UBC Theses and Dissertations

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UBC Theses and Dissertations

Group matrices Iwata, William Takashi 1965

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GROUP MATRICES • by W i l l i a m T. Iwata  A THESIS' SUBMITTED IN PARTIAL FULFILMENT OF • THE RI1QUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n t h e Department of Mathematics.  We ..accept t h i s t h e s i s as conforming t o t h e required standard from\andidates f o r the degree o f MASTER OF ARTS  THE UNIVERSITY OF BRITISH COLUMBIA September, 19^5  In p r e s e n t i n g the  this  thesis  fulfilment of  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  British  Columbia,  I agree that  the Library  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 . mission  f o r extensive  p u r p o s e s may his  in partial  representatives„  cation  of this  thesis  w i t h o u t my w r i t t e n  make  i t freely  I f u r t h e r agree that  copying o f t h i s  be g r a n t e d  shall  thesis  per-  f o r scholarly  by t h e Head o f my D e p a r t m e n t o r by  I t i s understood for financial  permission.  Department o f The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Columbia  that  gain  copying or p u b l i -  shall  n o t be a l l o w e d  ii  ABSTRACT  A new p r o o f i s  g i v e n o f Newman and T a u s s k y ' s r e s u l t :  if A  i s a unimodular i n t e g r a l n X n m a t r i x such t h a t A ' A i s a c i r c u l a n t , then A = QC where Q i s a g e n e r a l i z e d p e r m u t a t i o n m a t r i x 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 p r o v e d f o r unimodular i n t e g r a l skew c i r c u l a n t s .  C e r t a i n a d d i t i o n a l new r e s u l t s o f which' a r e : l )  a r e o b t a i n e d , the most i n t e r e s t i n g  Given any n o n s i n g u l a r  group m a t r i x A t h e r e  unique r e a l group m a t r i c e s U and H s u c h t h a t U i s p o s i t i v e d e f i n i t e and  A = UH;  2): I f  o r t h o g o n a l and H i s  A i s any unimodular  integral  k c i r c u l a n t , t h e n i n t e g e r s k and s e x i s t such t h a t A ' = P A is  symmetric, where P i s  exist  s and P A  the companion m a t r i x o f t h e p o l y n o m i a l x - l . n  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 f o r v a l u e s o f n < 6 :  t h e y a r e shown t o  be t r i v i a l f o r n = 1 , 2 , 3 and a r e e x p l i c i t l y d e s c r i b e d f o r n =  I  hereby c e r t i f y t h a t t h i s  abstract  is  satisfactory.  k,5,6.  iii  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 *  5.  Existence  o f N o n t r i v i a l Unimodular  '  8  Integral  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 Skew•Circulants  27  9. 10.  . Appendix Bibliography  36 •  .  37  ACKWOm^IXJEMENTS  I t i s a p l e a s u r e t o acknowledge my i n d e b t e d n e s s t o my s u p e r v i s o r Dr. R. C. Thompson f o r s u g g e s t i n g t h e s t u d y o f skew c i r c u l a n t s and o f c i r c u l a n t s i n g e n e r a l and f o r h i s encouragement and a d v i c e 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 o f order,, n  w i t h 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 c o n t a i n i n g K as a subring. elements  Let  R(G,F). denote a v e c t o r space over F which admits  g^,...,g  Ef  of  as a b a s i s a n d . i n w h i c h , a d d i t i o n a l l y ,  n p r o d u c t s a r e d e f i n e d by  n  •)' a . g . x x  L  n  ) b.g. J  L  a  i=l a r e i n F and g.  &  = ) J'  j=l  . ='g.g..  It  where  J X,J  a.,b. x'  j  Let  R„ „ denote the s e t o f  all  G, A.  ^  a  j _ i "*" ^(^•^)" where the s c a l a r s g  n  'i=l  a r e i n K.  -  1^ and 1^. be the i d e n t i t i e s o f G and K r e s p e c t i v e l y ; and l e t  1 = 1 ^ . ' 1^  denote the i d e n t i t y o f R^ ^ and o f G and o f K as w e l l except  under anomalous Since  i  L  .  &  i,j=l  •n  Let  a.b.g.  i s . " w e l l known t h a t t h e s e o p e r a t i o n s make  R(G,F): i n t o an a s s o c i a t i v e a l g e b r a .  elements o f the form  the  situations.  g^,...,g^ i s  It  is  clear that  a b a s i s f o r R(G,F):,  determined by the s c a l a r s  i n K.  R  v  i s a subring of  every element o f R^ ^ i s  R(G,F):.  uniquely  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.  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 . A ^matrix r e p r e s e n t a t i o n o f degree n o f G i s a homomorphism o f G i n t o  the f u l l l i n e a r group  L ^ ( F ) : , the n X n h o n s i n g u l a r m a t r i c e s over' F .  We i n t r o d u c e the l e f t r e g u l a r r e p r e s e n t a t i o n of G as f o l l o w s .  If  g e G, then n g S  i  =  I j=l  a  i  jCs)Sj  >  1  <  1  <  n  (l) •  - 2-  where each  a. .(g)  1  0 or 1.  S  Let  L(g),' -  the prime d e n o t i n g t r a n s p o s e . h(hg)  s L(h)L(g)., f o r h,g  m u l t i p l y eq.. 1  by h to  (a..(g).) ,  (2):  L(g). i s a p e r m u t a t i o n m a t r i x .  Moreover,  i n G, as the f o l l o w i n g computation shows.  Pre-  get n  h(gg  )• =  )  a  (g).hg  n  n  "  Z C I iJ<«> jk^ > ) k=I j=l = (hg)^ n ;  '  a  -I  a  &  ±^K  h  S  k .•  ••  k=l  n Thus  a (hg> i k  = ^  a(g)-a ( h ) , hence k  L(hg).' = L ( g ) / L ( h ) . ' , and so  j=l L(hg> = L ( l : ) L ( g ) . If i  L(g> = I ,  G is  i  =  a  (g). = 0, i f  ^ hence g i s  i 4 j,  and  the i d e n t i t y .  a _ ( g ) . = 1, i f Thus  i s o m o r p h i c to the group o f p e r m u t a t i o n m a t r i c e s L(g):,  g i n G, where L(g). i s elements o f G.  .-  then  n  = j ; and s o , g g  Lemma 1.  •  d e f i n e d r e l a t i v e t o the o r d e r i n g g ^ , . . . g ^ o f the }  VI  We s h a l l  c a l l L(g)- the l e f t regular, m a t r i x r e p r e s e n t a t i o n o f G  ( r e l a t i v e to a p a r t i c u l a r o r d e r i n g o f the elements o f G):. extend L(g): t o a r e p r e s e n t a t i o n o f the group r i n g  R  '.  We may for  every  G,i\.  n  k=l  n L(.u): =  £  a  L k  (  s  k  )  :  *  ^  k=l T h i s g i v e s u s , b y 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. .  F o r elements u,v '  i n R„ _ and a and b i n F G,F  L(uv). = L ( U ) L ( V > , L(au+bv): = a L ( u ) + bL(v): .  F o r e a c h g i n G the r i g h t  representation of G i s  given by  n s  i  s  =  X  b  ij( ) j' g  g  i = l,...,n.  (k)  3=1  and t h i s  o f G onto that G is  corresponds  n  t o the mapping  • R:g ->R(g> = (h, ,(g):):i  1 < i , j < n,  d i s t i n c t permutation matrices  o f degree n.  Eq_. k  i s o m o r p h i c to the m a t r i c e s R(g):, g i n G; t h e y form the  r e g u l a r m a t r i x r e p r e s e n t a t i o n of G.  implies right  Theorem, 1.  Any l i n e a r combination o f t h e m a t r i c e s o f the l e f t  regular  m a t r i x r e p r e s e n t a t i o n commutes w i t h any l i n e a r combination o f t h e m a t r i c e s of the r i g h t regular matrix representation.  Proof.  By e q . ' s 1 and k- r e s p e c t i v e l y we have f o r elements  (gg ,...,gg >  /  1  n  =-L (g>(g ,...,g > /  1  g and h i n G  (5>  /  n  P o s t - m u l t i p l i c a t i o n o f e q . 5 b y h g i v e s us  (gg h,...-,gg h): = L (g>(g h,...,g h> . /  1  n  /  /  1  ->  n  L'(g>R(hXg ,.. ,g > 1  t  /  n  where t h e l a t t e r r e s u l t f o l l o w s from e q . 6 . • P r e m u l t i p l y i n g t h i s by  g  and u s i n g e q . 5 produces  C&jh,..., g h } ' = L ( g ) : R ( h ) : ( g ~ g , . . . , g " g ) : ' /  1  1  n  1  /  n  = L'(g)R(h)L'(g- ):(g ,...,g ) 1  1  Comparing t h i s w i t h e q . 6 we g e t R(h): = l / ( g ) R ( h ) L ' ( g "'"):. L C g ^ C g " ) : = I = L(g):L(g):% 1  we get  n  /  .  Since  L(g)R(h> = R(h}L(g)., a s r e q u i r e d .  Any l i n e a r combination o f t h e l e f t r e g u l a r m a t r i x r e p r e s e n t a t i o n o f G over K i s c a l l e d a group m a t r i x o f G over K.  This  o f course  -5presupposes.an o r d e r i n g o f the elements o f G. m a t r i x L(g): i n eq.. 2 i n view, o f eq. 1. p o s i t i o n ' o f L ( g ) : ' ' p r e c i s e l y when g = g g-  when  We have a one a t the  gg. = g . ,  g' - g . g . 1  1  .  Thus i n L(u)' =  J  • , .  -  ."  /  h  g  ing exactly at those'positions  a l  n  p o s i t i o n o f L(g): p r e c i s e l y  L(.g ).,  G  '  k  a  appear-  Sk  k  (i>j)- f o r which g, = g . g .  words, a group m a t r i x o f G r e l a t i v e to g-^,...,g^  In  1  other  of G i s of the form  _ i > , 1 •< 1, j < n .  I W - =' (a  Theorem .2.  we have  n  k  S  r  (j,i)  .hence p r e c i s e l y when  (±,j)'  . Thus a one appears a t the  t  C o n s i d e r the p e r m u t a t i o n  (7>  J  Any m a t r i x over' F which commutes w i t h a l l m a t r i c e s o f the  r i g h t r e g u l a r m a t r i x r e p r e s e n t a t i o n o f G i s a group m a t r i x o f G; t h a t  is,  i t i s a l i n e a r combination o f the m a t r i c e s of the- l e f t .regular m a t r i x r e p r e s e n t a t i o n o f G.  Proof.  L e t C = (c.'.) , 1 < i , j < n, c. . i n K, be such t h a t ;  C = R(g )CR(g > , /  k  eq.'. k  k  k = l,...,n,  where  R ( g > = (b k  (g^):): Has d e f i n e d i n  i s the r i g h t r e g u l a r m a t r i x r e p r e s e n t a t i o n o f G.  Let  u-. be t h e  n - t u p l e row v e c t o r 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 e a c h k, we have,  - 6 -  c. . = u.Cu' = u.R(g. )CR(g, ) u . • /  1  n  s,t=l  T h i s sum may  where  and  s,t  g  a r e such t h a t  = g g  1  F o r , by eq.. k,  be s i m p l i f i e d .  k  _ 1 t  g. I  ^ so t h a t  -1 S  g  = g  g^"  1  J£  ^^(s^) ~  ' -1 = g. g . J "G  = ^B^ 1  H e n c e  ^  Thus  >  b v  * -1 g. = g g 1  - eq.  SiC  7; C i s  a group m a t r i x .  S i n c e the m a t r i c e s L ( g ) fc-rm a group i s o m o r p h i c t o GV, and s i n c e the m a t r i c e s a r e a l s o l i n e a r l y independent  over F, we  have Theorem 3-  to ' Theorem 3-  J^Cg); g  i  R(G,F): i s i s o m o r p h i c t h e a l g e b r a o v e r F generated b y the 4  G.  n  R  „ i s i s o m o r p h i c t o t h e r i n g generated over K by t h e  n  L(g}, g i n G. C o r o l l a r y 1.  The  i n v e r s e and the t r a n s p o s e o f a group m a t r i x i s a group  matrix. Proof.  The  i n v e r s e o f any m a t r i x i s a p o l y n o m i a l i n t h a t m a t r i x . Hence  by. Theorem 3 the i n v e r s e o f a group m a t r i x i s a group m a t r i x . Since  L ( g "^): = L ( g ) , , /  g i n G, the t r a n s p o s e o f a l i n e a r  combination  o f L ( g )',..'. , L ( g ) ; over F i s a g a i n a l i n e a r combination o f L ( g ^ )•,... ,L(g^): n  although i n a different  order.  m a t r i x i s a group m a t r i x .  T h i s proves t h a t t h e t r a n s p o s e o f a group  - 7-  3.  U n i t s and Unimodular Group M a t r i c e s . Elements u and v i n R  r i g h t units of  R  &  K  and r i g h t u n i t of R d e f i n e d over K. i s  ,  satisfying  respectively.  uv = 1 a r e c a l l e d l e f t  An element which i s b o t h a l e f t  'is c a l l e d a u n i t o f R^  v  and  s a i d to be unimodular i f  v  its  .  Any square m a t r i x  determinant i s  a unit  the  i n K.-  G i v e n t h e elements u,v  eq.. 3  i m p l y ; s L ( U ) ' L ( V ) . = (,L(UV) = L ( l ) . = 1^.  unimodular. exists  above, Lemma 2 a n d * d e f i n i t i o n given, i n Therefore,L(u)  C o n v e r s e l y , l e t L(u). be unimodular over K.  and by C o r o l l a r y 1 i t i s  is  Then L(u):  a group m a t r i x w i t h elements i n F.  In f a c t , L(u): ^ has elements i n K since, any element o f L(u). ^ i s the form  S ( d e t L(u)):  (det L(u):). L(u)  is  i n K where S i s  i n ' K.  a c o f a c t o r of L(u)  Thus an element v i n R  v  exists  of  and  such t h a t  = L(v),,-L(uv): = L ( u ) L ( v ) = I n ; and s o , by Theorem 3 uv = ! •  T h i s proves "Theorem 4'A.  Theorem k. . An element i s a l e f t u n i t of R_, „  — — —  •  •  c o r r e s p o n d i n g group m a t r i x i s C o r o l l a r y 2. Proof.  Theorem 5-  —  -  i f and o n l y i f  the  unimodular.  Every l e f t . ( r i g h t )  L(u)L(v). « I  G,i>-  unit is a unit.  L(v)L(u)..  The s e t o f a l l u n i t s o f R  G,iv v  under m u l t i p l i c a t i o n forms a  group i s o m o r p h i c " t o the m u l t i p l i c a t i o n group of a l l unimodular group . . m a t r i c e s o f 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 w i t h an element g o f order n, t h e group  ' n-l m a t r i x o f G over K r e l a t i v e t o t h e e l e m e n t s . l , g , . . . , g i s called a circulant  over K.  i-i-(j-i)  _  Let  i - j  g. = g  Thus,  1  \ . i = 1,. .. ,n.  