UBC Theses and Dissertations

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UBC Theses and Dissertations

Linear transformations on matrices. Purves, Roger Alexander 1959

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LINEAR  TRANSFORMATIONS  ON  MATRICES  by ROGER A L E X A N D E R B.A.,  University  A THESIS THE  of B r i t i s h  SUBMITTED  MASTER  FOR OF  THE  1957  Columbia,  IN PARTIAL  REQUIREMENTS  in  PURVES  FULFILMENT DEGREE  OF  OF  ARTS  t h e Department of MATHEMATICS  We  accept  required  THE  this  thesis  as conforming  to the  standard.  UNIVERSITY  O F ' B R I T I S H COLUMBIA  April,  1959  ABSTRACT  In  this  formations complex  o n M^,  w h i c h map  concerning  of n-square  sum  matrices  f o r some  of the r x r p r i n c i p a l  what  f o l l o w s , we  "direct  product"  use E  T(A)  over  the  i s the  transformations matrices; those  positive  preserve  integer  subdeterminants this  sum,  to transformations  = cUAV  trans-  of the s t r u c t u r e of  to denote  to refer  matrices  to non-singular  i s the determination which,  linear  The f i r s t  of the s t r u c t u r e of those  transformations  In  the algebra  non-singular  second  the  two p r o b l e m s  numbers, a r e c o n s i d e r e d .  determination  the  thesis  for a l l A  r,  of each  matrix.  and the  phrase  of the  form  in M n  or where  U,  T ( A ) = cUA'V V are fixed .  The  main  members  result  singularity  preservers  products.  The  r=l,  cases  i t i s shown  structure.  products.  which  Finally,  have  that  preserve  forthcoming  results paper  b y M.  of Mathematics, of M a t r i c e s :  Funct i o n s .  Marcus  proofs  no  Invariance  number.  nonk,  are  direct  separately.  If  significant a r e two  Eg, and which that  these  types  of  are not counter  r=3• will  also  a n d JR. P u r v e s  entitled  both  i f r >  there  i t i s shown  and t h e i r  n  i s a complex  i s that  do n o t g e n e r a l i z e t o t h e c a s e  These  Algebras  and c  are discussed  i t i s shown  direct  in M  and E ^ - p r e s e r v e r s ,  E^ p r e s e r v e r s  transformations  Journal  n  of the thesis  f  linear  examples  of M  r=l 2,3  that  I f r=2,  for a l l A  Linear  be  found  in a  i n the Canadian  Transformations  of the Elementary  of  Symmetric  In p r e s e n t i n g  this thesis i n partial fulfilment of  the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make it  freely  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 .  agree 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  I further •  copying of t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  t h a t c o p y i n g or 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  The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 8, Canada. Date  2-6  >  /?(Pf  permission.  ACKNOWLEDGEMENTS  Without it  i s doubtful  appeared.  the contribution whether  Over  a period  shown  a constant  to  the elementary  me  found for at  more  and s o l v i n g  mundane  education. the I  major  would  like  years  ideas  student  contained  he h a s patiently I have  and t o keep t o complete  with  deep  an  the fact  i n the thesis  my  have  of obtaining  along  Marcus  h i s techniques  new p r o b l e m s ,  requisites  to express  would  to explain  freely  efficient  For this, ideas  of five  mathematical  to reveal  a not e n t i r e l y  the  thesis  willingness  puzzling,  posing  this  of Marvin  that  are h i s ,  gratitude  t o Dr.  