By Samuel Schechter. (1) H = i-î -1 with 1 ^ i,j g n. [a, b,j

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1 On tne Inverson of Certan Matrces By Samuel Schechter 1. Introducton. Let a, a2,, an ; 61, 62,, 6 be 2n dstnct, but otherwse arbtrary, complex numbers. For the matrx H, of order n, (1) H = -î -1 wth 1 ^,j g n [a, b,j let G = H~ = \dj\. (The ndces, j, k wll range from 1 ton unless t s specfed otherwse.) If, for some constant p, (2) o< - bs» 1+ j - 1+ p * 0, then H s a segment of the well known generalzed Hubert matrx, and n ths case formulas for c have been gven by Savage and Lukacs [4], Smth [6] and Collar [1]. For (3) a, - bj = - j + p, Lnfoot and Shepherd [3] and Collar [2] gve formulas for c<}- and n both cases Collar exhbts dagonal matrces D, K such that G = DHTK. Collar [2] and Smth also evaluate the quanttes 23 ca 1 23 ca,j j These authors make use, n most cases, of the formula for the determnant [5] II (ay - ak)(h bj) (4) det H = ^7- - Il (aj - M * or requre the evaluaton of certan nvolved seres. The formulas of Collar and Smth are extended here to the general case (1). The method to be used does not depend on (4) but smply on Lagrange 's nterpolaton formula. Indeed (4) comes out as a by-product of formula (17) gven below. 2. Formula for the nverse. Let (5) A(x) = II(* - *), B(x) = U(x - O and denote the fundamental polynomals of the Lagrangan nterpolaton correspondng to the a, and 6,, respectvely, by (6) AAx) - A ^X) xl BAx) B(X) A'{at){x - at)' B'(b )(x - 6,) prme denotes dfferentaton. We then have Receved February 13, The work for ths paper was done at the AEC Computng and Appled Mathematcs Center, Insttute of Mathematcal Scences, under a contract wth the U. S. Atomc Energy Commsson. 73 Lcense or copyrght restrctons may apply to redstrbuton; see

2 74 SAMUEL SCHECHTER Theorem 1. The elements of G are gven by (7) tu = {a - b)aj(b)b{aj) and f H s symmetrc (8) dj = (ay - 6 )^J(6)A,(6J). Proof: For any polynomal p(x) of degree n-1 we may wrte (9) p(x) = Zp(a)A(x) or do) 4Ä = T, C A(x) x d (d c< = 5<*> Now let p(x) = Bk(x)A(bk) = Pk(x) and Then from (10) we obtan _ pt(a.) " ~ A'(af) /12) gt(a;)a(b<!) _ y ck 4(x)» a x If we set.r = 6, then, snce Bk(bj) = hk, the Kronecker delta, we get that Ch Skj = X a h. Thus Ck gves the desred nverse element, that s, Bk(a)A(bk), x, \t> \ a -u\ ck = -.,, N- = {at bk)bk(a)a{bk). If H s symmetrc: a by = ay &,, t follows that S»(ay) = ^ (&y) and the theorem s proved. We now obtan a smple formula for the sum of the c,y. Ths quantty arses n problems of aerodynamcs [2] and ts smplcty allows t also to be used as a check on some alleged nverse. Corollary. (13) X c y = X (ak bk) = s,3 k Proof. If we apply (9) to p(x) (14) By(<n)«(6») - Sfl,. = B3(x) and set x = bk we obtan that By symmetry ths s also vald f A, a and B, b are nterchanged. Thus we get that s = X ay X Ay(b,)ß,(oy) - XI & X Ay(6 )B (ay) = Z) a; ~ X &«Lcense or copyrght restrctons may apply to redstrbuton; see

3 ON THE INVERSION OF CERTAIN MATRICES 75 For the specal case of the Hubert segment (2), formula (13) gves Smth's [6] formula: s = n(p + n) and for the matrx of (3) we get the formula of Collar [2]: s = pn. 3. Row and column sums. Let the row and column sums of G be, respectvely, (15) ] c,y = a<, J2c] = ßj and defne the dagonal matrces Da, Dß by Da = \ct\, «2,, a ] Theorem Dß = [ß,ß,,&]. 2. The matrx H then satsfes the followng relatons: B'(b) A (ay) (17) B~1 = DaHTDß. Proof: Assumng (16) to be true we note that d can be wrtten n the form 1 A(b) B{a ) _ aßj (18) cu = b a,- B'(b) A'{a,) a 6, whch gves (17). To prove the formula for a n (16) we need only show that (19) **$»... y («y - b)a'{a ) However for any functon/(x) we may wrte (20) f[aua2,---,an] = S^'/U) /(ay) the left sde of (20) s the (n l)th dvded dfference of f(x) wth respect to the ak (see Mlne-Thomson [7] p. 9). If we apply (20) to the polynomal /(x) = B(x)/(x h,) of degree n 1 we have that 5U_1 f(x) = constant. Snce the coeffcent of xn~l m f(x) s 1, (19) s proved. An alternate proof of (19) may be obtaned, wthout dvded dfferences, by notng that the left hand sde of (19) s the sum of the resdues of the functon B(x)/A(x)(x b) at the a. (Ths, n fact, follows readly from (10).) However for a suffcently large crcle C about the orgn n the complex x-plane dx=l 2rJc (x b)a(x) whch proves (19). The proof for the ßj s obtaned n the same manner from (19) wth the roles of A, a and B, b nterchanged, and the theorem s proved. (The formula (19), ncdentally, represents an extenson of one of the formulas of Collar [2] for the Lcense or copyrght restrctons may apply to redstrbuton; see