Then g. g .  =  C - I = C . . and so t h e elements o f t h e . ^K, g  group m a t r i x a r e c o n s t a n t a l o n g each d i a g o n a l p a r a l l e l t o t h e main diagonal.  "  Let P be t h e companion m a t r i x o f t h e p o l y n o m i a l P  n  = I •n  and  n x -1.  Then  ' i+l  n-i+1  0  . 0  0  .. 0. o ..  i  :o  ..OV  0 11  1 00. . • o  0  ..0  0 0 ... 1  0  •Ml  (9)  where  1< i < n - 1.  i n P.  Moreover, Theorem 2 i n t h e s p e c i a l case o f c i r c u l a n t s becomes  Lemma 3.  I t f o l l o w s t h a t any c i r c u l a n t C i s a p o l y n o m i a l  The m a t r i c e s o f t h e l e f t and r i g h t r e g u l a r r e p r e s e n t a t i o n s  o f G - r e l a t i v e t o t h e elements l , g , . . . , g  n - 1  are c i r c u l a n t s .  Any m a t r i x  'commuting w i t h P i s a c i r c u l a n t . . I f t h e f i r s t row o f t h e c i r c u l a n t C i s g i v e n by ( c ^ , . . . , c .C=.  [c  n n  ) we w r i t e  (10)  - 9 -  f o r "brevity.  L e t the conjugate  t r a n s p o s e o f a m a t r i x A be denoted  •xby A . . Theorem 5.  L e t C = [6.,...,c ] 1 > n n  over t h e complex number f i e l d .  where  be a c i r c u l a n t of o r d e r n d e f i n e d  Let  T = ,n"  p, i s a p r i m i t i v e n t h r o o t o f u n i t y .  1 / /  ' (p^ ~ ^^~ ^), i < i , 2  i  1  1  Then  T*CT = aiag(e ,"...,e > 1  where the e i g e n v a l u e s ,  e ^ , . . o f  C  j < n  (ll)  ri  a r e g i v e n by t h e v e c t o r m a t r i x  equation' -  Proof. if  n  Since  .  ( e ^ . . . ^ ) / = n / T(c ,...,c ). 1  2  1  x - l = ( x - l ) g ( x ) where g(x). = 1 + x + ••• + x n  does n o t d i v i d e k.  .  /  n  11  (12>  \  Thus T i s u n i t a r y ; t h a t i s , T T = I .  g ( p ) = 0, k  For,  . . -xthe ( , j , i ; term o f T T i s g i v e n by n  n"  1  I  n  f(k-l>(j-l)p(k-l>(i-l). = n"  1  k=l  The RHS equals 1, i f i = j and Now,  I  k=l  equals g(p'  )• = 0, i f i ^ j . n  n  \. equals t h e j t h column o f T we get 3  L J  s i n c e P i s t h e companion m a t r i x o f p o l y n o m i a l x - l , t h e  e i g e n v a l u e s o f P a r e the r o o t s o f x - l ; if  pC^Ki-j)  namely, l , p , p , . . . , p 2  n  \  Thus  -10-  « n-V^pO-l  P,  p  2(j-l> ... (n-l):(5-l). ;  ) p  1 }  so t h a t , the j t h column o f T i s an e i g e n v e c t o r c o r r e s p o n d i n g t o the eigenvalue  i-1 p" o f P, n  Consequently  C =  T  *  ^  CT  c .P'-'  =1 3-1  Therefore-,- i f - we  j = l,...,n.  * r n-l\ Thus, T PT = d i a g ( l , p , . . . , p ):,  implies  (n-l)(j-l)v •>P >•  J  set n.  -  we  get  Jo.  (i-l):(j-l).  (13)-  eq. 11 and 12 a s . d e s i r e d ;  The p o l y n o m i a l s over K i n the n X n m a t r i x  ' P  =  010  ...  ;. 05 -1 0 . . . .  0  'o 1 0  a r e c a l l e d skew c i r c u l a n t s o f degree n over K.  Skew c i r c u l a n t s a r e not  group m a t r i c e s because i n any group m a t r i x the elements permutations  o f the elements  i n row .one, 1 < i < n.  In row  i are  However, t h e powers  -11-  of-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 t h e c y c l i c group o f  order 2n.  .  Since its  P_  i s t h e companion m a t r i x of t h e p o l y n o m i a l f (x) = x  eigenvalues  are  p,p ,...,p 3  2n-1  p  where  1 1  + 1,  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 . If 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; otherwise the  g(p^) = n .  Therefore,  i f T = n ^^(p^~^  ), 1 < i , ' j < n,  ( i , j ) element of T$T b e i n g  n  n V ^  --1l  • -p((22ii--ll)) (( kk -- ll )) p (( k k -- ll )) (( 22 jj --ll)) n  k=l  =  =  n ^-1 Y n - i V  of T ,  T T = 1^.  2(k-l)(j-i)  K=:.  •  -1  implies  p  / j - i \  M o r e o v e r , t h e p r o d u c t of P_ and  A... .the j t h column  yields  P j u =n-V?(p a-l,- ?(2J-l) ..., (n-l)(2J-l) . 2  p  >  p  )  ir  J  since  p  n  = -1  implies  p (2j-l) n  j t h column- o f T i s an e i g e n v e c t o r  p ^" , 2  1  j =l;...,n.  Therefore,  _ (_]_ )^j 1 _ o f P_ T*PT =  j  n  Q-^er  w o r r  corresponding to i t s  diag(p,p ,..:,p 3  2n_1  ).  i  s }  the  eigenvalue  Theorem 6 .  If  A = , .  .  a  j=i  j^_^  1  S  a  skew' c i r c u l a n t over K , t)hen  . "v  •  T A T = diag (  where  (e^...,e ) n  7  =  n^^T'Ca^, ...,a  e i  ,...,e >  ... .  n  (l-5>  ;  )' .'  • ' ,(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 z e r o 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 called nontrivial. exist:  We know- t r i v i a l unimodular' (skew); c i r c u l a n t s . always  see (eq_. lk) eg,. 9»'.. " I t i s shown i n [7] t h a t n o n t r i v i a l u n i m o d u l a r r  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 u n i m o d u l a r i n t e g r a l skew c i r c u l a n t s ? problem- i s n o t s e t t l e d . be t h e m a t r i x A A ' .  This  However, i f A were .such a m a t r i x . t h e n , so w o u l d  F o r , a d i a g o n a l element I n A A ' i s t h e sum o f t h e  squares o f t h e elements i n any row o f A ; a n d s o , o f f d i a g o n a l elements must occur i n A A ' since i t i s unimoduiar.  Therefore,  t h e . s o l u t i o n i s i n the  answer t o a n o t h e r 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 u n i m o d u l a r 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 ?  This question w i l l  .be t a k e n up i n t h e s e q u e l f o r v a l u e s o f n < 76. . 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 a n d • ,  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 o f o r d e r - n a n d a l l n X n  -13-  m a t r i c e s a r e assumed to be i n t e g r a l and unimodular. is  An  n. X n m a t r i x  c a l l e d a g e n e r a l i z e d p e r m u t a t i o n m a t r i x i f i t has e x a c t l y one non  zero element, +1 or -1, o c c u r i n g i n each row and column. Theorem 7«  . If  G i s a c y c l i c group o f o r d e r n and A ' A  i n t e g r a l group m a t r i x o f G, where A i s an integers,  n X n  i s a unimodular  matrix of r a t i o n a l  then A = QG where Q i s a g e n e r a l i z e d p e r m u t a t i o n m a t r i x and  C i s a unimodular group m a t r i x o f G. The  p r o o f proceeds by way o f Lemmas.  For n >  1,  let  [0,1,0,...,6J-  denote the m a t r i x i n eq. lk.  Lemma k.  L e t P and A be n X n unimodular m a t r i c e s o f r a t i o n a l  integers  such t h a t P'A'AP = A ' A .  (17):  Then a g e n e r a l i z e d p e r m u t a t i o n m a t r i x R e x i s t s such t h a t  RAPA R' = diag(P _1  where  n = n  n  + ••• + n  1  f t l  ,...,Pfi ):  (18):  s  and f o r e a c h i = l , . . . , s , . P  '  s  ' '  1,  Proof. it  o f the form  n.  1  1  1  and i s a one rowed submatrix o f t h e form ( l ) . o r ( - l ) n. >  i s n . X n .  [0,1,0,... ,0]  The m a t r i x B = APA  1  is  or  [0,1,0,...,0]-  1  —1  o r t h o g o n a l s i n c e (APA~ )'APA  i s a l s o an i n t e g r a l m a t r i x s i n c e A i s u n i m o d u l a r .  B = (b. .)-, 1 < i , j < n  i f n^ = 1, or i f  Therefore,  i s a g e n e r a l i z e d permutation matrix.  = I  :  -14-  L e t T be a l i n e a r t r a n s f o r m a t i o n o f a n n - d i m e n s i o n a l R  n  whose m a t r i x i s B r e l a t i v e t o a b a s i s e l'  where  e  space  ' o f R. ' n  Then, . '  tr i s a s u i t a b l e p e r m u t a t i o n on l , . . . , n and b^  = + 1. L e t  TT = ( j ( i > j ( 2 ) . . . . j ( r > . K j ( r + l > j ( r + 2 > . . . ( r > ) . ' ' 1  1  J  1  2  •••(j(r _ +l>j(r _ +2)>..-j(r _):> s  be a d e c o m p o s i t i o n into&s  1  B  (20).  s  1  d i s j o i n t c y c l i c products o f lengths, say,  '  n ,...,n r e s p e c t i v e l y where r, = 0, n. - r . - r . , i = 1,...,s 1' ' s . 0 l l .1.-1 n  r  m. n and where  n  and  j ( l ) . , . . . , j(n): i s a p e r m u t a t i o n o f 1, . . . , n w i t h j ( l ) . = 1.  T h i s g i v e s us a n o t h e r b a s i s o f R : n to  '( l'"-> n* f  f  =  K' 3(2):'"*' j(n) e  E  . - S(e ,...,e ). 1  ) :  (  when  we g e t  n = 1} and when n, > 1, k K  '  T ( t  ± }  =  b  j(i>, j ( i i ) i i - V i +  f  +  1  )  :  (22):  /  n  where S i s some p e r m u t a t i o n m a t r i x . Moreover b y e q . 21, 19 a n d 20 c o n s e c u t i v e l y f o r k = 1,...,s  2  <  1  •<  r  k-i+  V  -15-  with  I n o t h e r words except f o r change i n s i g n s T. permutes -y  ?  '  k-1  +V r. f  r  cyclically.  4.o>'">£  k - l  r  In matrix  n o t a t i o n t h i s amounts  k  to  (23)  where  H = d i a g (B^. ..,B ); g  i s a direct  whose t y p i c a l form i s t h e f o l l o w i n g :  sum o f  ^  X n^.  B = (+l). when n .k .'  matrices  B  = 1 and k  when n. > 1,  0  0 b  where o f course t h e  b. = + 1  V 0  1  0  b.'s a r e equal t o + 1. . L e t Z = d i a g ( l , b , b , b  ....,b ...b  n  i Then, s i n c e  b.  ."~ we g e t  '  1' 1  s  7  '  1  ).  n^/  -16-  0  1 0  ZB,-Z's k 0 K...b \  c o n s t r u c t a m a t r i x W,  i n form to Z,  v  n. l  of s b l o c k s  analogous  = ¥ a i a g (B ,...,B ).W' 1 s = d i a g (P  P  a d i r e c t sum  such t h a t  WHW"'  where  0 "k  1  Thus we may  1>.  ( i f n^. >  are the m a t r i c e s  n, 1  ,...,P  n  (2k).  ) s  d e f i n e d i n eq. 1.  ( T ( f >,..., T ( f » ' x  n  = S(T(  e ; L  But  >,..., T ( e »  /  n  « SB( ,.../e ):' ei  n  = SBS'^,...,^).' as a r e s u l t o f eq.'s eq. 23 we  22,  get H = SBS'.  19,  and  22 r e s p e c t i v e l y .  Comparing t h i s t o  T h e r e f o r e , by eq. 2k, WSBS'W' = d i a g (P  ,...,P n  l  and the lemma i s p r o v e d s i n c e R = FWS  i s a g e n e r a l i z e d permutation  and B =. A P A .  .  - 1  If,  i n Lemma k:  2) P i s a m a t r i x the o r d e r of P; [0,1,0,...,0]  l);. the m a t r i x A s a t i s f i e s the hypothesis  n  matrix  i n Theorem 7  r e p r e s e n t a t i o n o f G where n i s '  = I j ' 3)- the r i g h t hand s i d e s of eq. 18  , then Theorem 7 i s t r u e .  > s  v  o f the l e f t r e g u l a r m a t r i x i.e. P  n  equals  F o r , l e t L(h):, h i n G, be the  left  -irr e g u l a r m a t r i x r e p r e s e n t a t i o n w h i c h d e f i n e t h e group m a t r i x A'A relative, L  say, t o t h e o r d e r i n g g^,...,g  (h), h i n G  o f t h e elements o f G; l e t  he t h e l e f t r e g u l a r m a t r i e r e p r e s e n t a t i o n ' r e l a t i v e n-l  to 1, g, . ,.,g L (g) = o  where g i n G i s ' o f o r d e r n.  [0,1,0,...,0]^  Then,  (25).  a n d L(h> = S L ( h ) S ' , o  h i n G, where S i s a permutation? matrix, s u c h t h a t (g^,...,g S(l,g,...,g  ):'.  [0,1,0,...,0]; SRA  B u t / c o n d i t i o n s 2). and 3): imply RAL(g)A  whence, b y eq.'s  commutes w i t h L ( h ) j h i n G.  25  )' =  R" =  S R A L ( g ) A R ^ S ^ • L(g). and s o , _1  From Theorem .2, o b s e r v i n g t h a t t h e  l e f t and r i g h t r e g u l a r m a t r i x . r e p r e s e n t a t i o n s a r e i d e n t i c a l when G i s a b e l i a n , we i n f e r that SRA i s group m a t r i x C o f G r e l a t i v e t o g^,...,g . Put  Q = (SR).'.  T h i s proves Theorem 7 g i v e n assumptions l ) , 2) and 3)  above w h i c h a r e j u s t i f i e d as t h e next lemma.,,shows. Lemma 5.  L e t G,A,n  be d e f i n e d r a s i n Theorem 7-  m a t r i x , a l i n e a r combination L(h),  h i n G, o f G.  Let' A'A be a group  o f the l e f t regular representation matrices  Then a g e n e r a l i z e d permutation  matrix R exists  such  that  RAi(g)A~"*"RT =  [0,1,0,...,0]  (26):  where g i n G i s o f o r d e r n. Proof.  S i n c e G i s a b e l i a n Lemma 1 and 2 imply A'A and L(g): commute.  Therefore,  in particular.  P'A'AP = A'A, where  P = L(g).  This  us t o use eq. 18 i n Lemma \; t h a t i s , c o n d i t i o n l ) i s s a t i s f i e d .  permits Also  -18-  note'that • To  P •  - I  n  and P  4 I  r  i f r .< n. n  n  show c o n d i t i o n 3). h o l d s , l e t D = RA  i n eq. 18.  sums of powers from 1 t o n and .noting t h a t P  Then by  taking  = L ( g ) , i = 1, ...,n  1  and  1  n—1 L(l)  + L(g) +  ••• + L ( g  [1,1,.. .,1]  ); =•  D[l,...,l]  D  n  = diag  _ 1  n where B.  1  =  ) P  >1 P  1  ,  i = 1, . . . , s .  1  P  n. x  i= I  n. x  n = n.m.. 1  P  n. x  Thus rank  P  1  n  = I  implies  n  '  I n f a c t , when P  n.  is  1  so t h a t , i f m. = 2q., then B. = ' 1 ^x' x  ) P L n. . x J=l J  n  i s a circulant, '  2n  n •  .  1  = q_. ) P ^x. L . ., J=l  ^ = n. x ..  0.  n±  n When  again.  i s the s m a l l e s t p o s i t i v e i n t e g e r s u c h t h a t •  n  lt  1  a skew c i r c u l a n t , 2n^  2  (27)  (B ...,B•)•  U s i n g eq.. 18,  , so t h a t , f o r some i n t e g e r m., n.. ' 1'  = I  n.  J  i_i n.  1-get the s i m i l a r i t y r e l a t i o n  we  ) P . = m. ) P ^ = m. [1,... ,1] L n. x L n. x ' ' n J  B. = x  .  