, as already men- Geary (1948) and Whittle (1953) in dealing with time series, tioned, Craig (1933) was an initial paper on the independence with continuittg interest through to Mallows (1961), who is con- of quadraticforms (of "certain estimatesof variance", actually), cerned with "latent vectors of random symmetric matrices". followed by Craig (1943) "certain quadratic on forms". In be- 4.3. SolvingNormal Equations of FuIIRank Normal equa- tween, Hsu (1940) (1938, and Madow 1940) touched on the tions derived from applying least squiles to multiple regression subject,Aitken (1940) dealt with the independence of linear and data are usually of full rank. According to Kruskal & Stigler quadratic forms, and a decadelater, Aitken (1950) dealt with the (1997,pp. 9I-92),they havehad that nirme sinceGauBs (1823). independenceof two quadratic forms. (1949) Mat6rn also con- Their derivation and numerical solution have always been a tributes. Lancaster (1954) confines his attention to traces and sourceof great concern. Some early publications presentingnor- cumulants. The necessity condition for independenceis still a mal equationsare Wicksell (1930), Aitken & Silverstone(1942) hot topic; see, o.9., Driscoll & Krasnicka (1995), olkin (1997), and Bacon (1938) -a11 of whom use no maffices;but of course Driscoll (1999), and the recent MSc thesis by Dumais (2000), Tirrnbull & Aitken (1932) do. Until the acceptanceof matrix in- which includes 248 references. verses,methods for solving normal equations,being, 6 they Ero, 4.2. Multivariate Statisdcs. This branch of statistics has just simultaneouslinear equations,were simply successiveelim- spawned more matrix activity than any other except, perhaps, ination with back substitution----orthe Cramer's (1750) method for linear models. using determinants as described previously. Wilks (1932) used lots of determinantsin his work on the Despite its algebraic clarity, the use of a matrix inverse ini- multiple correlation coefficient; but he uses almost no matrices tially posedconsiderable arithmetic diffrculty. Although today's and certainly no matrix algebra. In conffast, Ledermann (1940) computers now make light work of that arithmetic, that has come deals with a matrix problem arising from factor analysis. And about in only the last forty years; and it is accomplished at in their discourseson canonical correlations,Bartlett (194L) and speeds that were utterly unimaginable then. During graduate Hsul (1941) use matrices aplenty. Bartky (1943) uses maffices student days in 1959, in a small computing group at Cornell, - sparingly,but gives I + M + p1z + . . . (I - M)-t without there was great excitement when we inverted a 10-by-10 ma- proof or a reference thereto. Bartlett (1947) makes consider- trix in seven minutes. After all, only a year or two earlier a able use of matrices in his long "Multivariate Statistics" paper friend had inverted a 40-by-40 maffix by hand, using electric wherein he writes that he has "avoided complicated analytical (Marchant or Monroe) calculators. That took six weeks! So it discussion of theory" but has made use of "matrix and vector is understandablethat statisticians were interested in computa- algebra". In doing so, he refers to Bartlett (1934) and to a paper tional techniques for inverting matrices-md they still are, for by Tukey, "Vector methods in analysis of varianco", describedas that matter, although at a much more sophisticated level than in having been on the progrimrme atPrinceton, November I ,1946. the pre-computer days. An early beginning to those sophisti- Also in the 1940s,Dwyer & MacPhail (1948) have a long paper cated ways was the Doolittle system for doing, and setting out, on matrix derivatives. the individual calculations. The date and location of the publica- From this time on, matrices gather momentum towards be- tion of this systemis perhapssurprising: Doolittle (1851) in the coming standard notation for multivariate statistics, peaking in 1878 U.S. Coast and Geodetic Survey Report. Numerous abbre- the classic book by Anderson (1958), and continuing through viations, improvements and comments followod, in a variety of to the present day in numerous books and papers, including the