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On the Minors of Absolutely Convergent DeterminantsAuthor(s): Leon W. CohenSource: Annals of Mathematics, Second Series, Vol. 34, No. 1 (Jan., 1933), pp. 125-129Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968344 .

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ON THE MINORS OF ABSOLUTELY CONVERGENT DETERM I NANTS.

BY LEON W. COHEN.*

The study of infinite determinants of the form (1) A = Idik+aikI subject to the conditions

(2) z aii< ? + ; [I I[aik t] ] = +X (i + k);

(2) i 11 1 = -+-=1; l<p ? 2 p q

and the associated system of linear equations was begun by von Koch' who considered the case p = 2. The extension to the general case was made by B6br2 by applying the theory of limited linear forms. The writer' also considered the solution theory and showed the convergence of the series

(3) [ a jAik ~2: [a JAR drp (i +k) k=1 i=1 i=1 k=1

where Aik is the minor of the element aik. In this note, we consider the minors

Ar, ... r,,; C. J, (ri i cj; ij =1, *, s)

obtained by striking out the rows ri and the columns cj from A and extend the result of (3) to these minors.

The matrix of the minor Ar1.. .r5; c1.. *C which is an absolutely convergent determinant, can, by a finite number of interchanges of rows and of columns, be brought into the following form:

I 11 aer 11 11a ir 11I

(i, j-,@,s) r + ri

flacrjf 11tder+aacr

(c 1t .; s) (c t ci; r t rj)

* Received, January 27, 1932. 1 H. von Koch, Palermo Rend. 28 (1909), pp. 255-266. 2 St. B6br, Math. Zeit. 10 (1921), pp. 1-10. 3 L. W. Cohen. Bull. Amer. Math. Soc. 36 (1930), pp. 563-572.

125



126 L. W. COHEN.

The determinant of this matrix differs from A,,...r; cl...c, at most in sign. In order to find a bound for A,,...r.; . c, we recall that the absolute value of A is at most4

P + IPI),

where the product is extended of over all products

p ajl a-,i, *a** aij,1 (n == 1, 2,*)

with distinct indices which can be formed from the matrix II aik I. A term of Ar,. . r;;c;...c has, except for sign and a factor which is a term of 3cr + a cr three possible forms:

(4) aeHH jh 1 h=1

Ht kt-e S

(7) = 1 ac+ j. aks +i . a i2 ...iS jk i iHa, j. )

k6) ahei1 ai 1 12 .. .h 1' h=l

where J* is a permutation of r , ra;

0 < U,<s; O ko<ki< ... < k. < s; o s

fjf(h) f (h) 1. h=1 =h #=s1

Hence we may write the inequality

! Ar r ;c cs_ P | c .

.s-I ~~~~tF ke-1 X

(7) + H i I aczhih I ~kt i nt; ktl tt= (j}') 1'lC'-< 1, t-7=1 kh7t-1+ t-1tl'tn=

x l al Cjh ! + ICh ilt *ih jnA , h = k,+I (j; I) h1Ii=lz--t n=

where the symbol I indicates that the sum is to be taken ovrer all (j; r)

permutations ji Y .. I j. of rl,***, rs and I F. Riesz, Les systemes d'6quations lineaires ... Paris, (1913), pp. 33-35.



MINORS OF ABSOLUTELY CONVERGENT DETERMINANTS. 127

Ch i1 in,,; jh , ap1a i ai1 aail I,

if the subscripts are distinct and zero otherwise. The first reduction we can make on (7) is a consequence of the follow-

ing:"

(8) [~ [2.~ aik ] = l < a (i t c),k

=1 Lik=i Qic | Ck~ ii * -; jkt | an'- r at L J

(i t Ck,, Jk, in the series on the right).

These follow from the Holder inequality, which we state in the form

(10) N(P'X (A i + p b) < I A I Nj(P) (ai) + MP'Nt) (be); (p > 1)

where Ni(a2) = [2 aiIPf']. Using (8), (9) and the notation of (10) in (7), we get, assuming <1

Ar. -r;c. cs < P{| ! !P l acAl t(J;r) h=l

( ) (J r) 1< kl <* * <k,, < rI lr I S h j I Ni~q (ack * P) i (aijk

x S

c q j jP aj

h = kn+1 aj ? ( - _i (j;r) h Nf ((Chi) s(ai)(j

Applying (10), we get because of its homogeneity

L- zj r __ PN (aIr .cr;kc) ...cj ] < _ ((r h-1

,? =, (1-a) (Jr) 1 ej <z~k s rl l rX {N(X) (ac,, ̂

x Nck N, (a~e i)Njk N, (ijk ) Ch NCh Njh) (~

+ ( 1 ) z rIX Nc(p) Niq (C) ~ q N(P>N) (aij ,)t (i fql+j).

+~~~'(i~~~a~~t' ~ a i NI lii Nc~ j (a A

(12) = 1 ~~~~~~~~ t~~~i k~~~h= +l +

A1Jk

iiV~~~~~ I2p (i.

5The inequality (8) is given by B6br, loc. cit. p. 5. The inequality (9) is given by the writer, loc. cit. p. 566.



128 L. W. COHEN.

Using (2) and (8) in (12), we get

[cl.,=~rlr=1 Arl -r;c c |2]q < p~1ry8 ; 1? +j (V u N (kg-kU- l)+8 -

(13) UJ; r) u-1 ora (j; r) likl< ...<ku,8

+ (_ i?)8s 08X1l

Ps! ors(i+ 1- (cjf rj; i,j = 1,2, , s).

The assumption made in (11) that x< 1 can be removed by the familiar artifice of a convergence factor. Because of the convergence required in (2), there is an integer m such that

(14) [4z [? aik qJ ] (i t k).

We choose a positive t less than 1 for which [4'! 0 rL q plq l.'P 1

(15) t 1Zi [2 | aik 4 I <-2 (t k)

and consider the determinant A' - dik + a' where

a t(dik+aik), i = dikmaXik~dik + aik, i=m +1m+2,* 2 .

From (14), (15) and the Holder inequality it is clear that (2) holds for A' and that r'< 1. Further

IAr, ... r.; el -C&I < t-t I Ar, ... r,,; c... C.

from which it follows that 00 00 q

z [ z I~Arl... r,,; C,...Co (ri tCj; sj1 *** )

converges. If, in (7), we use the inequality6

(16) | Ckt il . in;jk, I < W"1 I a, i I /q j ai* IP

we find by an analogous method that 00 00 1l r. [ _ IAr1 r,;c"...c. IP (r7 t Cj; ij 1,...,s)

1 * **r,= 81 * * ?8-

6 Loc. cit. (3), p. 567, p. 565.



MINORS OF ABSOLUTELY CONVERGENT DETERMINANTS. 129

converges and, if T1 <1, that the sum is at most

Pvs ! [ ( T--- -)]

The results may be summed up in the THEOREM. If the infinite determinant A - Icik+ aic! satisfies (2) and

A7.1 ... r,; l *c8 is the minor of A obtained by striking out the rows ri and the columns cj (ri t- c;; i, j-1, * s) then the series

z | z | Arl . }s; c .CsIq]

cr4 l~jr1 0_

r 0 .00 (r-l - e ic,,-1

(onvejqe. If a 1, then the two stums are respectively at most equal to

[Pass! ( + 1 j)f [P)] (1 + )

UNIVERSITY OF KENTUCKY.



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