B. i s 1 o r 0 a c c o r d i n g x  T  x  as P  .  n.  x  n  i  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 However, t h e rank o f the l e f t s i d e one  and  o n l y one  non  s i d e of eq. .27 i s 1 so t h a t on the r i g h t  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 k  V  side.  But D i s a unimodular m a t r i x of r a t i o n a l i n t e g e r s and  each.element o f D ^~ d i a g m^. =  d i v i d e s each element on the r i g h t  1,  - n and  eq.  27  (B-^, ... ,B )D,  hence  g  implies  m^. d i v i d e s 1.  so m^  Therefore  = [0,1,0,... ,0]  RAPA ^" = P -  "k  divides  .  -19-  Corollary  2.  I f A i s a unimodular  i n t e g r a l m a t r i x and A'A  is a  circulant,  then  (28).  A = QC where Q i s a g e n e r a l i z e d p e r m u t a t i o n m a t r i x and C i s a unimodular  integral  circulant. Theorem 8. circulant,  I f A i s a unimodular  i s a skew  then A = QC where Q i s a g e n e r a l i z e d p e r m u t a t i o n m a t r i x and  C i s a unimodular Proof.  i n t e g r a l m a t r i x and A'A  Let  integral  skew c i r c u l a n t .  [0,1,0,...,0]-  P =  .  Then, s i n c e  A (A i s by d e f i n i t i o n a  l i n e a r combination o f powers o f P, P'A'AP = A'A. can be used.  We  . Thus, eq. 18 in.Lemma k  s h a l l show s = 1 and t h e r e f o r e P  = P n  and t h i s would  1  e s t a b l i s h Theorem 8 s i n c e a m a t r i x w h i c h commutes w i t h a  nonderogatory  matrix i s a polynomial i n i t . Observe t h a t , i f P column o f the m a t r i x  i s a skew c i r c u l a n t , t h e n by a d d i n g the  n  sum / -1  P  n.  i  + P  n'.  +  + P  n  n.  I-'...  1  -1  t o e v e r y o t h e r column we  1 ^  i =  i  i  d i a g o n a l element  first  .. - 1 1  get a t r i a n g u l a r  m a t r i x w i t h -1 as the  first  and -2 f o r the r e m a i n i n g d i a g o n a l elements. . Thus ni  det  Y  P  J=l  J n  = (-l) 2 n i  n i  (29>  -20-  Also,  ^  ^ = 0,  P^  3=1  so t h a t i f m i s  X  mn-i .  1  R e t u r n i n g to eq.. 18,  i  odd  = 1,..., s  rijL 1  .  we  ,  X  see t h a t P  so t h a t each  P  = -I  implies P  n  n  n. x  = -I  , n. x  i s a skew c i r c u l a n t and n equals  i Therefore,  odd m u l t i p l e of n.. x  n  by eq. 18,  an  again  n det^P  =  0  det  J=l  Y  axag (P  ,...,P  n  ><  j=l  =  s  n  . ,  .  L> . ->  3  n. x  3=X  l Y P n  s  n  ) P  det  1=1  =  n „  ,  det  i=l  L  . -, •,  3  n.  i  n_n-s (-1).^;  =  30 and  29  where the l a s t two  equations f o l l o w d i r e c t l y from eq.'s  respectively.  But  eq. 29 a l s o i m p l i e s above t h a t the l e f t hand s i d e  \  Therefore  equals ( - l ) 2 n  7.  New  n  Results  s = 1 and Theorem 8 i s proved.  on Group M a t r i c e s  and Symmetric C i r c u l a n t s .  In what f o l l o w s , 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 . n o t a t i o n a l c o n v e n t i o n Theorem 8(Theorem 7 ) s t a t e s c i r c u l a n t o f the form  A'A  With t h i s  t h a t an p d i u  where A i s i u equals C'C  where  C  (skew) i s an i u  -21»  (skew) c i r c u l a n t .  I n view o f t h i s i t would be i n t e r e s t i n g t o note f o r  what v a l u e s o f n a r e p d i u (skew) c i r c u l a n t s i s an  o f the form C'C  where  C  i u (skew); c i r c u l a n t .  So f a r , v e r y l i t t l e n > 13.  i s known about t h i s f o r c i r c u l a n t s  I n an u n p u b l i s h e d work E.C.Dadei has shown i t t o be t r u e f o r  circulants  o f prime o r d e r l e s s than 100 > w i t h one e x c e p t i o n ; i n [6] i t  i s shown t o be f a l s e f o r n = 5  where equations  demonstrate t h a t t h e p d i u c i r c u l a n t form C'C  o f degree  where  C  [2,l,0,-l,-l,-l,0,l]g  i s an i u c i r c u l a n t .  study i n [7]  A = B'B .where  B  i s not of the  A r e s u l t o f Minkowski i n [5]  s e t t l e s t h e q u e s t i o n , i n g e n e r a l , f o r n < 7; n X n m a t r i x , then  11 and 12 a r e used t o  that i s , i f A i s a pdiu  i s an i u n X n m a t r i x , n < 7.  A  on t h e uniqueness o f t h e normal, b a s i s f o r normal c y c l i c  f i e l d s produced t h e r e s u l t t h a t a l l u i . c i r c u l a n t s a r e t r i v i a l f o r n = 2,3,4,6.  T h i s o f course i s c o n s i s t e n t w i t h Minkowski's r e s u l t .  A l s o f o r n = 5, [l].  an incomplete p r o o f appears i n [11]  R e c e n t l y i n a paper p r e s e n t l y . i n p r e s s  [12]  with corrections i n  R.C.Thompson s o l v e d  the q u e s t i o n f o r a l l v a l u e s o f n up t o 13 i n c l u s i v e by c o n s i d e r i n g a • more g e n e r a l problem which we s h a l l d e f i n e i n s e c t i o n 8. As f o r skew c i r c u l a n t s n o t h i n g has been w r i t t e n ' o n them.  In f a c t  I am i n d e b t e d t o Dr. R.C.Thompson f o r h i s c o n j e c t u r e s 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 p r o p o s i n g Theorem 8,  t h e p a r a l l e l t o C o r o l l a r y 2,  q u e s t i o n o f the e x i s t e n c e o f n o n t r i v i a l pdiu' skew c i r c u l a n t s .  and t h e  We s h a l l  d i s c u s s s e v e r a l cases i n t h e next s e c t i o n . I n s t e a d , we consider.whether the form P^C  where  every n o n t r i v i a l « i c i r c u l a n t i s o f  1 < k < n, P = [ 0 , 1 , 0 , . . . , 0 ]  and C i s a p d i u  -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 e i t h e r k = 1 or n = 2k.  T h i s i s o n l y a c o n j e c t u r e on my p a r t .  w i t h i t the f o l l o w i n g f a c t s are obtained. Let  ( c ^ , . . . , c ) be t h e f i r s t n  Lemma 6.  o f G d e f i n e d over  Then, w i t h o u t ambiguity we may w r i t e  1  ] n G  be a symmetric r e a l n o n s i n g u l a r group  m a t r i x w i t h p r i n c i p a l idempotent  decomposition  C = s ^ + . v .  and l e t  C  J ^ - I Q *  Let C = [c_....,c  — —  L e t G be a group o f o r d e r n.  row o f a group m a t r i x  the r i n g o f r a t i o n a l i n t e g e r s . C = [c^,...  However, i n consonance  e. denote t h e row sum o f t h e f i r s t i  + s E t  (31)  t  row o f E.. i  Then, f o r .  1) .  E ^ i s a symmetric r e a l group m a t r i x ;  2) .  t h e d i a g o n a l element o f E ^ i s a p o s i t i v e r a t i o n a l number e q u a l • to.r.n of  3) .  !  V)  .  = e. and e.e. «  x  n  j:  1  = c  n  1 . 1 c^+*«'+c'  •.  0, i f i i  x j  i f eigenvalue s  ' Note:  r . i s t h e rank of E . , t h e number o f e i g e n v a l u e s  C equal t o s^;  e.  x  where  + ••• + c , then e. = 0 f o r j i n' a  1  i s always an•• e i g e n v a l u e o f C.  Proof.  (E'.) = E'., i = 1,...,n and E'.E'. = 0, i =}= j .  C' = C  and t h e p r i n c i p a l idempotent decomposition  i m p l i e s E'^ = E^.  1 and .e, = 1.  T  Hence, s i n c e  o f C i s unique,  eq.31  I t i s known, e.g. see [ 8 ] , t h a t f o r p r i n c i p a l idempotent  decompositions a m a t r i x which commutes w i t h C commutes w i t h e v e r y Therefore,  s i n c e by  d e f i n i t i o n , C i s a l e f t regular representation,  1 implies a l l matrices the E  and  i n the r i g h t r e g u l a r r e p r e s e n t a t i o n s  so, by Theorem 2, the E^ a r e group m a t r i c e s  l ) ; s i n c e the E^ a r e r e a l by The  principal  d e f i n i t i o n of the  the main d i a g o n a l  of. G.  T h i s proves  decomposition.  and p o s s i b l y O's.  the c o r r e s p o n d i n g d i a g o n a l m a t r i x and of E^ i s constant,  Theorem  commute w i t h  idempotent decomposition r e q u i r e s t h a t E^ are  to a d i a g o n a l m a t r i x o f I's and  E^.  similar  By t a k i n g the t r a c e o f  E^  t a k i n g cognizance o f l ) , t h a t i s ,  2): f o l l o w s immediately.-.  Let  X = COl ( l , . . . be an n - t u p l e the E^'s  E. i  2  are  X = E:X . i  hence  column v e c t o r a l l of whose elements equal 1. idempotents, 3)  and  E.E.x i . ,j  :  follows  = 0.  d i r e c t l y from l ) and  (Note: >  F o r any J  E. x = E. (e'.x).-= • e. ( E . x ) "'= e.e.x, 1 X X l 1 . 1 1 '  i , . E.x ' i  whence  v  that  o f any  ' 1'  1' n  e. = e. 1  sum  the f a c t  = ( e , ...,e ).' =  v  a r e a consequence:of the f a c t t h a t the row  Then, s i n c e  n  .  These  e x,  1 ' n  results  1  group m a t r i x i s  independent o f the row.). From eq. 31,  Cx = S..E x +  ••• + s E x  By 3) i t i s p o s s i b l e f o r o n l y one ;  whence the p r e c e d i n g  But,  e^'s  ••• + c  to be non  = s e»+«v+s,e .  z e r o , say  e^,  e q u a t i o n reduces t o  •; c, •+•••+. c  1  o f the  so, c.+  •" • ,  since C i s nonsingular  =• s e n  n  .  n  11.  the l e f t  s i d e i s non  z e r o , so,  4 0'  Since  2 •e^ i s a row  sum  o f the r e a l m a t r i x E^,  e  =  >•0,  and t h i s  implies  =  1.  -2k-  Therefore  c, + * *• + c = s . 1 • n.. . 1 n  The next r e s u l t polar factorization Theorem 9.  . '••  i s an i n t e g r a l  c i r c u l a n t analogue o f the  theorem.  I f A i s an n X n n o n s i n g u l a r r e a l group m a t r i x then t h e r e  a r e unique r e a l m a t r i c e s H and U such t h a t A = UH where IS H i s a pd group m a t r i x and U i s an o r t h o g o n a l group m a t r i x . Proof.  Let  H = y"s E  +  1  C = A'A  ••• + / s E  are p o s i t i v e positive  . be the group m a t r i x i n eq. 31 and l e t where we  note t h a t t h e e i g e n v a l u e s s. o f C  T h e r e f o r e , by l ) . i n Lemma 6,  s i n c e C i s pd.  d e f i n i t e ^ group m a t r i x .  Moreover, A'A  where H i s t h e • o n l y p o s i t i v e by v i r t u e  o f t h e uniqueness  = K  (33)  2  d e f i n i t e matrix f o r which t h i s i s true o f e q u a t i o n 31•  I n [8] i t i s shown t h a t  f o r n o n s i n g u l a r A t h e r e a r e unique -real m a t r i c e s U and H o r t h o g o n a l and H A'A  = H  = H  2 q  H = H J  2  q  is positive  d e f i n i t e w i t h A = ^- Q  - w h i c h b y uniqueness  o f H i n eq. 33,  closure.  F o l l o w i n g t h i s , t h e terms  AyH,U  q  such t h a t U i s  But t h i s i m p l i e s i n turn  and t h e r e f o r e U i s a group m a t r i x by C o r o l l a r y  Q  H' i s a r e a l  implies,  1 and  i n Corollaries  4,5,6  multiplicative are  assumed t o be the group m a t r i c e s i n Theorem.9.  Corollary  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  H i s positive  ( d e t H);  definite.  = 1,  hence det H = + 1,  so, i t equals + 1 s i n c e  T h e r e f o r e det UH = det U = det A.  -25-  Corollary Proof.  k.  A i s normal i f f A = HU = UH.  C o n s i d e r t h e commutativity p r o p e r t y w i t h r e g a r d t o t h e idempotent  decomposition and t h e e q u a l i t y Corollary  5.  2  . T f i n Theorem 9, A = [  group m a t r i x and U = [u^,...,u^\ h = h, + • • • + h Jn Proof.  o f ITTJ and UH .. L i s an i n t e g r a l 1 n li and H = [ l ^ , . . . ,h ] , then  G  >. 1 and u 1  G  + • • • + u n  n  •= a, + • • • + a = + 1. l . .n —  L e t u = u + ••• + u and a = a. + 1 n 1  + a . The e q u a t i o n Ax = UHx n •  n  where x i s an n - t u p l e column v e c t o r a l l o f whose elements a = uh.  S i n c e A i s unimodular  -1 -1 f o r , xAA ;.x' = naa = I  unimodular  e q u a l 1, i m p l i e s  and i n t e g r a l i t s row sum equals + 1;  2  x  x  = n.  /  n  Since U i s orthogonal, u  = 1, because  2 nu  = x'U'Ux = x ' l ^ x = n.  +1 s i n c e + l H S i s p o s i t i v e Theorem 10..  Consequently,  definite.  I f A i s a unimodular  i n t e g e r s such t h a t P A Proof. . L e t K  KK = I and KPK = P'. ..... n.,.  c i r c u l a n t . t h e n t h e r e i s an  toy WO j  Hence KA'AK = ( A ' A ) ' = A'A  then  KQ, = KAKA"  [0,1,0,...,0]^.  :  i s an i n t e g r a l o r t h o g o n a l matrix,hence - 1  .  integral  be t h e n X n m a t r i x  In f a c t , i f Q = A K A  .  i s symmetric where P =  K =  Then  '  h « + 1 w h i c h p e r f o r c e equals  1  = A 'A  -1  -1 so t h a t AKA'  a g e n e r a l i z e d permutation  matrix.  so- t h a t K Q ' i s a c i r c u l a n t ,  and b e i n g  t r i v i a l implies there  k' s u c h t h a t 1 < k < n and KQ = + P .  Thus' by  K  , But  s i n c e the row  sum  ..  A' =  o f A and A  are  7  P  ±  A  K  i s . an  integer  3h  eq..  .  (35)  equal,  • A' a P A.  (36):  K  Suppose n i s odd. 2r = n-k  according  L e t r be an i n t e g e r such t h a t 2r = 2n-k  as k i s even or odd.  L e t • s be  or  the nonnegative  integer  (37)  ' s = 'n-r- , T h i s means r + k = n + Therefore  by eq.  s or r >  k = s  according  as k i s even o r  odd.  36  P A' = P t A = P A , R  and  so by  eq.  r  K  '  G  37 (P A)/ « A'P ~ S  N  = A'P  S  R  = P A'  = P A,  R  S  '..  