Marcus. I thank  like  the National  financial this  would  t o take Research  assistance  thesis.  given  this  opportunity  Council during  to  of Canada f o r  the w r i t i n g of  TABLE OF  1.  INTRODUCTION  2.  NON-SINGULARITY  3.  E -PRESERVERS  ^  45.  CONTENTS  1  PRESERVERS  2  6  r  Eg,  BIBLIOGRAPHY  17 37  1•  Introduction Let  the of  M  denote t h e algebra- of n-sguare m a t r i c e s  n  over  complex numbers, and GL^ t h e s e t o f n o n - s i n g u l a r members M . n  In t h e f i r s t  structure  of those  p a r t of t h i s  linear  t h e s i s we d e t e r m i n e  t r a n s f o r m a t i o n s - T of M  the  into M n  having  the p r o p e r t y In  those  t h a t T ( G L ) _• GL^. n  the second  linear  p a r t , we d e t e r m i n e  t r a n s f o r m a t i o n s of M  the s t r u c t u r e of  into M n  _ 4f p r e s e r v e  r  of are  The thesis: the  The r e m a i n i n g  i n the f i n a l  A » , J>(A), t r ( A ) ,  (A)  o f A, t h e rank  cases  subdeterminants.  of r = 1, 2 or 3  section.  following notation w i l l  transpose  w h i c h , f o r some n '  t h e sum o f t h e r x r p r i n c i p a l  t h e members o f M . considered  n  be u s e d  throughout  , det (A), denote  ±  the  respectively  o f A, t h e t r a c e o f A, t h e e n t r y  at of  p o s i t i o n ( i , j ) , t h e d e t e r m i n a n t o f A, where A i s a member M . In a d d i t i o n , 0 , I , E . , , denote r e s p e c t i v e l y t h e n , * n' n l j ' n x n zero m a t r i x , i d e n t i t y m a t r i x , t h e n x n m a t r i x w i t h a 1 at p o s i t i o n  ( i , j ) and z e r o s  of  p-tuples of p o s i t i v e  of  a l l p-tuples We w i l l  M  n  into M  n  The c o l l e c t i o n  i s d e f i n e d as t h e s e t  where 1 <  o f t e n be l e d t o l i n e a r  <••-< i ^ < n .  t r a n s f o r m a t i o n s of  o f t h e form = cUAV  or  T ( A ) = cUA»V  U, V f i x e d members o f M , and c a s c a l a r . * n'  formations The  integers Q  w =  T(A) for  elsewhere.  we w i l l  motivation  speak of l o o s e l y  as " d i r e c t  i s that, f o r the f i r s t  Such t r a n s products".  alternative,  regarding  -  the  members  £l]  the matrix  of  V  2>  and  T  of M  as n  n  first  transformation  then  mapping  i s itself  i s similar  to a matrix  with  from  linear  of  M.  to  a matrix  is  the matrix  Marcus  and J . L . McGregor whose  diagonal  E-j^z we  1.  l / ) . . . 1  l /  n  follows  canonical trace,  [ 2 ] we  vectors  U  (  2  entries  complete  )  - -(n-1)  column  U^ ^,...,U^ ^ 3  n  by a  of  A.  result  J(A) i s similar tr(A). the  If J(A)  similarity  •J  n  _  x  vector  Then  with  a l lentries  Normalize so t h a t  and  the set of  the ( i , i )entry  of  vectors UE-^U*  zero.  the proof  by d e f i n i t i o n .  form  n  ) i s orthonormal. i s not  have  to construct  are orthogonal.  u.,u._ w h i c h 11 i 2 We  from . zero  Set  k-component  and determine  to non-  a l ldiagonal  entries are n  proceed  as f o l l o w s .  