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Group matrices Iwata, William Takashi 1965

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GROUP MATRICES • by William T. Iwata A THESIS' SUBMITTED IN PARTIAL FULFILMENT OF • THE RI1QUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics. We ..accept t h i s t h e s i s as conforming to the required standard from\andidates f o r the degree of MASTER OF ARTS THE UNIVERSITY OF BRITISH COLUMBIA September, 19^5 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s representatives„ I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT A new proof i s given of Newman and Taussky's r e s u l t : i f A i s a unimodular i n t e g r a l n X n matr ix such that A 'A i s a c i r c u l a n t , then A = QC where Q i s a genera l i zed permutation matr ix and C i s a c i r c u l a n t . A s i m i l a r r e s u l t i s proved f o r unimodular i n t e g r a l skew c i r c u l a n t s . Cer ta in a d d i t i o n a l new re su l t s are obta ined, the most i n t e r e s t i n g of which' are: l ) Given any nonsingular group matr ix A there ex i s t unique r e a l group matrices U and H such that U i s orthogonal and H i s p o s i t i v e d e f i n i t e and A = UH; 2): I f A i s any unimodular i n t e g r a l k s c i r c u l a n t , then integers k and s ex i s t such that A ' = P A and P A i s symmetric, where P i s the companion matrix of the polynomial x n - l . F i n a l l y , a l l the n X n p o s i t i v e d e f i n i t e i n t e g r a l and unimodular skew c i r c u l a n t s are determined fo r values of n < 6: they are shown to be t r i v i a l f o r n = 1 ,2 ,3 and are e x p l i c i t l y descr ibed f o r n = k,5,6. I hereby c e r t i f y that th i s abstract i s s a t i s f a c t o r y . i i i TABLE OF CONTENTS Page 1. Group R i n g s 1 2. M a t r i x R e p r e s e n t a t i o n s and Group M a t r i c e s 1 3- U n i t s and Unimodular Group M a t r i c e s 7 k. C i r c u l a n t s and Skew C i r c u l a n t s * ' 8 5. E x i s t e n c e o f N o n t r i v i a l Unimodular I n t e g r a l C i r c u l a n t s and Skew C i r c u l a n t s 12 6. A New P r o o f on P o s i t i v e D e f i n i t e C i r c u l a n t s 12 7. New R e s u l t s on Group M a t r i c e s and Symmetric C i r c u l a n t s 20 8. P o s i t i v e D e f i n i t e S k e w • C i r c u l a n t s 27 9. . Appendix 36 10. B i b l i o g r a p h y • . 37 ACKWOm^IXJEMENTS I t i s a pleasure to acknowledge my indebtedness to my supervisor Dr. R. C. Thompson f o r suggesting the study of skew c i r c u l a n t s and of c i r c u l a n t s i n general and f o r h i s encouragement and advice i n p r e p a r i n g t h i s t h e s i s . 1. Group Rings • Let G be a f i n i t e group of order,, n w i th elements g^,...,g^ and l e t K be an i n t e g r a l domain and l e t ' F be a f i e l d conta in ing K as a subr ing. Let R(G,F). denote a vector space over F which admits the elements g^,...,g of Ef as a bas is and. in which, a d d i t i o n a l l y , n n n products are def ined by •)' a.g. ) b.g. = ) a.b.g. . where a . ,b . L x a x L J & J ' L i J & X , J x ' j i = l j= l i , j = l are i n F and g. . ='g.g.. I t i s . "wel l known that these operations make R(G,F): i n to an a s soc i a t i ve a lgebra. Let R„ „ denote the set of a l l G, A. • n elements of the form ^ a j _ g i "*"n ^(^•^)" where the sca lar s are i n K. ' i= l -Let 1^ and 1^ . be the i d e n t i t i e s of G and K re spec t i ve l y ; and l e t 1=1^.' 1^ denote the i d e n t i t y of R^ ^ and of G and of K as w e l l except under anomalous s i t ua t i on s . I t i s c l ea r that R v i s a subring of R(G,F):. Since g^,...,g^ i s a bas is f o r R(G,F):, every element of R^ ^ i s uniquely determined by the sca lar s i n K. We s h a l l r e f e r to R„ „. as a group r i n g of G,JK. G over K. 2 . Matr ix Representations and Group Matr i ces . A ^matrix representat ion of degree n of G i s a homomorphism of G into the f u l l l i n e a r group L^(F):, the n X n honsingular matrices over' F. We introduce the l e f t regu lar representat ion of G as fo l lows . I f g e G, then n g S i = I a ijCs)Sj > 1 < 1 < n ( l ) • j= l - 2 -where each a. .(g) 1 S 0 or 1. Let L(g),' - (a..(g).) , (2): the prime denoting transpose. L(g). i s a permutation matr ix. Moreover, h(hg) s L(h)L(g)., f o r h,g i n G, as the fo l l owing computation shows. Pre -mu l t i p l y eq.. 1 by h to get n h(gg )• = ) a (g).hg n n " Z C I a i J < « > & j k ^ h > ) S k k=I j= l ' = ( h g ) ^ n ; . • - I a±^K • • k=l n Thus a i k ( h g > = ^ a(g)-a k(h), hence L(hg).' = L(g)/L(h). ' , and so j= l L(hg> = L( l:)L(g). • . -I f L(g> = In, then a (g). = 0, i f i 4 j, and a_(g) . = 1, i f i = j ; and so, g g i = ^ hence g i s the i d e n t i t y . Thus Lemma 1. G i s isomorphic to the group of permutation matrices L(g):, g i n G, where L(g). i s def ined r e l a t i v e to the order ing g^,... }g^ of the elements of G. VI We s h a l l c a l l L(g)- the l e f t regular, matrix representat ion of G ( r e l a t i v e to a p a r t i c u l a r order ing of the elements of G):. We may extend L(g): to a representat ion of the group r i n g R '. f o r every G , i \ . n k=l n L(.u): = £ a k L ( s k ) : * ^ k=l This gives us, by Lemma 1 and the r u l e f o r m u l t i p l i c a t i o n i n R_, „ , G,Ji Lemma 2. For elements u,v i n R„ _ and a and b i n F . ' G,F L(uv). = L ( U ) L ( V > , L(au+bv): = aL(u) + bL(v): . For each g i n G the r i g h t representat ion of G i s given by n s i s = X b i j ( g ) g j ' i = l , . . . , n . (k) 3=1 and t h i s corresponds to the mapping • R:g ->R(g> = (h, ,(g):):i 1 < i , j < n, of G onto n d i s t i n c t permutation matrices of degree n. Eq_. k impl ies that G i s isomorphic to the matrices R(g):, g i n G; they form the r i gh t  regu lar matrix representat ion of G. Theorem, 1. Any l i n e a r combination of the matrices of the l e f t regu lar matr ix representat ion commutes w i th any l i n e a r combination o f the matrices of the r i gh t regular matrix representat ion. Proof . By eq . ' s 1 and k- r e spec t i ve l y we have f o r elements g and h i n G (gg1,...,ggn>/ = -L / (g>(g 1 , . . . ,g n > / (5> P o s t - m u l t i p l i c a t i o n of eq. 5 by h gives us (gg1h,...