S  w h i c h proves that' Mow  suppose n i s even.  the l e f t , of P  P A i s symmetric. Since  and. s.'==  the t r a c e o f AKA  n-r, we  same k i s even.  K  i s zero  = P  R + K  .  A = P  R + N  "  2 R  whence (P A)' = A'P " S  s so, P A  Hence, l e t t i n g  get from eq. 36 ' P V  and  = KP  i t f o l l o w s t h a t the number of elements i n the n o n t r i v i a l  i s z e r o , o r what i s the  k  '  i s symmetric. .  N  S  = A.'P " 1  A =  P A, S  .  r = .  on diagonal(s):  (n-k)/2  Corollary  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  k i n t e g e r k such t h a t A ' = P A  Theorem 11.  then t h e r e i s an  ' (where k i s even i f n i s even).  L e t A he a unimodular- i n t e g r a l  circulant.  Then t h e  e i g e n v a l u e s o f t h e symmetric m a t r i x KA a r e t h e square roots, o f t h e eigenvalues o f the p o s i t i v e  Proof. Hence  definite circulant  Observe t h a t K P . i s o b v i o u s l y symmetric 1  KA i s symmetric.  Positive  Definite  Skew  In t h i s s e c t i o n integral  skew  f o r each i = 1,...,n.  Then c o n s i d e r t h e p r i n c i p a l  decomposition o f KA and A'A = (KA).'KA;  8.  A'A/.  idempotent  and the p r o o f f o l l o w s .  Circulants.  B^ always denotes an n X n symmetric  unimodular  circulant.  • - By d e f i n i t i o n o f B >' we may w r i t e , f o r k > 1, n  B  '  ' -b.,I ]~ n  = [b .h. , b , ... ,b_ ,-h, ,-h, • cr 1' 2 ' k' k' k-1' n  n  (38):  w  '  i f n = 2k + 1, and ;  i f n = 2k + 2. of B n  B  n  B  v v 2 ' " ' ' V  [  o  ' " V - W  i  " ' - ^ ;  (39).  Then, by eq. l 6 i f B^ = A, we g e t f o r the I t h e i g e n v a l u e  e  . .  which, b y s u b s t i t u t i o n s ' o f f o r any n > 3 ,  " .„(21-l)(j- > a  1  t h e a.'s w i t h t h e b's i n eq. 38 o r 3 9 , y i e l d s  .  -28-  (2l  0  Lemma 7.  If  v ~ " (2i  v ~ -x  'L--o *i  i)j  iMn  i s t h e symmetric n X n skew c i r c u l a n t  or 39; where n = 2k+l o r ,2k +2  3)  (u>-  g i v e n b y eq.'s  38  then i t ' s e i g e n v a l u e s a r e g i v e n b y  j=i i  = 1,... ,n a n d  (43). ^  ' e. = e . ,, 1 n-i+1  v  y  f o r i = 1,2,...,k+l.  Proof.  E q . 42, o f c o u r s e , f o l l o w s d i r e c t l y from eq. 4 l . • Eq. 43 f o l l o w s  from eq. 39 ^ d t h e f a c t t h a t t h e e i g e n v a l u e s o f a symmetric r e a l a  are a l l r e a l . n - i + 1  For, p ( . ^ ^)' n  f o r i i n eq.  40  n-i+l ~  Y  Lemma 8.  ^  substituting  L j=l  (l-2i).( j - l ) . V  and so, b y comparing t h i s w i t h eq. 40, whether n i s odd o r even.  ^  a  we g e t ,  _ £  _ p-^ ^ i  +  matrix  e  i  =  e  j_ ~ n i+l e  This evidently implies  For n > 3 det B = ( n 1  e  2.  ) k  2  e(n). * + 1 —  ^  0  r  =  I>«".? 1.> k+  -29-  f  where.  :(n) =  V  2  e,_., "k+1  e  , i f n = 2k+2  k+l - '  i  f  n  =  2  k  +  1  G i v e n a square m a t r i x A we denote i t s t r a c e b y tr(A):.  From eq. 15  where A-B and eq. h-3 we have n Lemma 9.  For n > 3  tr(B ) = nb n  Q  = 2  £ e. + ( n > 6  '1=1  (  • k+l' 2 e  where  6(n) =  e  Lemma 10.  I f n.= 2k+l  and A  n  k + 1  ,  ;  i  f  n  = 2k + 2  I f n •= 2k + 1  = [a^,..-^a^]"  i s a unimodular i n t e g r a l  skew c i r c u l a n t w i t h e i g e n v a l u e s d e f i n e d as' i n eq. 15, then n e  k+l ~  +1.  a  0=2 Proof.  By eq. ±6, k e e p i n g i n mind t h a t  p  n  = -1, we g e t  'k+1 ~  a  L  l  +  a  j  p  >2  1  •  •n  = a  1  +  i=2  Therefore,  e  k+i  i  s  a  rational  integer; similarly with  >  k+1 e i g e n v a l u e o f t h e i n v e r s e o f A , which as w i t h A i s a unimodular n' n integral  skew c i r c u l a n t .  as d e s i r e d .  '  •'  C o r o l l a r y 7«  d i v i d e s 1 r. . , e,  v  :  x  '  k+1  1  .= + 1 k+1  —  if- -h - 2k+l, B^ i s as i n eq.. 38 then  e.  . = h  k+1  Proof.  e  Therefore, since  Since  + 2 O  Y  (-l) b  LJ  ^  = + 1.  j  '  j  —  B^ hy d e f i n i t i o n i s symmetric t h e c o r o l l a r y  follows d i r e c t l y  from Lemma 10. We now p r o c e e d t o show f o r which v a l u e s o f n i s B t r i v i a l o r n o n t r i v i a l . n O b v i o u s l y i t i s t r i v i a l f o r n = 1,2. Case 3: Proof.  B^. = I Let  3  = [a,b,-b]^  .  .  3  2  and so, s i n c e -1 + p - p  .  By Lemma 7 and  e  ±  = a + b (p-p )  e  2  = a - 2b = 1  2  = 0,  = a + b, whence, by Lemma 8  '  -31-  det  Therefore,  = e  Case k.  Since  2  B  B^ - [a,b,0,-b]^  equation a  Proof.  = (a + b ) ( a -  2  a = 1 + 2b i m p l i e s a + b =  i f b = 0 and a = + 1.  the  e  2 1  2  p  2b = 1  -  2b > = 1.  l + 3 " b = + l ; w h i c h holds  only  i s pd, a =. 1.  i s nontrivial for integral when b  4 0, a >  1.  solutions of  F o r example'  _  [3,2,0,-2]!".  V  By Lemma 7> eq.. k-2,  e^^ = a + b (p-p) .e = a + b (p-p)- = a + b (p-p> 5  5  9  5  2  ^ 2 and  so, s i n c e « (p-p  )  :  = 2, by Lemma. 2,  det B^. = ( e ^ ) • 2 we have  a  2  = ( a - 2b ) = 1 2  2  2  2 - 2b  = + 1 which equals + 1 s i n c e  3-^2  ^ ®'  C o n v e r s e l y i f . a and-b a r e s o l u t i o n s such t h a t a > 0, b 4 0.  2 a  Then  2 > 2b  i m p l i e s a > + /"2b  Therefore Case 5'  so t h a t  B^ i s a p d s i u skew c i r c u l a n t B  5  =  t ,b,c,-c,-b]^ a  a  2  1.  F o r example  e^' 2 * ^* e  >  and n o n t r i v i a l .  i s n o n t r i v i a l i f f a,b,c a r e s o l u t i o n s t o (kk):  - It-bc = 1.  (b-c):(l+b-c) where a >  a 4-/"2b > 0 and hence  = be  [3,2,1,-1,-2]^..  (U5)  '  -32-  By Lemma 7 and C o r o l l a r y 7  Proof.  e  1  « a + bX  - cX  e  g  = a + b X - cX-j^  2  (46)-'  2  e,.=  a - 2b + 2c = 1  3 . where  X-^ = p-p  4  3 2  Xg = p -p  and  and  p  i s t h e 10th p r i m i t i v e r o o t  2  4  3  Using the f a c t t h a t - 1 + p - p + p - p  of u n i t y .  f o r w a r d computation w i l l show t h a t Lemma 8  indeed, 2  =  e  j_ 2  ^  e  S a  n  i n  "k 6 e  e r  a n c  a straight ^ hence from  2 2 ~ ab-ac-3bc = 1.  l 2 ** G  e  =0  a  C  +  But  4  £ l  e  + e  2  -  The  l a t t e r equation  2  » 5a -20bc = 5  (47)  2  e ^ E g = 5'(b +c  - ab+ac - bc):.= 0.  gives  ( b - c ) ( b - c - a ) = -be  .  w h i c h reduces t o (b-c>(l+b-c> = be by eq. 46. . Eq. 47 i m p l i e s t h a t i f  C o n v e r s e l y i f a,b,c t h a t a > 1, then  B<_ i s n o n t r i v i a l then a > 1.  a r e i n t e g r a l s o l u t i o n s t o eq.'s  44 and 45 such .  