publications, including Horst (1932) and Dwyer (I94Id, l94lb, start of the Journal of Multivariate Analysis in I97I. Lg44) in the statisticalliterature. In his lgMpaper, Dwyer wrote "The reader should be familiar with elementary matrix theory llndeed Anderson (1983, p. 474) observes that "Hsu promoted the use of such as that outlined on pages l-57 of Aitken's book" (1939a). maffix theory in statistical theory as well as proving new theorems about matri- An ces". alternative approach was Bingham (1941) using the Cayl"y- fMAGE 24: April2000 page 27 Hamilton theorem and then Newton's equations.Other contribu- being used. Although the arithmetic of generulized inversesis tions included Hotelling (1943), Quenouille(1940), Fox (1950) scarcely any less than that of regular inverses, the use of general- and Fox & Hayes (1951). The arrival of computerssoon put an ized inverses is of enorrnoushelp in understanding estimability end to thesepencil-and-paper based methods, which by then had and its consequences. collected other variants of Doolittle such as abbreviatedDoolit- 4.6. Other Topics. Of the many other topics and places tle, Crout-Doolittle, and so on. where maffices have contributed substantially to statistics,only 4.4. Design of Experiments. In the 1950s a widely used- a few ire mentioned briefly here. book on experiment design was Cochran & Cox (1950, Znd 4.6. 7. Probability Theory andMarkov Chains. B ack-to-back ed.); it has but one, brief mention of maffices related to regres- papers using matrices for branching processesand queuing pro- sion. But only a year later Box & \Milson (1951) make sub- cessesare Hammersley & Morton (1954) and Bailey (1954), fol- stantial use of matrices for normal equations,their solution and lowed by the books by Kemeny & Snell (1960, 1962). resulting sums of squares. Likewise, Tocher (1952) has solid 4.6.2. Age Distibution Vectors. When the proportions (or = maffix usage. Both papers give, for order n, (I + oJ)-t numbers) in different agegroups of an animal population are ar- - I aJ l(1 + an); andTocher extends thisto (I + A E)J)-1 rayed in a vector, it is an age distribution vector (Lewis 1942). - I [(l + nA)-tn] A J. This is an exampleof a classof de- That vector is seldom constant over time. Leslie (1945, 1948), signs involving its own special matrices. There are many other by using age-specific birth rates as the first row of a matrix, examples,one of the most extensivebeing factorial designs,for and age-specific survival rates as the sub-diagonal of that ma- which Cornfield & Tukey (1956) developedfar-reaching use of trix (now called a Leslie maffix), shows how pre-multiplying an Kronecker (or direct) products of maffices for fixed effects mod- age-distribution vector for time t by a Leslie matrix yields (de- els. terministically) the age distribution vector at time t + 1. In a Nelder (1965a, b), Smith & Hocking (1978) and Searle & practical application of this, using monthly Leslie matrices and Henderson (1979) independently extendedthis to the dispersion a killing (diagonal) matrix, Darwin & Williams (1964) adapt matrix for models involving random effects. Another example this procedure to ascertainoptimum months of the year for poi- is Latin squares,for which Yates& Hale (1939) could haveben- soning rabbits on New Tnaland farms. Rabbits there ure a seri- efited from using maffices-but they did not. But Cox (1958) ously overpopulated pest, consuming grass needed for produc- certainly did-in abundance. Design features involving circu- ittg meat, wool and milk which sustain the country's economy lant matrices include autocorrelation and spectral density func- through exports. (Olkin tions (Wise, 1955), circular stationarymodels & Press, 4.6.3. Estimating Genetic Wortlt Programs for improving (Anderson, 1969) and partial factorial balance 1972); for these the genetic worth of an animal species through planned mat- Searle (1979) gives methods for inverting circulants of two and ings often include estimating the genetic worth of an untried three non-zero elements per row. animal from records of its ancestors. Searle (1963) shows that 4.5. Linear Models. H. O. Hartley makesan interesting com- the correlation between genetic worth and an