-,ggnh):/ = L/(g>(g1h,...,gnh>/. - > L ' (g>R (hXg 1 , . . t ,g n > / where the l a t t e r r e s u l t fo l lows from eq. 6. • Premul t ip ly ing th i s by g and us ing eq. 5 produces C&jh,..., g n h} ' = L / (g):R(h):(g~ 1 g 1 , . . . ,g" 1 g n ) : / ' = L ' ( g ) R ( h ) L ' ( g - 1 ) : ( g 1 , . . . , g n ) / . Comparing th i s w i th eq. 6 we get R(h): = l / ( g )R (h )L ' ( g "'"):. Since LCg^Cg" 1 ) : = I = L(g):L(g):% we get L(g)R(h> = R(h}L(g)., as requ i red. Any l i n e a r combination of the l e f t regu lar matrix representat ion of G over K i s c a l l e d a group matr ix of G over K. This of course - 5 -presupposes.an order ing of the elements of G. Consider the permutation matrix L(g): i n eq.. 2 i n view, of eq. 1. We have a one at the ( j , i ) p o s i t i o n ' o f L ( g ) : ' ' p rec i se l y when gg. = g., .hence p r e c i s e l y when g = g t g - . Thus a one appears at the (±,j)' p o s i t i o n of L(g): p r e c i s e l y when g' - g.g. 1 . Thus i n L(u)' = / a L(.gn )., we have a appear-1 J h r S k k ' Sk • , . - ." g k l n G ing exac t l y at those 'pos i t i ons (i>j)- f o r which g, = g.g. In other words, a group matr ix of G r e l a t i v e to g-^,...,g^ of G i s of the form IW- =' (a _i>, 1 •< 1, j < n . (7> 1 J Theorem .2. Any matr ix over' F which commutes w i th a l l matrices of the r i g h t regu lar matr ix representat ion of G i s a group matrix of G; that i s , i t i s a l i n e a r combination of the matrices of the- l e f t .regular matr ix representat ion of G. Proof. Let C = (c.'.) ;, 1 < i , j < n, c. . i n K, be such that C = R (g k )CR(g k > / , k = l , . . . , n , where R(g k> = (b (g^):): Has def ined i n eq.'. k i s the r i g h t regu lar matr ix representat ion of G. Let u-. be the n-tuple row vector i n which a one occurs i n column j and O's elsewhere. Then f o r f i x e d i , j , 1 < i , j < n, and each k, we have, - 6 -c. . = u.Cu' = u.R(g. )CR(g, ) /u. 1 • n s,t=l This sum may be s i m p l i f i e d . For, by eq.. k, ^^(s^) ~ ^ -1 ' -1 * -1 where s,t are such that g. g = g = g. g . Thus g. = g g I S J£ J "G 1 S i C and g 1 = g kg t _ 1^ so that g ^ " 1 = ^B^1- H e n c e > b v - eq. 7; C i s a group matrix. Since the matrices L(g) fc-rm a group isomorphic to GV, and since the matrices are also l i n e a r l y independent over F, we have Theorem 3-to ' Theorem 3- R(G,F): i s isomorphic 4the algebra over F generated by the J^Cg); g i n G. Rn „ i s isomorphic to the r i n g generated over K by the L(g}, g i n G. C o r o l l a r y 1. The inverse and the transpose of a group matrix i s a group matrix. Proof. The inverse of any matrix i s a polynomial i n that matrix. Hence by. Theorem 3 the inverse of a group matrix i s a group matrix. Since L(g "^): = L(g) , /, g i n G, the transpose of a l i n e a r combination of L(g )',..'. ,L(g n); over F i s again a l i n e a r combination of L(g^ )•,... ,L(g^): although i n a d i f f e r e n t order. This proves that the transpose of a group matrix i s a group matrix. - 7 -3. Uni ts and Unimodular Group Matr ices . Elements u and v i n R s a t i s f y i n g uv = 1 are c a l l e d l e f t and r i g h t un i t s of R & K , r e s p e c t i v e l y . An element which i s both a l e f t and r i g h t un i t of R v 'is c a l l e d a un i t of R^  v . Any square matrix def ined over K. i s s a i d to be unimodular i f i t s determinant i s a un i t t h e i n K.- Given the elements u,v above, Lemma 2 and *de f i n i t i on given, i n eq.. 3 imply;s L(U ) 'L(V ) . = (,L(UV) = L( l ) . = 1^. Therefore,L(u) i s unimodular. Conversely, l e t L(u). be unimodular over K. Then L(u): ex i s t s and by C o r o l l a r y 1 i t i s a group matr ix w i th elements i n F. In f a c t , L(u): ^ has elements i n K since, any element of L(u). ^ i s of the form S(det L(u)): i n K where S i s a cofactor of L(u) and (det L(u):). i s in' K. Thus an element v i n R v ex i s t s such that L(u) = L(v),,-L(uv): = L(u)L(v) = I ; and so, by Theorem 3 uv = ! • n This proves "Theorem 4'A. Theorem k. . An element i s a l e f t un i t of R_, „ i f and on ly i f the — — — • • G,i>-corresponding group matrix i s unimodular. C o r o l l a r y 2. Every l e f t . ( r i g h t ) un i t i s a u n i t . Proof. L(u)L(v). « I - L(v)L(u).. Theorem 5- The set of a l l un i t s of R v under m u l t i p l i c a t i o n forms a — G,iv group isomorphic"to the m u l t i p l i c a t i o n group of a l l unimodular group . . matr ices of G. over K.' - 8 -k. C i r c u l a n t s and Skew C i r c u l a n t s . When G i s a c y c l i c group with an element g of order n, the group ' n - l matrix of G over K r e l a t i v e to the elements.l,g,...,g i s c a l l e d a c i r c u l a n t over K. Let g. = g 1 \ . i = 1,. .. ,n. Then g. g . = i - i - ( j - i ) _ i - j Thus, C - I = C . . and so the elements of the . ^ K, g group matrix are constant along each diagonal p a r a l l e l to the main diagonal. " n Let P be the companion matrix of the polynomial x - 1 . Then P n = I and •n ' n-i+1 i + l 0 . 