Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler Academy of Scences of the Czech Republc, Inst of Computer Scence, Pod vodáren věží, 8 07 Praha 8, Czech Republc A R T I C L E I N F O A B S T R A C T Artcle hstory: Receved 7 Aprl 009 Accepted 6 August 009 Avalable onlne 6 September 009 Submtted by H Schneder Inspred by examples of small Hlbert matrces, the author proves a property of symmetrc totally postve Cauchy matrces, called AT-property, and consequences for the Hlbert matrx 009 Elsever Inc All rghts reserved AMS classfcaton: 5A48 5A57 Keywords: Hlbert matrx Cauchy matrx Combned matrx AT-property Introducton As s well known [5], a Cauchy matrx (maybe even not square) s an m n matrx assgned to m + n parameters x,, x m, y,, y n (one of them can be consdered as superfluous) as follows: C =, =,, m, j =,, n x + y j For generalzed Cauchy matrces, addtonal parameters u,, u m, v,, v n, have to be consdered (one of whch agan superfluous): u v j Ĉ = x + y j Research supported by the Insttutonal Research Plan AV 0Z000504 E-mal address: fedler@mathcascz 004-795/$ - see front matter 009 Elsever Inc All rghts reserved do:006/jlaa0090804
5 M Fedler / Lnear Algebra and ts Applcatons 4 (00) 5 56 If we restrct ourselves to the square case, t s well known that C s nonsngular f and only f, n addton to the general exstence assumpton that x + y j /= 0 for all and j, the x s are mutually dstnct as well as the y j s are mutually dstnct In fact, there s a formula [,4] for the determnant of C(m = n),k,>k(x x k )(y y k ) det C = n,j= (x () + y j ) Clearly, such formula s easly establshed also for the generalzed Cauchy matrx Ĉ Snce every submatrx of a Cauchy matrx s also a Cauchy matrx, the formula () allows us to fnd the nverse matrx to C Thus C =[γ j ], where l/=(x j + y l ) k /=j(y + x k ) γ j = (x j + y ) l/=j(x j x l ) k /=(y y k ) () In ths note, we shall be nterested n real symmetrc Cauchy matrces, n partcular n the postve defnte and totally postve case, and n the matrx C C, where means the Hadamard entrywse product Recall that a real matrx s totally postve f all ts submatrces have postve determnant A smple corollary of Eq () s Theorem A A symmetrc Cauchy matrx (for whch y = x for each ) C = x + x j s postve defnte f and only f all the x s are postve and mutually dstnct It s totally postve f and only f ether 0 < x < < x n, or 0 < x n < < x Corollary B If C s a postve defnte Cauchy matrx then there exsts a permutaton matrx P for whch PCP s totally postve Let us menton that the famous Hlbert matrx (eg []) (more precsely, the fnte secton thereof) H n = + j s clearly a Cauchy matrx In fact, the results for small Hlbert matrces were an nspraton for the author to present ths note The second nspraton was the followng noton If A s a nonsngular matrx, then t makes sense to defne the Hadamard product A (A T ) We shall call t the combned matrx of A Let us recall three well known propertes of the combned matrces Theorem C All row sums of every combned matrx are equal to one The combned matrces of a nonsngular matrx A and (A T ) concde Multplcaton of a nonsngular matrx by nonsngular dagonal matrces from any sde does not change the combned matrx A less known property was presented n []: Theorem D Let A =[a j ] be a real symmetrc postve defnte matrx, let A =[α j ] Then the combned matrx M = A A of A wth entres m j has the followng propertes: M I s postve semdefnte, Me = e; here, I s the dentty matrx and e s the vector of all ( ones max m ) ( m ) () Remark It seems stll an open problem to characterze the set of all combned matrces of n n postve defnte matrces For n, and gve a complete characterzaton [4]
M Fedler / Lnear Algebra and ts Applcatons 4 (00) 5 56 5 Remark It s easy to see that the combned matrx of an M-matrx as well as of an nverse M-matrx s an M-matrx The possble dagonal entres of such matrces were characterzed n [] The characterzaton s smlar to that n, above: For each, m, and max(m ) (m ) In the sequel, we shall use the followng two denttes: Lemma E Let for n, x,, x n be ndetermnates Then the followng holds: Ifnseven, then n x k + x j k= j /=k dentcally Ifnsodd, then x k x j = 0 (4) n x k + x j k= j /=k dentcally x k x j = (5) Proof One can use the Lagrange denttes, but we shall apply a drect proof Multply the left-hand sde of (4)by >j(x x j ) We obtan a homogeneous polynomal of degree ( ) n It s easly seen that ths polynomal s dvsble by each x x j, /= j, thus by >j(x x j ),of ( ) n degree agan The left-hand sde of (4) s thus an ntegral constant To determne t, consder the term xn n x n n x of the hghest weght of ndces For n even, we get zero, for n odd, we get one Results We frst ntroduce a new noton whch seems to be rather artfcal Let G =[g k ] be an n n square matrx wth nonnegatve dagonal entres We say that G has the alternate trace property, shortly AT-property, f n ( ) k { for n odd, g kk = 0 for n even k= Observe that the dentty matrx as well as any combned matrx of a nonsngular dagonal matrx have the AT-property Our man task wll be the followng result: Theorem The combned matrx of every symmetrc totally postve Cauchy matrx has the AT-property Proof Suppose frst that the Cauchy matrx C corresponds to the n-tuple x k satsfyng 0 < x < < x n accordng to Theorem A Then the formulae () yeld for the dagonal entres m of C C, due to postvty, m = ( ) n x + x k x x k k /=