4 76 SAMUEL SCHECHTER sum of a seres. Extensons of other formulas gven n [2] can lkewse be obtaned from (20) by specalzng/(a;).) 4. Remarks. 1.) We note that f H s symmetrc then a = /3, and f we set D = Da = Dß then (21) G = DHD 2.) From (16) and (17) one mmedately obtans the formula for the determnant of H up to a sgn. That s (22) (det H)2 = II -\ Ol ß and for H symmetrc we get that (23) det#= (-)'»<"-»"n^. o (a,) The sgns may readly be determned, by nducton, by usng (6) and the formula for the (n + 1, n + 1) element of G +1 : Cn+, n+-det Hn+ = det Hn Hn = H and Hn+1 ={ r} wth 1 ^,j ^ n + 1. [a bj) 3.) Formula (8) may also be appled to the problem of obtanng a least-squares ft of a functon f(x) on, say, (0, 1) by a functon of the form X Tí**'- (All var ables are here assumed to be real.) The normal equatons for ths problem yeld a matrx of the form (1) wth 6 = a 1, and the y may be obtaned by applyng G to the moments of f{x) wth respect to the xa. For fttng a functon of two varables the problem to determne 7tJ such that s solved by / / [fx, y) X 7a xhf fdxdx = m 'o J o,y K = DHDFD'H'D' mm H = {a + ay + 1/ ' H' = {a/ + a/ + l/ ' K ' Í7ljí '» /. F ={/o /o f(x>y)xaya'dxdy}> and Ö, D' are the dagonal matrces correspondng to H, H', respectvely, defned above. Lcense or copyrght restrctons may apply to redstrbuton; see

5 ON THE INVERSION OF CERTAIN MATRICES 77 Although the elements c,y get qute large n the case of the Hubert matrx, t may happen that for sutable choces of the a ths may not be the case. An explct soluton for y can also be obtaned from (7) for the equatons /o (fx) -Eî"]^ = 0, k= 1,2,- gven the moments of f(x) wth respect to the xa'. 4.) If H s real and symmetrc and f a, > 6,-, = 1, 2, -, n then t follows from (4) that all the prncpal mnor determnants are postve and H s postve defnte. In ths case a, > b for all and j; that s, all the elements of H are postve, snce f a < ay, then 0 < a, 6 < ay by. Thus B(a ) > 0 and, snce A(x) has n smple zeros at the a,, A'(a) alternate n sgn. From (8) t then follows that, f a, < ay for < j then ( 1) 'dj > 0 so that G has the same checkerboard dstrbuton of sgns as the nverse of the Hubert segment. From (16) t follows that n ths case the a, also alternate n sgn and that ( l),+nay > 0. Let X and /x denote the smallest and largest egenvalue of H respectvely, and let M= mn(,a*~fev.y kjt* \\ak a / Then t follows readly that A (b) ^ M?"1 > 1 and that If the a, ncrease wth then X g mm s mm I-=- I Cu \a O/ mn(x(-l)+jc,y)_1 ^ X. «? We then get for the P-condton [8] of H : - ^ max (-=- J max c, 2ï max [A (6,)]2 ^ max M2n~2 X, \a bf,, so that the P-condton of H may get very large. Ths number has been estmated for the Hubert segment (2) wth p = 0 by Todd [8]. New York Unversty, New York, New York 1. A. R. Collar, "On the recprocaton of certan matrces," Proc. Roy. Soc. Edn., v. 59, 1939, p A. R. Collar, "On the recprocal of a segment of a generalzed Hubert matrx," Proc. Cambrdge Phlos. Soc, v. 47, 1951, p E. H. Lnpoot & W. M. Shepherd, "On a set of lnear equatons, II," Quart. J. Math., Oxford, v. 10, 1939, p I. R. Savage & E. Lukacs, "Tables of nverses of fnte segments of the Hubert matrx," Contrbutons to the Soluton of Systems of Lnear Equatons, O. Taussky, Edtor, Natonal Bureau of Standards Appled Mathematcs Seres 39, 1954, p U. S. Govt. Prntng Offce. 5. G. Pólya & G. Szego, Aufgaben und Lehrsätze aus der Analyss, v. 2, Sprnger, Berln, 1925, (Reprnted by Dover Publcatons, New York, 1945), p R. B. Smth, "Two theorems on nverses of fnte segments of the generalzed Hubert matrx," MTAC, v. x, no. 65, January L. M. Mlne-Thomson, The Calculus of Fnte Dfferences, Macmllan, London, J. Todd, "The condton of the fnte segments of the Hubert matrx," Contrbutons to the Soluton of Systems of Lnear Equatons, O. Taussky, Edtor, Natonal Bureau of Standards Appled Mathematcs Seres 39, 1954, p U. S. Govt. Prntng Offce. Lcense or copyrght restrctons may apply to redstrbuton; see