0 i :o . . O V 0 . . 0. o .. 0 1 1 (9) 1 00. . • o 0 . . 0 0 0 . . . 1 0 •Ml where 1< i < n - 1. I t follows that any c i r c u l a n t C i s a polynomial i n P. Moreover, Theorem 2 i n the s p e c i a l case of c i r c u l a n t s becomes Lemma 3. The matrices of the l e f t and r i g h t regular representations of G-relative to the elements l , g , . . . , g n - 1 are c i r c u l a n t s . Any matrix 'commuting with P i s a circ u l a n t . . I f the f i r s t row of the c i r c u l a n t C i s given by (c^,...,c ) we write .C=. [c n n (10) - 9 -f o r "brevity. Let the conjugate transpose of a matrix A be denoted •x-by A . . Theorem 5. Let C = [6.,...,c ] be a c i r c u l a n t of order n defined 1 > n n over the complex number f i e l d . Let T = , n " 1 / / ' 2 ( p ^ i ~ 1 ^ ^ ~ 1 ^ ) , i < i , j < n where p, i s a p r i m i t i v e nth root of unity. Then T*CT = aiag(e1,"...,eri> ( l l ) where the eigenvalues, e ^ , . . o f C are given by the vector matrix equation' - . ( e ^ . . . ^ ) / = n 1 / 2 T ( c 1 , . . . , c n ) . / . (12> Proof. Since x n - l = (x-l)g(x) where g(x). = 1 + x + ••• + x 1 1 \ g(p k) = 0, i f n does not divide k. Thus T i s unitary; that i s , T T = I . For, . . -x-the (,j,i; term of T T i s given by n n n " 1 I f ( k - l > ( j - l ) p ( k - l > ( i - l ) . = n" 1 I pC^ K i - j ) k=l k=l The RHS equals 1, i f i = j and equals g(p'L J )• = 0, i f i ^ j . Now, since P i s the companion matrix of polynomial x n - l , the eigenvalues of P are the roots of x n - l ; namely, l,p, p 2,...,p n \ Thus i f \. equals the j t h column of T we get 3 -10-P, « n - V ^ p O - l p 2 ( j - l > ; . . . ) p ( n - l ) : ( 5 - l ) . 1 } so that, the j t h column of T i s an eigenvector corresponding to the i-1 * r n - l \ eigenvalue p" of P, j = l , . . . , n . Thus, T PT = d i a g ( l , p , . . . , p ):, n Consequently C = ^ c .P'-' implies * T CT = 1 3-1 J ( n - l ) ( j - l ) v • > P >• Therefore-,- i f - we set n. - Jo. ( i - l ) : ( j - l ) . we get eq. 11 and 12 as.desired; The polynomials over K i n the n X n matrix (13)-P = ' 0 1 0 . . . 0 ; . 'o 05 1 -1 0 . . . . 0 are c a l l e d skew c i r c u l a n t s of degree n over K. Skew c i r c u l a n t s are not group matrices because i n any group matrix the elements In row i are permutations of the elements i n row .one, 1 < i < n. However, the powers -11-o f - P c o n s t i t u t e a m a t r i x r e p r e s e n t a t i o n f o r the c y c l i c group o f order 2n. . S ince P_ i s the companion m a t r i x of the p o l y n o m i a l f (x) = x 1 1 + 1, i t s e igenvalues are p , p 3 , . . . , p 2 n - 1 where p i s . t h e 2nth p r i m i t i v e r o o t 2n of u n i t y . I f h(x) = x - 1 , then h(x) = ( x - l ) ( x + l ) g ( x ) where g(x) = 2 2n—2 k\ 1 + x + ••• + x . Therefore g(p ) = 0 , i f n does not d i v i d e k ; o therwise g(p^) = n . T h e r e f o r e , i f T = n ^^(p^~^ ) , 1 < i , ' j < n , the ( i , j ) element of T$T b e i n g n • n -1 V - ( 2 i - l ) ( k - l ) n ( k - l ) ( 2 j - l ) = ^-1 Y p 2 ( k - l ) ( j - i ) n - l ^ p(2i - l ) (k- l ) p (k- l ) (2 j - l ) = n - i V k=l • K=:. -1 / j - i \ i m p l i e s T T = 1^. Moreover , the product of P_ and A... .the j t h column of T , y i e l d s P j u =n -V?(p 2a- l,- p ?(2J- l) > . . . , p (n- l ) (2J- l) ) . i r J s i n c e p n = -1 i m p l i e s p n ( 2 j - l ) _ (_]_ )^j 1 _ j n Q-^er w o r r i s } the j t h column- of T i s an e i g e n v e c t o r of P_ c o r r e s p o n d i n g t o i t s e igenvalue p2^" 1, j = l ; . . . , n . T h e r e f o r e , T*PT = d i a g ( p , p 3 , . . : , p 2 n _ 1 ) . Theorem 6. I f A = , a j ^ _ ^ 1 S a skew' c i r c u l a n t over K, t)hen . . j = i • . "v T A T = d i a g ( e i , . . . , e n > ... ;. (l-5> where ( e ^ . . . , e n ) 7 = n^^T'Ca^, ...,a )' .' • ' ,(l6). 5. • E x i s t e n c e o f N o n t r i v i a l Unimodular I n t e g r a l C i r c u l a n t s and Skew C i r c u l a n t s . , • ..' v , . ' . ' - • • A -unimodular i n t e g r a l (skew), c i r c u l a n t - i s ' c a l l e d t r i v i a l , i f a l l elements, i n any row a r e zero except f o r a s i n g l e + . l j o t h e r w i s e , i t i s c a l l e d n o n t r i v i a l . We know- t r i v i a l unimodular' (skew); c i r c u l a n t s . always e x i s t : see (eq_. lk) eg,. 9»'..r "It i s shown i n [7] t h a t n o n t r i v i a l unimodular c i r c u l a n t s e x i s t i f n 4 2,3,4,6. . ' •• What about n o n t r i v i a l unimodular i n t e g r a l skew c i r c u l a n t s ? T h i s problem- i s not s e t t l e d . However, i f A were .such a m a t r i x . t h e n , so would be the m a t r i x A A ' . F o r , a d i a g o n a l element I n A A ' i s the sum o f the squares o f the elements i n any row o f A ; and s o , o f f d i a g o n a l elements must occur i n A A ' s i n c e i t i s unimoduiar . T h e r e f o r e , t h e . s o l u t i o n i s i n the answer t o another q u e s t i o n : . F o r w h i c h v a l u e s o f n do n o n t r i v i a l unimodular skew c i r c u l a n t s ' e x i s t when t h e y a r e p o s i t i v e d e f i n i t e ? T h i s q u e s t i o n w i l l .be t a k e n up i n the s e q u e l f o r v a l u e s o f n < 7-6. . A New P r o o f o f a- Theorem on P o s i t i v e D e f i n i t e C i r c u l a n t s and • , Skew C i r c u l a n t s . " • . I n t h i s s e c t i o n G i s always a c y c l i c group of o r d e r - n and a l l n X n -13-matrices are assumed to be i n t e g r a l and unimodular. An n. X n matrix i s c a l l e d a genera l ized permutation matr ix i f i t has exact ly one non zero element, +1 or -1, occur ing i n each row and column. Theorem 7« . I f G i s a c y c l i c group of order n and A 'A i s a unimodular i n t e g r a l group matrix of G, where A i s an n X n matr ix of r a t i o n a l i n teger s , then A = QG where Q i s a genera l ized permutation matr ix and C i s a unimodular group matr ix of G. The proof proceeds by way of Lemmas. For n > 1, l e t [0,1,0,...,6J-denote the matrix i n eq. lk. Lemma k. Let P and A be n X n unimodular matrices of r a t i o n a l integers such that P'A'AP = A 'A . (17): Then a genera l i zed permutation matrix R ex i s t s such that RAPA _ 1 R' = d i a g (P f t l , . . . , P f i s ) : (18): where n = n n + ••• + n and f o r each i = l , . . . , s , . P i s n . X n . 