i s the  K  a  different  n  and has non-zero  D<*> - J J  that  matrices  Let J(A) denote the Jordan  J(A) i s diagonal  is  lemmas  zero.  If  and  product  non-singular.  of M  transform  as i n  i s the direct  non-singular  i s a member  Proof.  where  i n three  If A  different  vectors  preservers  establish  matrices  2.1. A  of T  column  U.  We  Lemma  dimensional  representation  Non-singularity  singular  2  2 -  If A  by  induction.  is in M  The c a s e  n=l  ^ and J ( A ) i s d i a g o n a l  - 3 -  with  zero  trace  from  zero.  and column  two  non-zero  all  diag  form  eigenvalues  B obtained  there  diagonal  zero  (l,V) j(A) diag  (l,V  -  1  such  )  matrix.  Then,  1  has  product  diagonal  (A,, V B V  - 1  ) trans-  entries.  c a n assume  o f J(A) i s 1 and t h e submatrix  t h e zero  VBV  i s a similarity  we  least  By t h e  that  = diag  of J(A),  J(A) i s not d i a g o n a l ,  j ( A ) has a t  i s zero.  The  of A and has a l l non-zero  entry  since  i s a V'"£  entries.  i s different  by d e l e t i n g t h e f i r s t  i f i t s trace  i s the ( l , l )entry  If  not  the ( l , l )entry  o f J(A) i s . n o t  hypotheses  non-zero  where X  c a n assume  The s u b m a t r i x  row  induction  we  t h e (1,2)  B, d e f i n e d  as b e f o r e ,  we h a v e  as above i s a V in M n  such  that  the product, P = diag  has of  a l l non-zero  (l,V) j(A) diag  e n t r i e s on t h e d i a g o n a l ,  - 1  )  i f the ( l , l ) entry  P i s not zero. If  (1,1)  t h e (1,1) e n t r y  entry  o f VBV  1  i s zero,  there  non-zero  represents diagonal Lemma  1°  diagonal  entries.  diag  n  a similarity  _  1  * b  1  J  Then  ) P diag  transform  as t h e  that  \i f  l  i s taken  the product ( i f  1  , I ^ )  o f A and has a l l non-zero  entries.  2.2;  — — — — — —  /0  (U, I  and b ^  i s a U such  U  has  (^V  F o r any A £ 0 i n t h e M n  there  i s a Z in M n  - 4 -  such  that  A + Z h a s no e i g e n v a l u e s  eigenvalues  Proof. P  lemma  ^AP h a s a l l d i a g o n a l  Define  X =  x.. ij  entries  1  = -  (P  1 +  _ 1  itself  t o 1.  2.3.  I t i s then  We  some  In  A  i > j  equal  to 1 while  i=l,...,n,  easily  P ^AP + X h a s  none  verified  Any T w h i c h p r e s e r v e s  have  of which are  that  Z = PXP""  1  non-singularityi s  that i f  x,  other  a subset  - [T(I)]"  - A) = 0  words,  T(A)] = 0  of the d i s t i n c t  have  that  1  T(A  T ( A ) = 0.  +  x.  eigenvalues  eigenvalues  no e i g e n v a l u e s [T(I)]-  f o rthat  the distinct  i s a n A /= 0 s u c h  + Z, Z  1  then det(xl  there  zero.  property.  det[xl  are  from  that  non-singular.  Proof.  for  different  AP).. ' i j  (P ^ A P ) ^  the required  Lemma  such  rt  i < j  X has n e i g e n v a l u e s  eigenvalues  has  i s a P in M  i=l,....n  x. . = 0  equal  there  (x,.)as f o l l o w s . x..=  Then  with the  o f Z.  By t h e a b o v e  — — -  i n common  of  o f A. Choose  i n common. Z) = [ T ( I ) ] "  We 1  [T(l)]~ T(A)  Now  1  suppose  a Z such have  T(Z).  