54 M Fedler / Lnear Algebra and ts Applcatons 4 (00) 5 56 The AT-property then follows mmedately from Lemma E If C corresponds to postve x s n reverse order than n Theorem A, the result follows from the fact that the matrx JCJ, where J s the skew dentty matrx, has the same AT-property as C Remark By Corollary B, the assumpton that C s totally postve can be removed; of course, the correspondng property would be more complcated Remark 4 By Theorem C, the same asserton as n Theorem holds for totally postve generalzed Cauchy matrces Theorem The combned matrx of every prncpal submatrx of the Hlbert matrx has the AT-property In addton, f a square submatrx of the Hlbert matrx has consecutve rows and consecutve columns, then ts nverse as well as ts combned matrx have ntegral entres The dagonal entres of the combned matrx are squares of ntegers Proof The frst part s a corollary to Theorem snce any prncpal submatrx of the Hlbert matrx s a totally postve Cauchy matrx To prove the second part, observe that by (), t suffces to show that n the case of consecutve rows and columns and the substtuton of the correspondng ntegers, the rato l/=(x j + y l ) k /=j(y + x k ) l/=j(x j x l ) k /=(y y k ) s an nteger Change n the denomnator the summaton ndex l to k and k to l Then the whole rato can be wrtten as the product of four ratos k<j y + x k x j x k k>j y + x k x j x k l< x j + y l y y l l> x j + y l y y l Each of the ratos s an nteger snce the numerators are consecutve ntegers In the case that = j and x k = y k for each k, two and two of the above ratos concde Let us add an alternatve proof of Theorem whch makes the AT-property more understandable Observaton Let C be a nonsngular Cauchy matrx Then there exst dagonal nonsngular matrces D and D, such that C = D C T D (6) Proof In the notaton of (), we can rewrte () n the form l/=(x j + y l ) k /=j(y + x k ) γ j = (x j + y ) l/=j(x j x l ) k /=(y y k ) = U j V, x j + y where U j = (x j + y j ) k /=j V = (x + y ) k /= x j + y k x j x k, y + x k y y k The nonsngular matrces D = dag(v,, V n ), D = dag(u,, U n ) fulfl then (6)
M Fedler / Lnear Algebra and ts Applcatons 4 (00) 5 56 55 Observaton A matrx Q satsfyng dag Q = dagq has the AT-property f and only f the matrx QS has the trace property, e { for n odd, trqs = 0 for n even, where S = dag(,,,, ( ) n ) Proof Indeed, f q, q,, q nn are the common dagonal entres of Q and Q, then q, q,, ( ) n q nn are the dagonal entres of QS We complete the proof that C has the AT-property Let C be a symmetrc totally postve Cauchy matrx By Observaton and the symmetry of C, there exsts a nonsngular dagonal matrx D 0 such that C = D 0 CD 0 It s well known that the nverse of a nonsngular totally postve matrx has the checkerboard sgn-pattern Therefore, there exsts a dagonal matrx D wth postve dagonal entres, such that C = DSCSD The matrx Q = D CD has thus the property that (QS) = QS, e, QS s nvolutory Therefore, the egenvalues of QS are and only On the other hand, Q s postve defnte, so that the matrx Q SQ whch has the same egenvalues as QS s congruent to S Thus the egenvalues of QS are and wth the same multplcty f n s even, the multplcty of beng greater by one f n s odd Also, the dagonal entres of Q and ts nverse concde By Observaton, Q has the AT-property The fact that Q Q = C C now completes the proof It also shows the relatonshp wth the nvolutory property of QS Example The submatrx 4 5 G = 4 5 6 5 6 7 of the Hlbert matrx has the nverse 00 900 60 900 880 00 60 00 575 Thus 00 5 6 G G = 5 576 50 6 50 5 s an ntegral matrx It clearly has the AT-property The condton (6) also holds wth U = V = 0, U = V = 0, U = V = 05 The nvolutory matrx QS s then dag( 0, 0, ( ) 05Gdag 0, 0, 05, e 0 5 4 5 4 5 4 4 5 4 5 The trace condton s fulflled Observe that the Hadamard power of QS s the modulus of G G Remark 5 It seems of nterest that the real postve defnte matrces A for whch equalty n () s attaned [, Theorem ] have the property that (up to multplcaton by a nonsngular dagonal
56 M Fedler / Lnear Algebra and ts Applcatons 4 (00) 5 56 matrx from both sdes and smultaneous permutaton of rows and columns) AS s nvolutory, S beng a dagonal matrx dag(±, ±,, ±), satsfyng trs = n Indeed, t was proved n [] that such matrx has the form (up to multplcaton by a nonsngular dagonal matrx from both sdes and smultaneous permutaton of rows and columns) B b A = b T, β where β s a number, B an (n ) (n ) matrx of the form I + (β )uu T, u a unt real vector, and b the β -multple of u IfS s the dagonal matrx dag(,,, ), then B b AS = b T β s nvolutory snce A B b = b T β Addendum An amusng corollary of Lemma E s the followng property of the tableaux of the numbers t j = +j j for /= j, t =,, j = 0,, : 4 5 4 4 5 6 Choose any prncpal mnor" of even order; then the sum of all the products n odd rows s equal to the sum of all the products n even rows If the mnor has odd order, the frst sum exceeds the second by one References [] M-D Cho, Trcks and treats wth the Hlbert matrx, Amer Math Monthly 90 (98) 0 [] M Fedler, Relatons between the dagonal elements of two mutually nverse postve defnte matrces, Czechoslovak Math J 4 (89) (964) 9 5 [] M Fedler, Relatons between the dagonal entres of an M-matrx and of ts nverse, Mat-fyz časops SAV (96) 8 [4] M Fedler, TL Markham, On the range of the Hadamard product of a postve defnte matrx and ts nverse, SIAM J Matrx Theory Appl 9 (988) 4 47 [5] G Pólya, G Szegö, Zweter Band, Sprnger, Berln, 95