1 s ' ' ' n. 1 1 1 and i s a one rowed submatrix of the form (l). or ( - l ) i f n^ = 1, or i f n. > 1, of the form [0,1,0,... ,0] or [0,1,0,...,0]-1 1 —1 Proof. The matr ix B = APA i s orthogonal s ince (APA~ )'APA = I : i t i s a lso an i n t e g r a l matrix s ince A i s unimodular. Therefore, B = (b. .)-, 1 < i , j < n i s a genera l ized permutation matr ix. -14-Let T be a l i n e a r transformation of an n-dimensional space R whose matrix i s B r e l a t i v e to a bas i s e e ' of R. Then, n l ' ' n . ' where tr i s a su i t ab l e permutation on l , . . . , n and b^ = + 1. Let TT = (j( i>j(2) . . . . j(r 1 >.Kj(r 1+l>j(r 1+2>... J(r 2>).' ' -•••(j(r s_ 1+l>j(r B_ 1+2)>..-j(r s_):> (20). be a decomposition into&s d i s j o i n t c y c l i c products of lengths, say, n n , . . . , n r e spec t i ve l y where r, = 0, n. - r. - r. n , i = 1,...,s and 1' ' s . 0 ' l l .1.-1 r m. n and where j ( l ) . , . . . , j(n): i s a permutation of 1, . . . , n w i th j ( l ) . = 1. This gives us another bas i s of R : to n ' ( f l ' " - > f n * = K ' E 3 ( 2 ) : ' " * ' e j ( n ) ) : ( 2 1 ) : . - S ( e 1 , . . . , e n ) . / (22): where S i s some permutation matr ix. Moreover by eq. 21, 19 and 20 consecut ive ly f o r k = 1,...,s we get when n = 1} and when n, > 1, k K ' T ( t ± } - = bj(i>, j ( i + i ) f i + i - V i < 1 •< r k- i + V -15-with In other words except f o r change i n signs T. permutes ?-y ' +Vfr. 4.o>'">£ c y c l i c a l l y . In matrix notation t h i s amounts k-1 r k - l r k to (23) where H = diag (B^. ..,B g); i s a d i r e c t sum of ^ X n^ . matrices B whose t y p i c a l form i s the following: B = (+l). when n = 1 and .k .' k when n. > 1, 0 b. 0 b 0 V 1 0 where of course the b.'s are equal to + 1. . Let Z = diag (l,b n,b,b ....,b ...b ). i ."~ ' 1' 1 s 7 ' 1 n^/ Then, since b. = + 1 we get -16-ZB,-Z's k 0 1 0 0 K...b 0 1 "k \ ( i f n^ . > 1>. Thus we may construct a matrix W, a d i r e c t sum of s blocks analogous i n form to Z, such that WHW"' = ¥ aiag (B ,...,B ).W' 1 s = diag (P ,...,P ) v n, n 1 s (2k). where P are the matrices defined i n eq. 1. But n. l ( T ( f x >,..., T ( f n » ' = S(T( e ; L >,..., T ( e n » / « SB( e i,.../e n): ' = SBS ' ^ , . . . , ^ ) . ' as a r e s u l t of eq.'s 22, 19, and 22 r e s p e c t i v e l y . Comparing t h i s to eq. 23 we get H = SBS'. Therefore, by eq. 2k, WSBS'W' = diag (P ,...,P > n l n s and the lemma i s proved since R = FWS i s a generalized permutation matrix and B =. APA - 1. . v I f , i n Lemma k: l);. the matrix A s a t i s f i e s the hypothesis i n Theorem 7 2) P i s a matrix of the l e f t regular matrix representation of G where n i s ' the order of P; i . e . P n = I j ' 3)- the r i g h t hand sides of eq. 18 equals [ 0 , 1 , 0 , . . . , 0 ] , then Theorem 7 i s true. For, l e t L(h):, h i n G, be the l e f t - i r -regular matrix representation which define the group matrix A'A r e l a t i v e , say, to the ordering g^,...,g of the elements of G; l e t L (h), h i n G he the l e f t regular matrie representation' r e l a t i v e n - l to 1, g, . ,.,g where g i n G i s ' of order n. Then, L o ( g ) = [0,1,0,...,0]^ and L(h> = SL o(h)S', (25). h i n G, where S i s a permutation? matrix, such that (g^,...,g ) ' = S(l,g,...,g ):'. But/conditions 2). and 3): imply RAL(g)A R" = [0,1,0,...,0]; whence, by eq.'s 25 SRAL(g)A _ 1R^S^ • L(g). and so, SRA commutes with L(h)j h i n G. From Theorem .2, observing that the l e f t and r i g h t regular matrix.representations are i d e n t i c a l when G i s abelian, we i n f e r that SRA i s group matrix C of G r e l a t i v e to g^,...,g . Put Q = (SR).'. This proves Theorem 7 given assumptions l ) , 2) and 3) above which are j u s t i f i e d as the next lemma.,,shows. Lemma 5. Let G,A,n be d e f i n e d r a s i n Theorem 7- Let' A'A be a group matrix, a l i n e a r combination of the l e f t regular representation matrices L(h), h i n G, of G. Then a generalized permutation matrix R exists such that RAi(g)A~"*"RT = [0,1,0,...,0] (26): where g i n G i s of order n. Proof. Since G i s abelian Lemma 1 and 2 imply A'A and L(g): commute. Therefore, i n p a r t i c u l a r . P'A'AP = A'A, where P = L(g). This permits us to use eq. 18 i n Lemma \; that i s , condition l ) i s s a t i s f i e d . Also -18-note'that P n - I and P r 4 I i f r .< n. • • n n To show condition 3). holds, l e t D = RA i n eq. 18. Then by taking of powers from 1 to n and .noting that P 1 = L ( g 1 ) , i = 1, ...,n and sums n—1 L ( l ) + L(g) + ••• + L(g ); =• [1,1,.. .,1] we 1-get the s i m i l a r i t y r e l a t i o n D [ l , . . . , l ] n D _ 1 = diag (Blt...,B•)• (27) n where B. = ) P J , i = 1, ...,s. Using eq.. 18, again. P n = I implies 1 i_i n. n >1 1 P = I , so that, f o r some integer m., n = n.m.. In f a c t , when P i s n. n.. ' 1 ' 1 1 ' n. 1 1 1 a skew c i r c u l a n t , 2n^ i s the smallest p o s i t i v e integer such that n • 2n • . 