ancestor-basedes- ment in the publisheddiscussion of Tocher(1952, p.96), sug- timate of it is of the form x'A-lx where x' is closely related gesting that it was Barnard who introduced the unifying princi- to rows of A. As a result, through knowing how elements of ple of developing "analysis of variance by analogy to regression A- 1 involve cofactors and the determinant of A, & recurrence analysis". This is, of course, the vital connectionfor what used relationship for the correlation (based on the number of genera- to be called "fitting constants" and is now usually known as lin- tions of ancestors)wns established.Conclusions are that going ear models. It is the foundation on which enonnous numbers beyond grandparents conffibutes very little to one's estimate of of research papers have been built-and also numerous books, the untried animal. (Horse-racing enthusiasts wanting to buy an so many in fact that commenting on them is well beyond the untried yearling should take heed!) paper. And certainly during the last forty or more scope of this 4.6.4. Inverting A + UBV. In statistics, the matrix A * years, all of them use matrices as the lingua franca. The earli- UBV arises in various forms. For example, one special case est book to do so seemsto be Kempthorne (1952) and a paper is the dispersion matrix D (p) - pp' for a multinomial random of the same decadeis Searle (1956). Nevertheless,seven years variable, where p is a vector of probabilities and D(p) is the latet the book by Williams (1959) on regressionhad only a tiny diagonal matrix of elements of p. Also, it occurs as an inffaclass mention of maffices. correlation matrix (1 - p)l+ pJ, it arisesas discriminant analysis One of the greatestcontributions to understandingthe appar- (Bartlett 1951) and it turns up when elements of a matrix are ent quirkiness of normal equations of non-full rank (as is cus- altered (Sherman & Morrison 1950). linear models), which have an infinity of solutions, tomary with What is interesting about the inverse of A + UBV is that (1962). (1920) is due to Rao Using the work of Moore and Pen- although rose (1955), he showedhow a generalizedinverse maffix yields a solution to the normal equations and how that solution can (A + UBV)-I - A-1 - A-tU(B-t + VA-1U)-1VA-t, be used to establish estimable functions and their estimators- and these results are invariant to whatever generalized inverse is as the standard result is widely known, there are many variations page28 fMAGE 24: April2000 of it, as well as numerous special cases. Henderson & Searle and Rao (1952) seemto be the first books having substantialsec- (1981b) give a number of theseas well as much of the history. tions in matrix notation and making considerableuse of matrix 4.6.5. Partitioned Inverses. The value of generalized in- algebra. Rao (1952) begins with some thirty pages of matri- 'oThe verses in linear models analysis has already been mentioned. ces, having written in his preface problems of multivariate They are particularly useful when dealing with partitioned mod- analysis resolve themselves into an analysis of the dispersion els E(y) - Xr91 * XzFz. Then the matrix Q* = matrix and reduction of determinants." Thereafter came Ander- son (1958),Graybill (1961),Rao (1965),Searle (1971), Graybill (1976), Seber (1977), ffid many others too numerous to men- (; +( (D-cA-B)-(-cA- 'r) tion, all using matricesextensively. So do Kotz et al. (1985) in :) :") the Encyclopedia of Statistical Sciences(Vol. 5), and the Ency- clopedia of Biostatistics, the latter having "Matrix Algebra" as is a generahzeAinverse of a major entr5/. Now to the occurrence / a' B\ of statisticsin matrix books. Those by Frazer, Duncan & Collar (1938), Aitken (1939a) and Fer- Q- | I rtn (1941) have no statistics. Bellman (1960) has two chap- D/ \C ters (lUVo) on Markov matrices and probability theory. And provided the Graybill (1969,1983) and Searle(1966, 1982) books have plenty of statistics,as one would expect from their titles. rank(a) - rank(A) * rank(D - CA-B), Then there are the specialized books on generalized inverses, three in the same year: Pringle & Rayner (1971), Rao & Mi- cf. Marsaglia & Styan (1974). For non-singular Q, the gen- tra (1971), both with plenty of