that  - 5 -  From  this, equation  eigenvalues  of [ T ( l ) ]  eigenvalues  of A  eigenvalues  of [ T ( l ) ]  eigenvalues  o f Z.  the  lemma  Theorem then  is  + Z.  of  M  such  and  lim B ^ r — > <*» . By we  T(Z) are a subset  1  This  that  distinct  the  of the  contradicts the choice  distinct  distinct of Z and  n o n - s i n g u l a r i t y and T ( l ) = I  set of n d i s t i n c t  s e t of e i g e n v a l u e s .  eigenvalues  I f T(A) does  then  not  have  =T(A).  the continuity  of T  - 1  ,  as B ^ — T ( A ) ,  T  -  1  ( B ^ ^ )-> A . r  a s r -—> oo  =  B  5> e . v . T ( A )  r  e.v. T ~ ( B ) •  the eigenvalues  singular  of the  distinct  a l l eigenvalues.  e.v.  If  have  the  x  have  Corollary  also  that  (r) e i g e n v a l u e s , l e t [B } be a s e q u e n c e o f members (r) that B has n d i s t i n c t eigenvalues f o r a l l r  -. n d i s t i n c t  and  B u t we  I f T(A) has a  A h a s t h e same  Then  T(Z) are a subset  1  If T preserves  T preserves  n  can conclude  proven.  2.1.  Proof.  we  1  r  -> e . v . A  of T(A) are the eigenvalues  o f A.  2.1. T preserves U,  V  such  1  or  n o n - s i n g u l a r i t y , then  that  either T(A) =  UAV  T(A) =  UA»V  there  exist  non-  -  Proof. [T(l)] there  If T preserves T  1  preserves  exists  that  [T(I)]  3 •  —i  this  1  fora l l A  in M  n  fora l l A  in M  n  b y m u l t i p l y i n g on t h e l e f t  s e c t i o n we  transformations  r x r principal a group.  first  T on  which  there  the trace  with  zeros  i f r >  preserve of every  are linear  but which  i s a p r o j e c t i o n onto  a l l matrices  show t h a t  subdeterminants  If r=l,  preserve  example of  1  by T ( l ) .  .  linear  which  = UAtT  Preservers In  form  T(A)  follows  3 °f [ 3 ]  either  1  the corollary  the  _ 1  By T h e o r e m  [ T ( I ) ] " * T ( A ) = UA'U"  and  E  non-singularity, the transformation  eigenvalues.  a U such  or  6 -  have  t h e sum o f member o f  transformations no  the subspace  a t some  2 the  inverse. of  fixed  An  consisting  set of o f f - d i a g o n a l  positions. If of  then  3.1. *  we  n  If r > ~  Suppose  shall  Lemma  diagonal  2.1  A  entries.  r  r  (A) t o d e n o t e  (A) = E T ( A ) f o r a l l A r v  r  T(A) = 0 , r  E  and E ( A ) = E ( T  E ( A + X) = E ( T ( A r  write  t h e sum  subdeterminants.  2 and E  T i s non-singular  Proof.  By  i s in M  the r x r principal  Lemma " ''  (1)  A  A  r  0.  _  1  (A)).  Then  + X)) = E ( T ( X ) )  i s similar  in M n  r  to a matrix  =  with  E (X). r  a l l non-zero  - 7 -  Let  X =  ( x . . ) be d e f i n e d a s x,,  = x  i=l,...,r-l  x. . = 0 ii x. . = ij  i= r,•..-n ' '  0  i < j  x. . =  -(P  IJ  where By r—  P "*"AP h a s a l l d i a g o n a l  (l)E  (P"" AP  must  entries  entries  be z e r o .  y  i > j  i j  J  entries different Therefore  we  from  zero.  the coefficient  of  c a n show  assuming  diagonal  n-r+1  t h e same  that  n or 2 <  e n t r i e s which  sum e x c e p t  that a  " B S  Then  includes B replaces =  by c h o o s i n g  X diagonal  To  to zero. show T  1  Thus  E_(Y) It  follows  = E T(T  transformations  form  a group.  such  a group  We  E - 1  r  easily  4»  f  (X.  Then,  B, a n d  from  =  E T r  lemma  preserve  This  a n d T has an  inverse.  :  this  now p r o c e e d  f o rr >  b e t h e sum o f  a and excludes  be z e r o ,  (Y))  from  which  l e t  e n t r i e s o f P "''AP a r e e q u a l ,  A must  preserves  r , l e t a,B be a n y  0  a = 8, a n d a l l d i a g o n a l  equal  linear  Thus,  (n-r) + 1  t h e sum o f a n y ( n - r ) + 1  n-r+1 <  s  and  o f P "**AP.  e n t r i e s o f P -^AP.  diagonal  have  i s t h e sum o f t h e l a s t  o f P "*"AP i s z e r o . Now,  two  This  on t h e d i a g o n a l  suitably,  we  AP) , .  1  x  Sp  + X ) = 0.  1  _ 1  _ 1  (Y).  that  t h e s e to f  E , f o r some  to determine  r >  2,  the s t r u c t u r e of  i s the content  of  -  Theorem  3.1.  If  E  E  for  (A)  =  T(A)  X  in  some  a l l A  r  M  such  n  By for  some  that  in M  r >  4/  Lemma  UV  a  deg(det(xA  UAV  for  a l l A  in  M  T(A)  =  UA»V  for  a l l A  in  M  =  e  I where we  rank  Let  A  £  B))  <  1  of 0  rank be  a  for  can  has  eigenvalues  i t cannot  that  A  eigenvalue  than  deg  T  exist  U,  V  n n  which  preserves  first  which  to  B  A  most  in  one  follows  member  in M  four  of  of  M  i f and  n  E^,  lemmas E^,  preserves  matrices  at  1  at  Jordan  rank  n  .  for 1.  Then  only  i f  superdiagonal  If A  entry.  immediately.  There  position  ( l , l )being  form.  non-zero  have more  ( l , l )p o s i t i o n .  superdiagonal.  As  n-1,  277") .  Our  fixed  is  zero  the  at  A  this  i n the  A  assume  eigenvalues  greater  r <  A  1.  We  on  1  any  zero  where  a  transformation  1 then  the  (mod  is non-singular.  linear  +  r 6 =a. 0  know t h a t  rank  be  there  =  maps m a t r i c e s  Proof.  , then  4 <  T(A)  2,  r >  3.2.  of  that  either  Lemma 3 « 1  show t h a t  such  It  and  to  -  J.  or  is  for  3  at  mean  is  If A If A  one.  cannot  (1,2)  also  would  than  be  the  of has  has  a  1  and  on  a  rank  a l l  non-  Assume  impossible  (2,2); that  is  this  the without  1 of  elseA  1. has  det(xA  only  one  non-zero  +  <  for  B)  1  a l l  entry B.  i t follows  1  easily  is  -  In t h e  other  has  a t most one  the  s e t of n o n - z e r o  A i s i n Jordan  9 -  direction,  non-zero  we  first  eigenvalue.  e i g e n v a l u e s A,,  form  with  Suppose A  , ...,A,.  e i g e n v a l u e A.. 1  Choose B d i a g o n a l w i t h  (i.. ij). f  (ij/ij) Then  j=l#"««#k  and  d e t ( x A , ++  l's  0*s  k >  in  x,  1,  det(xA  and  so A has  We diagonal  TT  B) B) =  entries such  If  there  form of  that  1.  Let  one  0 and  zero.  there  with  that  in position j at  positions  (\.  x)  -  ( TT // X. )x j=l j  j  non-zero  eigenvalue  show t h a t  k  Let  i  be  Q  at p o s i t i o n  the  B =  n > 2 and  (b,.)  and  be  X*  largest (i ,i Q  (1,2-) of t h e  e i g e n v a l u e s X,0,  1  a l l super-  i s a 1 at p o s i t i o n  i s a zero  of a m a t r i x  rank  a t most  are  has  w o u l d have d e g r e e g r e a t e r t h a n  assume X t  now  integer n=2,  + B)  and  A  elsewhere.  j=l If  show t h a t  A  +l). Jordan  is clearly  defined  as  follows b . . = ii  0  b. , = 1  i£  11  l  o  b,,  i =i  1 = 1  i  o  i£  o i  +1 o  +1  +1,1  ' o  = 0  elsewhere.  Then det(xA If X t must  be  a zero  0, at  + B)  in order  = -A.x  -  t h a t deg  ( i , i +l).  x. det(xA  Repeating  + B) this  <  1,  there  procedure,  we  can  if  7V. *  show t h a t  1.  