1 P 2 n i = I so that, i f m. = 2q., then B. = ) P J = q_. ) P ^ = 0. n. n. ' 1 ^x' x L n. x^. L n. x x . n x . ., x J=l J=l .. n n± When P i s a c i r c u l a n t , B. = ) P .J = m. ) P ^ = m. [1,... ,1] n. ' x L n. x L n. x ' ' n x . T x . n x i Thus rank B. i s 1 or 0 according as P i s a c i r c u l a n t or a skew c i r c u l a n t . x n. x However, the rank of the l e f t side of eq. .27 i s 1 so that on the r i g h t side one and only one non zero component e x i s t s ; say, m [ l , . . . , l ] k "k a r i s i n g from a c i r c u l a n t P . Thus n divides each element on the r i g h t V\ k side. But D i s a unimodular matrix of r a t i o n a l integers and so m^ divides each.element of D ~^ diag (B-^, ... ,B g)D, hence m^ . divides 1. Therefore m^ . = 1, - n and eq. 27 implies RAPA-^" = P = [0,1,0,... ,0] . "k -19-C o r o l l a r y 2. I f A i s a unimodular i n t e g r a l matrix and A'A i s a c i r c u l a n t , then A = QC (28). where Q i s a generalized permutation matrix and C i s a unimodular i n t e g r a l c i r c u l a n t . Theorem 8. I f A i s a unimodular i n t e g r a l matrix and A'A i s a skew c i r c u l a n t , then A = QC where Q i s a generalized permutation matrix and C i s a unimodular i n t e g r a l skew c i r c u l a n t . Proof. Let P = [0,1,0,...,0]- . Then, since A (A i s by d e f i n i t i o n a l i n e a r combination of powers of P, P'A'AP = A'A. . Thus, eq. 18 in.Lemma k can be used. We s h a l l show s = 1 and therefore P = P and t h i s would n 1 e s t a b l i s h Theorem 8 since a matrix which commutes with a nonderogatory matrix i s a polynomial i n i t . Observe that, i f P i s a skew c i r c u l a n t , then by adding the f i r s t n column of the matrix sum / -1 I - ' . . . 1 ^ P + P + n. n'. i i + P n i = n. i -1 1 .. -11 to every other column we get a t r i a n g u l a r matrix with -1 as the f i r s t diagonal element and -2 f o r the remaining diagonal elements. . Thus n i det Y P n J = ( - l ) n i 2 n i (29> J=l -20-Also, ^ P^ ^ = 0, so that i f m i s odd 3=1 X mn-i rijL . 1 1 . , X Returning to eq.. 18, we see that P n = - I implies P n = -I , n n. n. x x i = 1,..., s so that each P i s a skew c i r c u l a n t and n equals an i odd multiple of n.. Therefore, by eq. 18, again x n d e t ^ P 0 = det Y axag (P ,...,P n >< J=l j = l n s „ = n det ) P 3 . , L> n. 1=1 . -> x 3=X s nl = n det Y P 3 . , L n. i = l . -, •, i n_n-s = (-1).^; where the l a s t two equations follow d i r e c t l y from eq.'s 30 and 29 r e s p e c t i v e l y . But eq. 29 also implies above that the l e f t hand side equals ( - l ) n 2 n \ Therefore s = 1 and Theorem 8 i s proved. 7. New Results on Group Matrices and Symmetric C i r c u l a n t s . In what follows, the l e t t e r s i , u, p, d, s stand f o r i n t e g r a l , unimodular, p o s i t i v e , d e f i n i t e , symmetric, r e s p e c t i v e l y . With t h i s n o t a t i o n a l convention Theorem 8(Theorem 7)states that an pdiu (skew) c i r c u l a n t of the form A'A where A i s i u equals C'C where C i s an i u -21-» (skew) c i r c u l a n t . In view of t h i s i t would be i n t e r e s t i n g to note f o r what values of n are pdiu (skew) c i r c u l a n t s of the form C'C where C i s an i u (skew); c i r c u l a n t . So f a r , very l i t t l e i s known about t h i s f o r c i r c u l a n t s of degree n > 13. In an unpublished work E.C.Dadei has shown i t to be true f o r ci r c u l a n t s of prime order l e s s than 100 > with one exception; i n [6] i t i s shown to be f a l s e f o r n = 5 where equations 11 and 12 are used to demonstrate that the pdiu c i r c u l a n t [ 2 , l , 0 , - l , - l , - l , 0 , l ] g i s not of the form C'C where C i s an i u c i r c u l a n t . A r e s u l t of Minkowski i n [5] s e t t l e s the question, i n general, f o r n < 7; that i s , i f A i s a pdiu n X n matrix, then A = B'B .where B i s an i u n X n matrix, n < 7. A study i n [7] on the uniqueness of the normal, basis f o r normal c y c l i c f i e l d s produced the r e s u l t that a l l ui. c i r c u l a n t s are t r i v i a l f o r n = 2,3,4,6. This of course i s consistent with Minkowski's r e s u l t . Also f o r n = 5, an incomplete proof appears i n [11] with corrections i n [ l ] . Recently i n a paper p r e s e n t l y . i n press [12] R.C.Thompson solved the question f o r a l l values of n up to 13 i n c l u s i v e by considering a • more general problem which we s h a l l define i n section 8. As f o r skew c i r c u l a n t s nothing has been written'on them. In f a c t I am indebted to Dr. R.C.Thompson f o r his conjectures on skew c i r c u l a n t s , e s p e c i a l l y f o r proposing Theorem 8, the p a r a l l e l to C o r o l l a r y 2, and the question of the existence of n o n t r i v i a l pdiu' skew c i r c u l a n t s . We s h a l l discuss several cases i n the next section. Instead, we consider.whether every n o n t r i v i a l « i c i r c u l a n t i s of the form P^C where 1 < k < n, P = [ 0 , 1 , 0 , . . . , 0 ] and C i s a pdiu -22-c i r c u l a n t ; and a d d i t i o n a l l y , i f P^C i s symmetric, then either k = 1 or n = 2k. This i s only a conjecture on my part. However, i n consonance with i t the following facts are obtained. Let G be a group of order n. Let (c^,...,c n) be the f i r s t row of a group matrix C of G defined over the r i n g of r a t i o n a l integers. Then, without ambiguity we may write C = [c^,... J ^ - I Q * Lemma 6. Let C = [c_....,c ] be a symmetric r e a l nonsingular group — — 1 n G matrix with p r i n c i p a l idempotent decomposition C = s ^ + . v . + s t E t (31) and l e t e. denote the row sum of the f i r s t row of E.. Then, f o r i i . 