statistics,and Boullion & Odell - - t eralized inverse a becomes the regular inverse Q . In the (1971) with a modest tilt to statistics. Shortly thereafter came ' caseof Q = X'X for X = (Xt Xz), the rank condi- Ben-Israel & Greville (I974) with virtually none. = fion alwaysholds, and Q* a-. The use of Q- in providing Whilst this incursion of statisticsin maffix books was taking f, = a:X'y then leads immediately to very useful results such place, we may note that it was also happeningin linear algebra p1, as 9z and R(prl9), the reduction in sum of squaresfor journals. In Linear Algebra and its Applications (begun in 1968 pr. fitting Fr adjustedfor and often referred to as T Ar{) and Linear and Multilinear Alge- bra (started in 1972) the early issuesshowed little evidence of 5. BOOKS statistics, whereasnowadays there is quite a good representation of statistics. In particular, LAA periodically now has special is- Tracing the intermingling of matrices and statistics in published sueson linear algebra and statistics;the 1990 special issue (vol. books could be a long process.Here it will be brief, in terms of 127) had 656 pages. Many papersin theseissues stem from the a dichotomy: the occurrenceof matrices in statisticsbooks, and now almost annual Workshop on Matrices and Statistics. Started of statisticsin matrix books. in 1990, the only yeius of this decadein which the workshop has Fisher (1935 et seq.) had no matrices in any of its nu- not met are 1991 and 1993 (the next Workshop is scheduled for merous editions. Many sets of numbers are laid out in rect- Hyderabad,India, December 2000). So matrices in statisticsare angular iurays, but they ire designs, not matrices. Snedecor alive and well. (1937), Kendall (1943-1952), Mood (1950), ffid Mood & Gray- bill (1963) had either no matricesor almost none, but by Kendall 6. OMISSIONS & Stuart (1958, Vol. I, lst ed.), their chapters15 and 19 had To thosewhose workhas not beenmentiored, my apologies. For matrix notation for the multi-normal distribution, for quadratic a hint at the vastness(in both time and geography) of the litera- forms, and for least squares.But the Snedecorbook took much ture, they are referred to hrntanen & Styan (1988); ithas 1,596 join longer to the matrix crowd. It is in Snedecor & Cochran literature references on matrices and statistics and many, many (1989, 8th ed.), where the prefaceheralds the arrival of matrices more have been added in supplements during the last eleven with the following near-apologia: years. Seealso Anderson, Das Gupta & Styan (1977) and Styan (1998) for more references. " A significant changein this edition occurs in the notation used to describe the operations of multi- ACKNOWLEDGEMENTS ple regression.Matrix algebra replacesthe original summation operators, and a short appendix on ma- It is a pleasure to thank Professor George P. H. Styan for his trix algebra is included." editorial help in finalizing this paper. An earlier version of this paper was presentedas an invited talk at the Eighth International Cram6r (1946) was early in having a whole chapter "Ma- Workshop on Matrix Methods for Statistics (Tampere, Finland, trices, determinants and quadratic forms", referencing Cochran August 1999), &rd is Paper No. BU-I444-M in the Department (1934) for the last of thosethree topics. But Kempthorne (1952) of Biometrics at Cornell University. fMAGE24: April2000 page 29 REFERENCES A. Ben-Israel & T. N. E. Greville (1974). GeneralizedInverses: Theory and Applications. Wiley, New York. [Reprinted (with corrections) A. C. Aitken (1937). Studiesin practical mathematics,I: The evaluation 1980: Robert E. Kreiger, Huntington, NY.l with applications of a certain triple product matrix. Proceedings of the inverse matrix. the Royal Societyof Edinburgh,ST, L72-L81. M. D. Bingham (1941). A new method for obtaining Journal of the American StatisticalAssociation,36, 530-534. A. C. Aitken (1938). 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A memoir on the theory of matrices.Philosophical A,62 (1948),277.1 Transactionsof the Royal Societyof l-ondon,l{8,17-37. A. C. Aitken (1949). On the Wishart distribution in statistics. K. L. Chung, with the cooperation of C.-S. Cheng & T.