B,_ i s a n o n t r i v i a l p d unimodular skew c i r c u l a n t . F o r , :  2 2 4 e e „ + e, = 5a - 20bc 2 3  .  1n  w h i c h by e q u a t i o n 44 equals get , £j. 2 e  =  • H  e n c e  5'  "'Thus'solving f o r t h e i n t e g e r  C o r o l l a r y 7,-  e  -j_ 2  = + 1 and so by Lemma 8  e  w  e  B,- i s  -33-  5 divides  unimodular; moreover, s i n c e  e = e.,-1, 12 3 '  t h e d i f f e r e n c e e-.-e  n  3  0  = 1 ; which i s eq. 46. To show t h a t a l l t h e eigenvalues o f &g a r e - p o s i t i v e we note t h a t X-^ and Xg a r e t h e - r o o t s o f the p o l y n o m i a l  ~2 X(x): = x -x-1  which means  .  So t h a t _  e-i  £9  -  —  , b-e  a  +  a +  h+c „.  o  , h-c  and so by eq.'' 45 h-c > 0.  " h+c -  p  v5  o  +  We must show ' a + ~ ^ > + ^jp  /5  o  _  /"5 •  • •  ? y eq. 44, s i n c e a > 1, be > 0  Hence we o n l y need t o show  i s f a l s e when b,c > 0. . By s q u a r i n g b o t h . s i d e s get  ' a  2  Case 6.  Hence  <  and transposing  /5  terms we  " ;  2  + a(b-c) < (b-c)  But t h e r i g h t hand s i d e i s n e g a t i v e contradiction.  a +  - be .  according  t o eq. 45.  This i s a  £-^,£g > 0 and so, B,_ i s pd.  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 integral  s o l u t i o n s o f t h e equations  a-2c = 1 ( ^ ) where  a >  2  - 3 h  2  = l  1. F o r example'[5,4,2,0,-2,-4]g  .  (48> --(4> 9  Proof.  We have b y Lemma 7 5 2 H= a+b(p-p ). + c ( p -p ):  e e  :?  2  = a-2c *5  2  ^4-  e = a-bCp-p?): + c(p -p );. 5  But b y Lemma-9 and eg. '-4-3 i n Lemma 7 tr(B ) = 6  .  6a = 4(a+c(p -p ):> + 2(a-2c) 2  1+  implies c = To show c 4'°-  L  B  H  B  .  Then  ^ 6 6> 6 .  _1=  •  Bg  / 2 >  e t Hg = [ l , 0 , - l , 0 , l , O j g  •" . 6 6 6  Since  c(p -p ) .  B  H  B  _1  = (a-2c>H B 6  i s an i n t e g r a l m a t r i x  _ 1 6  ••  .  a"~2c d i v i d e s one, and so., e  2  = a-2c = 1 2  s i n c e Bg i s pd. and s o ,  Therefore,  i f c = 0, Bg = I g . 2  4  I t f o l l o w s t h a t p -p  2  = (a+c): - 3 h .  {I  By Lemma '8, e e . = 1. •1 3 1  C o n v e r s e l y , suppose a,b,c a > 1.  Then  a-2c = 1 - i m p l i e s  s a t i s f y equations 48 and 4-9 c ^ 0 so t h a t  such t h a t  -35-  0  Therefore,,  +1 = 0.  -0  (51):  T _ ^ equals one b y eq. 49 so t h a t b y Lemma 8, Bg i s  E  E  unimodular.  -  i+/3 A s o l u t i o n t o eq.- 51 i s of u n i t y .  5 i-/3  Thus  p  =  *^  p = •  which i s a 12th p r i m i t i v e r o o t  y  implies  = a+/3b + c a-/3h + c.  =  3 By eq. 48,  ,c > 0 and hence b y eq. 50 i t f o l l o w s  a + c> + so t h a t £-j_>3 ^ 0.  7.  = a + bT  e e "0^ =  6  P~P > ^ 2  = =  2  2  = 1  s o l u t i o n of the cubic  d?  - c? + 2  1  = a + bTl  ' 5 2  ^  ''^3  when  p  2  ^  =  3-2  Tl^ - T| + T)  a  know.fiicar*:  = a + bTl^ - c T l + dT|  x^ - -x and  thig drily f a c t s  For A^ = [a,b,c,d,-d,-c,-d]~  ^  where  /3b  T h i s proves t h a t Bg i s p d .  £  Case  that  2x  2  - 71 +• dT^ c  •34''' '  a  r  e  s  b u t t o n s o f the equation  + 1  i s the 14th p r i m i t i v e root o f u n i t y .  equation i s  " i + ( | /7 )  cos Q  cos" ( ^ 1  )) .  ^  -36-  Result:  ¥ e have shown t h a t [ 3 , 2 , 0 , - 2 ] ^ ,  [5,4,2,0,-2,-4]^  [3,2,1,-1,-2] , and  a r e 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 d i a g o n a l elements 3 , 3 , 5 i n t h e s e m a t r i c e s a r e m i n i m a l f o r t h i s class of n o n t r i v i a l matrices. t o be o f t h e form C'C  Hence i t i s i m p o s s i b l e f o r these' m a t r i c e s  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 s i n c e t h e d i a g o n a l elements o f C'C would o t h e r w i s e exceed  9.  3,3,5-  Appendix.  Let  C = ( v n m  n  ) where'm and n a r e i n t e g e r s . m y  Then m + n and m - n  a r e square i n t e g e r s i f and o n l y i f t h e r e i s a unique m a t r i x A o f r a t i o n a l integers of the form  '' J  consequence  The r e l a t i v e l y prime s o l u t i o n s o f the e q u a t i o n  x  2  o f the f a c t :  2 2 * + y . = z . with  r > s > .0, ( r , s ) =  such t h a t CV= A'A.  y . even a r e x = r  2  T h i s comes as d i r e c t  2 . 2 -2 - s , y = 2 r s , z = r + s , where  ! . . ' [ '  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 t h e c o n d i t i o n s on m and n are relaxed.  F o r example  60 V _Y8E:J\ S4>?\ V60 65 J " V77V V i '  r.65  The case f o r 2 X 2  skew c i r c u l a n t s t u r n s out to,be  trivial.  -37-  BIBLIO GRAPHY 1.  . E.C.Dade and 0. Taussky, Some hew of r a t i o n a l integers,  r e s u l t s connected w i t h m a t r i c e s  P r o c . Symposium i n Pure Math., o f the  Am.Math.Soc, 8(1965)., 78-88. 2.  G. Higman, The u n i t s o f group r i n g s , Proc.London M a t h . S o c ,  46  (1940)., 231-248. 3.  D. H i l b e r t , T h e o r i e des corps de nombres a l g e b r i q u e s , P a r i s , ( 1 9 1 3 ) , l 6 4 .  4.  M. Kneser, K l a s s e n z a h l e n d e f i n i t e s Q u a d r a t i s c h e r Formen, A r c h i v der Mathematik, V I I I  5.  (1957), 241-250.  H. Minkowski, Grundlagen f u r e i n e t h e o r i e der q u a d r a t i s c h e n Formen m i t g a n z z a h l i g e t K o e f f i z i e n t e n , Gesammelte Ahhandlungen  6.  M. Newman and 0. Taussky, C l a s s e s o f P o s i t i v e D e f i n i t e Unimodular Circulants,  7.  1 (l91l),3-144..  9 (1956);, 71-73.  M. Newman and 0. Taussky, On a g e n e r a l i z a t i o n o f the normal b a s i s i n a h e l i a n a l g e b r a i c number f i e l d s , Comm. on Pure and A p p l i e d Math. 9  (1956>, 85-91. 8.  S. P e r i l s , The Theory o f M a t r i c e s ,  Cambridge  9.  0'. Taussky, M a t r i c e s o f r a t i o n a l i n t e g e r s , B u l l . Amer .Math. Soc. ,66  (I960), 327-345. 10.  1952.  • .  0. Taussky, Normal m a t r i c e s i n some problems i n a l g e b r a i c number t h e o r y , P r o c . I n t e r n . 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, C l a s s e s o f D e f i n i t e Group M a t r i c e s , P a c . J o u r n . o f Math.  13.  R.C.Thompson, Normal Matrices, and the Normal B a s i s i n A b e l i a n Number Fields,  14.  12  (1962)1,  1115-1124.  R.C.Thompson, Unimodular Group M a t r i c e s w i t h R a t i o n a l I n t e g e r s as Elements,  14(1964)., 719-726.  •  . .  

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