1) . E^ i s a symmetric r e a l group matrix; 2) . the diagonal element of E^ i s a p o s i t i v e r a t i o n a l number equal • to.r.n where r . i s the rank of E., the number of eigenvalues of C equal to s^; . •. 3) .! e. = e. and e.e. « 0, i f i i j : x x x j 1 V) i f eigenvalue s n = c n + ••• + c , then e. = 0 for j i 1 and .e, = 1. ' 1 . 1 n' a T 1 Note: c^+*«'+c' i s always an•• eigenvalue of C. Proof. (E'.) = E'., i = 1,...,n and E'.E'. = 0, i =}= j . Hence, since C' = C and the p r i n c i p a l idempotent decomposition of C i s unique, eq.31 implies E'^ = E^. I t i s known, e.g. see [8], that f o r p r i n c i p a l idempotent decompositions a matrix which commutes with C commutes with every E^. Therefore, since by d e f i n i t i o n , C i s a l e f t regular representation, Theorem 1 implies a l l matrices i n the r i g h t regular representations commute with the E and so, by Theorem 2, the E^ are group matrices of. G. This proves l ) ; since the E^ are r e a l by d e f i n i t i o n of the decomposition. The p r i n c i p a l idempotent decomposition requires that E^ are s i m i l a r to a diagonal matrix of I's and p o s s i b l y O's. By taking the trace of E^ and the corresponding diagonal matrix and taking cognizance of l ) , that i s , the main diagonal of E^ i s constant, 2): follows immediately.-. Let X = COl ( l , . . . be an n-tuple column vector a l l of whose elements equal 1. Then, since the E^'s are idempotents, 3): follows d i r e c t l y from l ) and the f a c t that 2 E. X = E:X and E.E.x = 0. (Note: For any i , . E.x = (e n , ...,en ).' = e nx, i . i i . ,j > J ' i v 1' ' 1' 1 ' hence E. x = E. (e'.x).-= • e. (E.x) "'= e.e.x, whence e. = e. . These r e s u l t s 1 X X l v 1 . 1 1 ' 1 1 are a consequence:of the f a c t that the row sum of any group matrix i s independent of the row.). From eq. 31, Cx = S..E x + ••• + s E x so, c.+ • • • + c = s e»+«v+s,e . By 3); i t i s possible f o r only one of the e^'s to be non zero, say e^, whence the preceding equation reduces to •; c, •+•••+. c =• s n e n . 1 •" • , n 11. But, since C i s nonsingular the l e f t side i s non zero, so, 4 0' Since 2 •e^ i s a row sum of the r e a l matrix E^, e = >•0, and t h i s implies = 1. -2k-Therefore c, + * *• + c = s n . 1 • - n.. . 1 . '• The next r e s u l t i s an i n t e g r a l c i r c u l a n t analogue of the polar f a c t o r i z a t i o n theorem. Theorem 9. I f A i s an n X n nonsingular r e a l group matrix then there are unique r e a l matrices H and U such that A = UH where IS H i s a pd group matrix and U i s an orthogonal group matrix. Proof. Let C = A'A . be the group matrix i n eq. 31 and l e t H = y"s 1E + ••• + / s E where we note that the eigenvalues s. of C are p o s i t i v e since C i s pd. Therefore, by l ) . i n Lemma 6, H' i s a r e a l p o s i t i v e d e f i n i t e ^ group matrix. Moreover, A'A = K 2 (33) where H i s the•only p o s i t i v e d e f i n i t e matrix f o r which t h i s i s true by v i r t u e of the uniqueness of equation 31• In [8] i t i s shown that f o r nonsingular A there are unique -real matrices U and H q such that U i s orthogonal and H q i s p o s i t i v e d e f i n i t e with A = ^-Q- But t h i s implies A'A = H q 2 = H 2 - which by uniqueness of H i n eq. 33, i n turn implies, H = H QJ and therefore U i s a group matrix by C o r o l l a r y 1 and m u l t i p l i c a t i v e closure. Following t h i s , the terms AyH,U i n C o r o l l a r i e s 4,5,6 are assumed to be the group matrices i n Theorem.9. C o r o l l a r y 3. I f det A = + 1, then det H = 1 and det U = det A. (A,H,U are real)-2 Proof. By eq. 33 (det H); = 1, hence det H = + 1, so, i t equals + 1 since H i s p o s i t i v e d e f i n i t e . Therefore det UH = det U = det A. -25-C o r o l l a r y k. A i s normal i f f A = HU = UH. Proof. Consider the commutativity property with regard to the idempotent decomposition and the eq u a l i t y of ITTJ and UH2.. Co r o l l a r y 5. .Tf i n Theorem 9, A = [ L i s an i n t e g r a l unimodular 1 n li group matrix and U = [u^,...,u^\G and H = [ l ^ , . . . ,h ] G , then h = h, + • • • + h >. 1 and u n + • • • + u •= a, + • • • + a = + 1. J- n 1 n l . . n — Proof. Let u = u n + ••• + u and a = a. + + a . The equation Ax = UHx 1 n 1 n • where x i s an n-tuple column vector a l l of whose elements equal 1, implies a = uh. Since A i s unimodular and i n t e g r a l i t s row sum equals + 1; -1 -1 2 f o r , xAA ;.x' = naa = x I n x / = n. Since U i s orthogonal, u = 1, because 2 nu = x'U'Ux = x ' l ^ x = n. Consequently, h « + 1 which perforce equals +1 since +lHSis p o s i t i v e d e f i n i t e . ' . . Theorem 10.. I f A i s a unimodular i n t e g r a l circulant. then there i s an integer s such that P A i s symmetric where P = [0,1,0,...,0]^. Proof. . Let K be the n X n matrix : K = toy WO j -1 Then KK = I and KPK = P'. Hence KA'AK = (A'A)' = A'A so that AKA' ..... n.,. i s an i n t e g r a l orthogonal matrix,hence a generalized permutation matrix. In f a c t , i f Q = AKA - 1 then KQ, = KAKA"1 = A 'A - 1 so- that KQ'is a c i r c u l a n t , and being t r i v i a l implies there is. an integer k' such that 1 < k < n and KQ = + P K . Thus' by eq.. 3h , .  