-P. Chiang, Biometrika,36, 59-62. eds. (1983). Pao-Lu Hsu Collected Papers. Springer-Verlag,New York. A. C. Aitken (1950). On the statistical independenceof quadratic forms in normal variates.Biometrika,3T ,93-96. W. G. Cochran (1934). The distribution of quadratic forms in a normal system, with applications to the analysis of covariance. Proceed- A. C. Aitken & H. Silverstone (1942). On the estimation of statistical ings of the Cambridg, Philosophical Society,30, 178-191. parameters. Proceedings of the Royal Society of Edinburgh Section A,61, 186-L94. W. G. Cochran (1938). The omission or addition of an independent variate in multiple linear regression. Journal of the Royal Statisti- D. A. Anderson (1972). Designs with partial factorial balance. 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The Annals of Mathematical Statistics,9, 48-55. T. W. Anderson, S. Das Gupta & G. P. H. SUan (1972). A Bibliography of Multivariate Statistical Analysis. Oliver & Boyd, Edinburgh. A. T. Craig (1943). Note on the independence of certain quadratic [Reprintad 1977: Robert E. Kreiger, Huntington, NY.] forms. The Annals of Mathemntical Statistics, 14, 195-197 . H. M. Bacon (1938). Note on a formula for the multiple correlation G. Cramer (1750). Introduction d I'Analyse des Lignes Courbes coefficie nt. The Annals of Mathematical Statistics, 9 , 227 -229 . Algibriques. Frdres Cramer & Cl. Philibert, Geneva. [Reprinted 1968: Landmarks of Science, Readex Microprint.l N. T. J. Bailey (1954). On queuein! processeswith bulk service. Jour- nal of the Royal Statistical Society Series B, 16, 80-87. H. Cramdr (1946). Mathematical Methods of Statistics.Princeton Uni- versity Press. W. Bartl{y (1943). Multiple sampling with constant probability. The Annals of Mathematical Statistics, 14,363-377 . J. H. Darwin, & R. M. Williams (1964). The effect of time of hunting on the size of a rabbit population. New Zealand Journal of Science, M. S. Bartlett (1934). The vector representationof a sample. Proceed- 7 ings of the Cambridge Philosophical Society, 30, 327-340. ,341-352. de la Vallde Poussin(1911). Sur la mdthodede I'approximation M. S. Bartlett (194I). The statistical significance of canonical correla- C. J. minimum. Annales de la Socidtd Scientifique de Bruxelles, 35, 1- tions. Biometrile, 32, 29-37 . 16. M. S. Bartlett (1947). Multivariate analysis (with discussion). Journal M. H. Doolittle (1881). Method employed in the solution of normal of the Royal Statistical Society Series 8,9, 176-197 . equations and the adjustrnent of a triangulation. In Report of the Bartlett (1951). An inverse matrix adjusfrnentarising in discrimi- M. S. Superintendent of the U. S. Coast and Geodetic Surttey Showing Annals Statistics,22,107-1 11 . nant analysis.The of Mathematical The Progress of the Work During the Fiscal YearEnding With June, E. T. Bell (1937). Men of Mathematics. Victor Gallancz, London 1878,Government Printing Office, Washington,pp. 1 15-120. [Pa- and Simon & Schuster, New York. [Reprinted 1986: Touchstone per No. 3 presentedby Chas. A. Schott, Assistant in charge of Books, London.] Office.l E. T. Bell (1953). Men of Mathemntics, Volume 2. Pelican Books: M. F. Driscoll (1999). An improved result relating quadratic forms and Penguin, London. [Reprint of secondhalf of Bell (1937).] chi-squaredistributions. The American Statistician, 53, 273-27 5 . R. Bellman (1960). Introduction to Matrix Analysis. McGraw-Hill, M. F. Driscoll & B. Krasnicka (1995). An accessibleproof of Craig's New York. [2nd Edition: 1970. Reprinted 1995 (lst ed.), 1997 theorem in the generalcase. The American Statistician,49,59-62. 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The Annals of Mathematical Statistics, 15, 82-89. cesses. Journal of the Royal Statistical Society Series B, 16,76- P. S. Dwyer (195l). Linear Computations.Wiley, New York. 79. P. S. Dwyer & M. S. MacPhail (1948). Symbolic matrix derivatives. H. V. Henderson & S. R. Searle (1979). Vec and vech operators for The Annals of Mathematical Statistics,19, 517-534. matrices, with some uses in Jacobians and multivariate statistics. The CanadinnJournal of Statistics,T, 65-81. R. W. Farebrother (1997). A. C. Aitken and the consolidation of mafrix theory. Linear Algebra and its Applications,264,3-t2. H. V. Henderson& S. R. Searle(1981a). The vec-permutation mafix, the vec operator and Kronecker products: a review. Linear and R. W. Farebrother (1999). Fitting Linear Relationships: A History of M ultilinear Alg ebra, 9, 27 1-28 8. the Calculus of Obserttations 1750-1900. Springer-Verlag, New York. H. V. Henderson& S. R. Searle(1981b). 