that  10  -  are  no  l*s  above  the  diagonal,  0. Now  is  there  -  a s s u m e A, =  Once there  again,  is  a  1  0  and  let  at  that  i  be  the  the  position  (1,2)  entry  largest  integer  ( i ,'i +l) . o' o  If  i  B  =  (b..)  such >  2  as  b.. ii  =  0  i = l , 2 , i +1,i . ' * o ' o  b ^  =  1  elsewhere  b.  .  1+1,1  o  b  A  o  v  define  of  21  the  diagonal.  =1  . '  on  o  -  1  =  0  elsewhere  off  the  diagonal.  Then det(xA In 2  <  i <  cannot  this  n-1, be  a  way,  can 1  (2,3)  =  shown  implies  that  the  Lemma  3.3  and  A  £  for  a l l B  0  a in  Jordan  thus  A  Let  r  an  M  ,  3  that  and  fixed  x . 2  in positions  be  ]  /  I  A  n  A  3  show t h a t  there  ) .  deg(det(xA form  i s of  member then  To  (i,i+l),  set  d i a g ( E  have  entry,  =  eliminated.  We  zero  B)  a l l l*s  be  at B  B  +  of  rank  integer of  M^.  has  at  A  +  B))  has  <  at  1  for a l l  most  one  non-  1.  such If most  that deg one  3 <  r <  E (xA  +  r  non-zero  n, B)  <  1  eigenvalue,  -  Proof.  Assume A  A,, , . . . .A, . 1 ' n Then  11 -  i s i n Jordan  form  with  Let B = diag (z...... z ). \ 2 ' n f  E (xA  + B) =  r  TT  S  /.  . \,_ „  w= ( 1 . . . . . 1  We  have  r  ;eQ  r  k=l  with  k  S  x  k  — t =0  i s t h e sum  t members  and S  over  TIV  a l l subsets  set.  + z. )  k  k  • TTox*  —> a e s " S t ^ w  i s t h e empty  q  (xA..  k=l  TT (XA.. + z. ) - *y c v where  eigenvalues  Bew-S?  of w =  We  ( i  c a n now  l  t  . . . , i ) r  write  r  t=0 and  for t >  a  -  t  y  Z w£Q  T  for  2,  any c h o i c e This  we  have  x rn  ^  weQ  y * S,<Zt —  TT v TT z  =  ^  X  ^  ve<2„ .„ r^-tn  From symmetric  this  = o  P  .  "TT TT  t  -  z  c a n be c o n s i d e r e d  z. ;^ a s f o l l o w s : s  z  BEW-S  a  PEW-  w  of z.,....z I n  sum  rn  aeS  .  S. <ZW t - ^  we  pev P  "N ^ -  Z  B  (  ^  as a p o l y n o m i a l  TT x ;  TT  y *  N  s <S(Q - v ) t nn v  can conclude  i n the  aeS.  a  1  7  that  the t - t h elementary  f u n c t i o n o f a n y n - ( r - t ) o f t h e A.^,...,A.  n  i s zero.  -  12 -  In  particular,  the second  of  any n-(r-2)  o f t h e A»'s i s z e r o .  are  equal  and  that  k  they  are equal  n'-(r-2) <  = n-(r-2),  we  We  Then,  can choose  r >  a set of i .  3  i f a l l t h e A,*s  3 and  A,  £ A.^  c  setting  eigenvalues  d for j=l,...,k-l.  3  have 0  = E ( A . ,A.. ,...,A, ^ l 9  c  1  x  k  _  ) = A. E ( A , . i c i i 1  +  E  1  0  ,...,A. i k  1  )  1  k - l  = E (A,,,A,. ...A. ) = A-.E^A.. , ...,A., l k - l a ± i i  )  9  x  1  1  k  + E (A.  -  1  , ... ,A.  ?  1  A  then  )  -  (A.. , . .. ,A.  *  We  function  Assume t h a t  since  i . £ c,  1  symmetric Now  to zero.  n.  A.. ,...,A,. where l . k - l x  elementary  )  k-1  have (A. - A, ,) E.. (A.. , . . . ,A.. ) = 0 - c d 1 i, ' ' l , '  1  If A,^,  r >  i£ c , d  3.1.  Now  3 then  n  k-1 <  are equal  we  k-1  n-2  to zero  and t h e e i g e n v a l u e s by t h e argument  o f Lemma  can w r i t e A, A, , = 0 e d  and  i t follows  that  A  can have  a t most  one  non-zero  eigenvalue. If  r = 3 we  have  that  the second  symmetric  function  A.  of  any  ( n - l ) o f t h e A., i s z e r o . j  E . (A,, , • • ., A,. , A.. . , . . . ,A, ) i i * j - 1 j +!' ' n  L e t E.