A ' = ± P K A . (35) But since the row sum of A and A 7 are equal, • A' a P K A . (36): Suppose n i s odd. Let r be an integer such that 2r = 2n-k or 2r = n-k according as k i s even or odd. Let•s be the nonnegative integer ' s = 'n-r- , (37) This means r + k = n + s or r > k = s according as k i s even or odd. Therefore by eq. 36 P R A ' = P r t KA = P G A , ' and so by eq. 37 ( P S A ) / « A ' P N ~ S = A ' P R = P R A ' = P S A , '.. S which proves that' P A i s symmetric. ' Mow suppose n i s even. Since the trace of AKA = K P K i s zero on the l e f t , i t follows that the number of elements i n the n o n t r i v i a l diagonal(s): of P k i s zero, or what i s the same k i s even. Hence, l e t t i n g r = (n-k)/2 and. s.'== n-r, we get from eq. 36 . . . ' P V = P R + K A = P R + N " 2 R A = P S A , whence ( P S A ) ' = A ' P N " S = A.'P1" s and so, P A i s symmetric. . C o r o l l a r y 6. I f A i s a unimodular i n t e g r a l c i r c u l a n t then there i s an k ' integer k such that A' = P A (where k i s even i f n i s even). Theorem 11. Let A he a unimodular- i n t e g r a l c i r c u l a n t . Then the eigenvalues of the symmetric matrix KA are the square roots, of the eigenvalues of the p o s i t i v e d e f i n i t e c i r c u l a n t A'A/. Proof. Observe that KP 1.is obviously symmetric f o r each i = 1,...,n. Hence KA i s symmetric. Then consider the p r i n c i p a l idempotent decomposition of KA and A'A = (KA).'KA; and the proof follows. 8. P o s i t i v e D e f i n i t e Skew Ci r c u l a n t s . In t h i s section B^ always denotes an n X n symmetric unimodular i n t e g r a l skew c i r c u l a n t . • - By d e f i n i t i o n of Bn>' we may write, f o r k > 1, B = [b .h. ,b n, ... ,b_ ,-h, ,-h, -b., ]~ (38): n • cr 1' 2 ' ' k' k' k-1' ' I n w ' i f n = 2k + 1, and ;B n B [ v v 2 ' " ' ' V o ' " V - W i " ' - ^ ; (39). i f n = 2k + 2. Then, by eq. l 6 i f B^ = A, we get f o r the I t h eigenvalue of B n -e. . " a.„(21-l)(j-1> which, by sub s t i t u t i o n s ' o f the a.'s with the b's i n eq. 38 or 39 , y i e l d s f o r any n > 3, . -28-'L--o0*i v ( 2 l~ i ) j-x v(2i~ iMn"3) (u>-Lemma 7. I f i s the symmetric n X n skew c i r c u l a n t given by eq.'s 38 or 39; where n = 2k+l or ,2k +2 then i t ' s eigenvalues are given by j=i i = 1,... ,n and ' e. = e . ,, (43). 1 n-i+1 ^ v y f o r i = 1,2,...,k+l. Proof. Eq. 42, of course, follows d i r e c t l y from eq. 4 l . • Eq. 43 follows from eq. 39 a ^ d the f a c t that the eigenvalues of a symmetric r e a l matrix are a l l r e a l . For, p ( n. ^ + ^ ) ' _ p-^  ^ i a ^ ^ su b s t i t u t i n g n - i + 1 f o r i i n eq. 40 we get, _ Y (l-2i).( j - l ) . £ n - i + l ~ L V j= l and so, by comparing t h i s with eq. 40, e i = e j _ ~ e n i + l ^ 0 r = I>«".?k+1.> whether n i s odd or even. This evidently implies Lemma 8. For n > 3 2 det B = ( e ) e(n). * + 1 n 1 2 . k — -29-where. f :(n) = 2 "k+1 e,_., , i f n = 2k+2 V ek+l - ' i f n = 2 k + 1 Given a square matrix A we denote i t s trace by tr(A):. From eq. 15 where A-B and eq. h-3 we have n Lemma 9. For n > 3 t r ( B n ) = nb Q = 2 £ e. + 6(n> '1=1 ( where 6(n) = • 2 e k + l ' i f n = 2k + 2 e k + 1 , ; I f n •= 2k + 1 Lemma 10. I f n.= 2k+l and A n = [a^,..-^a^]" i s a unimodular i n t e g r a l skew c i r c u l a n t with eigenvalues defined as' i n eq. 15, then n ek+l ~ a 0=2 +1. Proof. By eq. ±6, keeping i n mind that p n = -1, we get 'k+1 ~ a l + L a j p > 2 •n = a 1 + 1 • i=2 Therefore, ek+i i s a r a t i o n a l integer; s i m i l a r l y w i t h > k+1 eigenvalue of the inverse of A , which as with A i s a unimodular n' n i n t e g r a l skew c i r c u l a n t . Therefore, since ev divides 1 r.: .x, e, . = + 1 ' k+1 ' k+1 — as desired. •' 1 C o r o l l a r y 7« if- -h - 2k+l, B^ i s as i n eq.. 38 then e. . = h + 2 Y ( - l ) j b = + 1. k+1 O LJ ^ ' j — Proof. Since B^ hy d e f i n i t i o n i s symmetric the c o r o l l a r y follows d i r e c t l y from Lemma 10. We now proceed to show f o r which values of n i s B t r i v i a l or n o n t r i v i a l . n Obviously i t i s t r i v i a l f o r n = 1,2. Case 3: B^ . = I . Proof. Let = [a,b,-b]^ . By Lemma 7 and 3 . 3 e± = a + b (p-p 2) e2 = a - 2b = 1 2 ' and so, since -1 + p - p = 0, = a + b, whence, by Lemma 8 -31-det = e 1 2 e 2 = (a + b ) 2 ( a - 2b > = 1. Therefore, a = 1 + 2b implies a + b = l+3"b = + l ; which holds only i f b = 0 and a = + 1. Since B i s pd, a =. 1. Case k. B^ - [a,b,0,-b]^ i s n o n t r i v i a l f o r i n t e g r a l solutions of V 2 p _ the equation a - 2b = 1 when b 4 0, a > 1. For example' [3,2,0,-2]!". Proof. By Lemma 7> eq.. k-2, e^^ = a + b (p-p5) .e2 = a + b (p5-p9)- = a + b (p5-p> ^ 2 and so, since « (p-p ): = 2, by Lemma. 2, det B^ . = ( e ^ ) 2 = ( a 2 - 2b 2) 2 = 1 • 2 2 we have a - 2b = + 1 which equals + 1 since 3-^2 ^ ®' Conversely i f . a and-b are solutions such that a > 0, b 4 0. Then 2 2 a > 2b implies a > + /"2b so that a 4-/"2b > 0 and hence e^'e2 * > ^ * Therefore B^ i s a pdsiu skew c i r c u l a n t and n o n t r i v i a l . Case 5' B5 = t a,b,c,-c,-b]^ i s n o n t r i v i a l i f f a,b,c are solutions to a 2 - It-bc = 1. (kk): (b-c):(l+b-c) = be (U5) where a > 1. For example [3,2,1,-1,-2]^.. ' -32-Proof. By Lemma 7 and C o r o l l a r y 7 e 1 « a + bX - cX 2 e g = a + bX 2 - cX-j^  (46)-' e,.= a - 2b + 2c = 1 3 . 4 3 2 where X-^  = p-p and Xg = p -p and p i s the 10th p r i m i t i v e root 2 3 4 of unity. Using the f a c t that - 1 + p - p + p - p =0 a st r a i g h t forward computation w i l l show that ej_e2 ^ S a n i n"k e6 e r a n c ^ hence from Lemma 8 = indeed, 2 2 2 e l G 2 ** a ~ C + ab-ac-3bc = 1. But 4 £ l e 2 + e 2 » 5a2-20bc = 5 (47) - e ^ E g = 5'(b +c - ab+ac - bc):.= 0. The l a t t e r equation gives (b-c)(b-c-a) = -be . which reduces to (b-c>(l+b-c> = be by eq. 46. . Eq. 47 implies that i f B<_ i s n o n t r i v i a l then a > 1. Conversely i f a,b,c are i n t e g r a l solutions to eq.'s 44 and 45 such . that a > 1, then B,_ i s a : n o n t r i v i a l pd unimodular skew c i r c u l a n t . For, 2 2 . 