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Knobloch, eds.), Academic Press, Communities ses.Bio metrika, 2'1, 16 1-190. Boston, pp. 43-89. S. Kotz, N. L. Johnson& C. B. Read, eds. (1985). Encyclopedia of & W. Ledermann (1994). Matrix theory. In Com- I. Grattan-Guinness Statisrtcal Sciences, Volume5: Lindeberg Condition to Multitrait' the History and Philosophy of the Math- panion Encyclopedia to Multimethod Matrices. Wiley, New York. emntical Sciences (I. Grattan-Guinness,ed.), Routledge, London, G. W. H. Kowalewski (1909). Einfiihring in die Determinantentheorie pp.775-786. einschliefilich der unendlichen und der Fredholmschen Determi' F. A. Graybill (1961). An Introduction to Linear Statistical Models, nanten. Veit & Co., l-eipzig. t4th Revised Edition: 1954.1 Volume 1. McGraw-Hill, New York. [Revised as Graybill (1976).] 'nor- W. H. Kruskal & S. M. Stigler (1997). Normative terminology: (1969). to Matrices with Applicartons in F. A. Graybill Introduction mal' in statistics and elsewhere. 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Proceedings of the Royal Societyof London SeriesA, D. W. Smittr & R. R. Hocking (1978). Maximum likelihood analysis of 283,147-t62. the mixed model: the balanced case. Communicationsin Statistics- J. A. Nelder (1965b). The analysis of randomtzed experiments with A, Theoryand Methods,T,1253-1266. orthogonal block stnrctue, II: Treaffnent stnrcture and the general G. W. Snedecor (1937). Statistical Methods. Iowa State University variance. Royal analysis of Proceedings of the Society of London Press, Ames. [5ttr Edition: 1956. 6ttr Edition by Snedecor & SeriesA,283, 163-178. Cochran(1967).1 I. Olkin (1997). A determinantal proof of the Craig-Sakamoto theorem. G. W. Snedecor & W. G. Cochran (1967). Statistical Methods,6tlt Applications, -223. Linear Algebra and its 264, 217 Edition. Iowa State University Press, Ames. tSttr Edition: 1989. I. Olkin & S. J. Press (1969). Testing and estimation for a circular 1st-5th Editions by Snedecor ( I 937 -1956).1 stationary model. TheAnnals of Mathematical Statistics, 40, 1358- G. P. H. Styan, ed. (1998). Three Bibliographies and a Guide. Prepared 1373. for the Seventh International Workshop on Matrices and Statistics, R. A. Penrose(1955). A generalized inverse for matrices. Proceedings in celebration of T. W. Anderson's 80th birthday (Fort Lauderdale, of the Cambridge Philosophical Society,51, 406413. Florida, December 1998). Dept. of Mathematics & Statistics, R. M. Pringle & A. A. Rayner (1971). GeneralizedInverse Matrices McGill University, Monfr6a1. with Applications to Statistics.Grffin, London. J. J. Sylvester (1884). Sur la rdsolution gdndrale de l'6quation S. Puntanen& G. P.H. Styan (1988). A PersonalGuide to the Literature lin6aire en matices d'un ordre quelconque. Comptes-Rendusde l'Acoddmie des SciencesParis,99,4O94tZ & 432436. in Matrix Theory for Statistics and Some Related Topics. Report A205, Dept. of Mathematical Sciences, University of Tampere, K. D. Tocher (1952). The design and analysis of block experiments Tampere, Finland. (with discussion).Journal of the Royal Statistical Society Series B, t4,45-100. page32 IMAGE April 2000 H. W. Tirnbutl & A. C. Aitken (1932). An Introduction to the Theory from Eigenwert and the second part, Wert, means value and is af Canonical Matrices. Blackie, London. [Reprinted 196I: Dover, cognate to the English word worth of similar meaning. Wort is New York.l an archaic term for herb, today found in compounds, as Fare- B. A. Welch (1935). Some problems in the analysis of regression brother points out. My amateur sleuthing in the Oxford English among k samplesof two variables.Biometrika,zT,145-160. Dictionary (OED) has traced worth and wort back to similar but P. Whittle (1953). The analysisof multiple stationarytime series.Jour- different Old English (OE) word forms. If the two words have