(A.,) i j  i= l,2. '  We  first  denote show  -  A, j E ^ (A, j )  j=l,...,n  is  13  -  zero. A  A.  E (A. ,... A. ) 2  1  /  = Tv-.E^A,..) +  n  E (A..) 2  = A.. E . (A.. ) J 1 J Summing  these  n n  E  equations  2 ^ 1 ' * *"  =  2  2 ^ 1 ' *' *  E  A  Since  n >  2  E  (A. ,...,A, )  *~ l Setting  j=l,...,n.  =  0  and  A,.E.(A..) =  n  X  j  0,  j  A.. j=l we  have A. J  = A.. s J  2  A,. (Tv.. J J Thus  the  non-zero  were more function  than of  completes  Lemma  3.4.  integer for  such  a l l B  Proof. rank any  1,  in M  We A  A  n+3  can  has  principal  function.  case  that  0  non-zero  n-1  Let  =  e i g e n v a l u e s of  one  any  the  s)  of  t h e A, s  fixed  r <  n  + 3 .  i f and  a t most  A one  only  deg  If there  second  not  member Then  i f A  be  symmetric  zero.  This  non-zero  E^(xA  of +  B)  „ a n d r an n+3 d e g ( E ( ' x A + B) < 1  i s of  i s i n Jordan  subdeterminant  Thus  would  T  a  assume  equal.  eigenvalue the  0 be  r <  are  r=3.  that  £  A  +  <  1.  B)  M  r  rank  form.  entry.  (xA  of  1.  If A  i s of  It follows i s a t most  a  that linear  -  14 -  We now p r o c e e d by i n d u c t i o n on t h e " o n l y S ( l ) i s t r u e by lemma 3 - 2 .  of t h e lemma, S ( n ) . be E r  r  in M  ,. n+4  i f * part Let A  We have an r s u c h t h a t 4 < r < n + 4  —  (xA + B) i s l i n e a r ' •  in  v  x  and  —  f o r each B i n M  .,. n+4  If  = n + 4# A i s o f rank 1 by lemma 3 « 2 . If  zero and  r < n + 4# by lemma 3 - 3 A has a t most  eigenvalue the (2.3)  A..  In A t a k e  A. t o be i n p o s i t i o n ( l , l )  t o be e ( = l o r 0 ) .  entry  positions  (3/2)  and r - 3  positions  ( i , i ) i > 3,  l * splaced with  one non-  zeros  L e t B have a 1 a t  i n any o f t h e d i a g o n a l i n a l l remaining  positions.  Then E (xA  + B) = -A.- £• x  r  I f A, ± 0 and £ = 0 , row and column  2  o r A. = £ = 0 , we have t h a t  o f A have o n l y  entries.  B to those matrices  with  column, we s e e t h a t  t h e h y p o t h e s e s o f S(n+3)  Therefore  zero  zero  the second I f we  restrict  e n t r i e s i n t h e s e c o n d row and  the submatrix of A o b t a i n e d  are s a t i s f i e d .  by d e l e t i n g row and  column 2 o f A i s o f rank 1 and hence A i s a l s o . I f A. = 0 , we  show f i r s t at ( 1 , 2 )  where £ ^ i s t h e e n t r y Let  B be d e f i n e d  t h a t A cannot and  the entry  as f o l l o w s : b  h  21 i  ±  b..  =  b  n+4,n+3  =1 = 0  3 < i < r-2 elsewhere.  have £ ^ = £,-, = 1 at (n+3,n+4).  -  + B) = e^c^x  Then E ( x A r  Then t h e i n d u c t i o n the  15 -  and we pan assume  argument  applies  ~ °-  as b e f o r e t o g i v e  r a n k of A as 1.  Lemma 3 . 5 .  I f f o r some  r  such that  4 < r < n  E T ( A ) = E (A) f o r a l l A i n M t h e n m a t r i c e s o f r a n k r r n a r e mapped t o m a t r i c e s o f r a n k Proof.  L e t A be o f r a n k  1.  lemma, d e g ( E ( x A  + B) < 1  E T(A) = E (A),  deg E^(xT(A)  r  r  r  non-singular, T(A)  i s of rank  Lemma 3 . 6 .  for  this  Proof.  Let  1 by T. Then by t h e p r e c e d i n g  f o ra l l B i n ^ . + T ( B ) ) < 1.  r  such t h a t  4 < r < n  n  A be o f rank r .  