4 e n e „ + e, = 5a - 20bc 1 2 3 which by equation 44 equals 5' "'Thus'solving for the integer e-j_e2 w e get , £j.e2 = • H e n c e C o r o l l a r y 7,- = + 1 and so by Lemma 8 B,- i s -33-unimodular; moreover, since 5 divides the difference e-.-en e 0 = e.,-1, 3 1 2 3 ' =1; which i s eq. 46. To show that a l l the eigenvalues of &g a r e - p o s i t i v e we note that X-^  and Xg are the-roots of the polynomial ~2 X(x): = x -x-1 which means . So that _ , b-e h+c „. e-i - a + o _ o /5 , h-c " h+c -£ 9 — a + p + o v 5 • • We must show ' a + ~ ^ > + j^p /"5 • ?y eq. 44, since a > 1, be > 0 and so by eq.'' 45 h-c > 0. Hence we only need to show a + < /5 i s f a l s e when b,c > 0. . By squaring both.sides and transposing terms we get ' " ; 2 2 a + a(b-c) < (b-c) - be . But the r i g h t hand side i s negative according to eq. 45. This i s a contradiction. Hence £-^,£g > 0 and so, B,_ i s pd. Case 6. Bg = [a,b,c,0,-c,-b]g i s n o n t r i v i a l i f f a,b,c are i n t e g r a l solutions of the equations a-2c = 1 . (48> ( ^ ) 2 - 3 h 2 = l --(49> where a > 1. For example'[5,4,2,0,-2,-4]g Proof. We have by Lemma 7 5 2 H-e = a+b(p-p:?). + c(p -p ): e 2 = a-2c *5 2 ^4-e 5 = a-bCp-p?): + c(p -p );. But by Lemma-9 and eg. '-4-3 i n Lemma 7 . t r ( B 6 ) = 6a = 4(a+c(p2-p1+):> + 2(a-2c) implies / 2 > c = c(p -p ) . To show c 4'°- L e t Hg = [ l , 0 , - l , 0 , l , O j g . Then •" .B6H6B6_1= ^B6H6>B6_1. • = (a-2c>H 6B 6 _ 1 • . Since Bg i s an i n t e g r a l matrix a"~2c divides one, and so., e 2 = a-2c = 1 2 4 since Bg i s pd. Therefore, i f c = 0, Bg = Ig. I t follows that p -p and so, By Lemma '8, e1 e . = 1. • 1 3 2 2 = (a+c): - 3h . {I Conversely, suppose a,b,c s a t i s f y equations 48 and 4-9 such that a > 1. Then a-2c = 1-implies c ^  0 so that -35-0 -0 +1 = 0. (51): Therefore,, E T _ E ^ equals one by eq. 49 so that by Lemma 8, Bg i s unimodular. -i+/3 A s o l u t i o n to eq.- 51 i s p = • y which i s a 12th p r i m i t i v e root 5 i-/3 of unity. Thus p = *^ implies = a+/3b + c = a-/3h + c. 3 By eq. 48, ,c > 0 and hence by eq. 50 i t follows that a + c > + /3b so that £-j_>£3 ^  0. This proves that Bg i s pd. Case 7. For A^ = [a,b,c,d,-d,-c,-d]~ thig drily f a c t s know.ficar*: ^ ^ = a + b T a - c ? 2 + d? e 2 = a + bTl^ - cTl 1 + dT|2 e = a + bTl 2 - c71 +• dT^ 6 ' 5 2 • 3 4 ' ' ' where "0^  = P~P > ^ 2 = = ^ ''^3 = ^ ' a r e s b u t t o n s of the equation 3 - 2 x^ - -x - 2x + 1 and Tl^ - T|2 + T) = 1 when p i s the 14th p r i m i t i v e root of unity. s o l u t i o n of the cubic equation i s " i + (| /7 ) cos Q c o s " 1 ( ^ ) ) . -36-Result: ¥e have shown that [3,2 ,0,-2]^, [3,2,1,-1,-2] , and [5 ,4,2 ,0,-2 ,-4]^ are p o s i t i v e d e f i n i t e 'unimodular skew c i r c u l a n t s . The diagonal elements 3 ,3 ,5 i n these matrices are minimal f o r t h i s c l a s s of n o n t r i v i a l matrices. Hence i t i s impossible f o r these' matrices to be of the form C'C where C i s a n o n t r i v i a l unimodular p o s i t i v e d e f i n i t e i n t e g r a l skew c i r c u l a n t since the diagonal elements of C'C would otherwise exceed 3 ,3 ,5-9. Appendix. Let C = ( m n ) where'm and n are integers. Then m + n and m - n v n m y are square integers i f and only i f there i s a unique matrix A of r a t i o n a l integers of the form '' J such that CV= A'A. This comes as d i r e c t consequence o f the f a c t : The r e l a t i v e l y prime solutions o f the equation 2 2 2 * 2 2 . 2 -2 x + y . = z . with y . even are x = r - s , y = 2rs, z = r + s , where r > s > .0, ( r , s ) = ! . . ' [ ' The above p r o p o s i t i o n can be v i o l a t e d , i f the conditions on m and n are relaxed. For example r .65 60 V _Y8E:J\ S4>?\ V60 65 J " V77V V i ' The case f o r 2 X 2 skew c i r c u l a n t s turns out to,be t r i v i a l . -37-BIBLIO GRAPHY 1. . E.C.Dade and 0. Taussky, Some hew r e s u l t s connected with matrices of r a t i o n a l integers, Proc. Symposium i n Pure Math., of the Am.Math.Soc, 8(1965)., 78-88. 2. G. Higman, The units of group r i n g s , Proc.London Math.Soc, 46 (1940)., 231-248. 3. D. H i l b e r t , Theorie des corps de nombres algebriques,Paris,(1913),l64. 4. M. Kneser, Klassenzahlen d e f i n i t e s Quadratischer Formen, Archiv der Mathematik, VIII (1957), 241-250. 5. H. Minkowski, Grundlagen fur eine theorie der quadratischen Formen mit ganzzahliget K o e f f i z i e n t e n , Gesammelte Ahhandlungen 1 (l91l),3-144.. 6. M. Newman and 0. Taussky, Classes of P o s i t i v e D e f i n i t e Unimodular  C i r c u l a n t s , 9 (1956);, 71-73. 7. M. Newman and 0. Taussky, On a gener a l i z a t i o n of the normal basis i n ahelian algebraic number f i e l d s , Comm. on Pure and Applied Math. 9 (1956>, 85-91. 8. S. P e r i l s , The Theory of Matrices, Cambridge 1952. 9. 0'. Taussky, Matrices of r a t i o n a l integers, B u l l . Amer .Math. Soc. ,66 (I960), 327-345. • . 10. 0. Taussky, Normal matrices i n some problems i n algebraic number theory, Proc. Intern. Congress, Amsterdam, 1954. 11. 0. Taussky, Unimodular i n t e g r a l c i r c u l a n t s , Math..Z., 63 (1955).,286-289. 12. R.C.Thompson, Classes of D e f i n i t e Group Matrices, Pac.Journ.of Math. 13. R.C.Thompson, Normal Matrices, and the Normal Basis i n Abelian Number F i e l d s , 12 (1962)1, 1115-1124. 14. R.C.Thompson, Unimodular Group Matrices with Rational Integers as Elements, 14(1964)., 719-726. • . . 

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