nal of the Royal Statistical Society Series B, I 5, I 25-139 . the sameroot, the OED gives no clue of this. S. D. Wicksell (1930). Remarks on regression. The Annals of Mathe- But here's the curious thing. The dictionary traces the word matical StatisticJ,1, 3-13. root back to the same OE word as wort. The corresponding S. S. Wilks (1932). On the sampling distribution of the multiple corre- obsolete German form is Wurz (according to Duden Herlcnnfts- lation coefficient. The Annals of Mathemntical Statistics,3, 196- 203. wiirterbuch) meaning both root and herb. This appears in mod- (root) (spice). in- E. J. Williams (1959). RegressionAnalysis.Wiley, New York. ern German as Wurzel and Wiirze Now that's teresting, os eigenvalue was consistently known as latent root J. Wise (1955). The autocorrelation function and the specffal density function. Biometrika, 42, 151-1 59. when I was a student in Scotland in the 1950s. So, in some higher sense,Farebrother is right after all with his translation. F. Yates & R. W. Hale (1939). The analysis of Latin squareswhen two or more rows, columns or treafinents are missing. Journal of the If you wish to retain the senseof the German eigen (meaning Royal Statistical Society Supplement, 6, 67:7 9 . own) without committing bilingual murder, why not use pnoper value or pnoper root, for the older meaning of proper isprecisely ShayleR. SSaRLE: biometrics@ cornell.edu one's own. But to me latent root says it all: a hidden root. Yet Departments of Biometrics and Statistical Science I'll make the pessimistic prediction that eigenvalueis now used Cornell University, Ithaca, NY 14853, USA in so many areasof mathematics and by so many people in other scientific disciplines that it is here to stay. MR Lookup-AReference Tool for Linking REFERENCES R. W. Farebrother (1999). The root of the matter. Image, no. 23, p. 8. & W. Ledermann (1994). Matrix theory. In Com' Authors can now accessMathemntical Reviqvs (MR) Database I. Grattan-Guinness panion Encyclopedia to the History and Philosophy of the Math- to verify and create references that link to reviews and original ematical Science.tG. Grattan-Guinness,ed.), Routledge, London, sources. Users input basic reference information into the form pp.775-786. at the Web site www.ams.org/mrlookup Hans ScHnUDER: hans@ math.wisc.edu and MR delivers electronic publication-ready references with Dept. of Mathematics, University of Wisconsin-Madison live links to reviews in MathsciNet and to original articles. The Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, USA MR Lookup tool enables authors and publishers of source arti- cles to build links in a simpler, single, consistentformat-free of charge. The older literature from the past 60 ye:rs becomespart reviews of the papers. And, as of the web immediately through Let us Root for Root those older articles become available online on JSTOR or other publisher sites, new links will be added automatically from the original articles, without any action by review on MathSciNet to No matter what the proper translation is for the original German those sourcepublishers. words into eigerwalue and eigerwector,these are a poor choice. Eigenroot ismuch better than eigenvalue; after all, an eigenvalue Annette Etr{gns o N, P romotions M anag e r : awe@ ams.org is a root of an equation and as such the word root carries with it PO Box 6248 American Mathematical Sociery, the implications that it can be zero, positive or negative, real or Providence, N 02940-6248, USA complex. Moreover, for lecturing, root and vector without the word eigen bring good clarity to the subject, with no confusion such as is possible between value and vector; easier, too, for root is in keeping with the early Eigenvaluesand LessHorrible Words studentswriting notes. Finally, use of latent root and latent vector. So let us root fot root. Grattan-Guinness& Irdermann (L994, p. 785), as reported by ShayleR. SSaRLE: [email protected] Farebrother (1999, p. 8), are right to object to the term eigen- Departrnents of Biometrics and Statistical Science value as a horrible hybrid of German and English, but Fare- Cornell University, Ithaca, NY 14853, USA brother's etymology of this word is not quite right. It derives