We can w r i t e  i =l where C. i s o f r a n k  T  exists  Since T i s  1.  r  Since  Since  h o l d s f o r a l l members o f M , and t h u s n'  I f f o r some  a l l A in M  1  1 = 1, • • .r,.  1  Then  r  r  i =l  i =l  and p r e s e r v e s E  we have  -  We of  this  rank  are  now  in  section.  we  have  By  T  write,  has for  position  to  prove  c o r o l l a r y 2.1,  the  since  main  T  theorem  preserves  either  or Since  a  -  16  an  T(A)  =  T(A)  = UA'V. V  inverse  a l l B  in  UAV  U,V  are  non-singular  and  we  can  M n  E (U(VBV~ )V)  =  1  r  Setting  UV  =  P  our  E^VBV" )  condition E (PB)  =  r  If  will  =  or  tr(C (P)  I)  on  of be  the  C (B), r  r  r  -  r  different entry  E (B)  tr(C (PB))  t  I)  from  we  can  zero.  C^CB) the  0  is  1  matrix of  which  set  we  the  P  =  e  l 6  I, For  of  theorem  r6  that  the  r  B  such  entries entries  subdeterminant  equal  to  1. -  conclude  0 ( i , j ) entry that are  the  at  (j,i)  zero.  except  is  for  B those  (j,i) position  of  Then  I)  C (B))  C (P)  =  r  r  *  0.  I which  implies  [4]  ==0(27r) .  later 3»1  we  =  a l l zero  r  contradiction  C (B))  choose  tr((C (P) By  r  a l l other  with  diagonal  tr(C (B))  assume  Then and  E ^ (B)  becomes  or  (C^CP) -  =  1  as  reference  we  state  this  part  of  the  proof  - 17 -  Lemma  3.7.  Suppose  E  (UAV) = E r  where UV  r  = e  4•  l  6  i s fixed  I  E  In and M  n  used  E  12  +  for  E  E  °  l s  f  section,  , f o r some  Then  r  a  n  we  a  linear  4.  r >  determined  For r = l,2,3,  In t h e c a s e s  lemma  i s not true.  k  2  a  n  d  E  3«4  3^ 12 E  +  necessary  t r a n s f o r m a t i o n of  4 fails.  F o r r=3,  34  n.  conditions that  for r >  clear.  r <  3"  sufficient  ing  and 2 <  the preceding  .preserve  in M  r6=0(2ir).  where  E^,  (A) f o r a l l A r  E  34^  +  B  ^  the  r=l,2,  reason-  this i s  The 4 x 4 i s  l  i  n  e  a  matrix  r  a l l B i n M. .  4  The there  question  are linear  r=l,2,3#  subsets  now  arises  i s whether  transformations which  and which  We  that  are not d i r e c t  are going  to look  preserve  E  f o r some  r  products.  f o r such  of t h e s e t of l i n e a r  or not  transformations  transformations  on M  i n two  .  The  n first  of these  which  arise  example is  here  i s the set of l i n e a r  from  permutations  Hadamard That  of entries  i s the interchange  the set of l i n e a r product  transformations  o f two  transformations  of every  matrix  with  (c,  . a .. )  The  consisting  some  i s C : ( a . , ) -—>  of matrices.  entries. C  S  fixed  An second  of the  matrix.  -  for  a l l ( a . ..) i n M  C  ( c , .) h e r e .  13  ^  is  C  1J  a member  formations  E^:  Here  preserve  n  of  is a  .  We  any S w h i c h the trace.  ( ^j)h  E^:  Lemma  c  a  s  U.l.  linear now  on  the set of pairs  at  a subset  preserve  leaves  involves  entries  on t h e m a i n f o r any C  two  l  T  entries  of S  .  such  order  have moves  two  that  the entries  leaves  Eg  w h e r e ir^  7Tg i s a  permutation a n d 7T^,  entries,  located  positions,  entries.  on a d i a g o n a l zero  matrix  at ( i , j ) ,  entries  that  SD  be  zero.  these  on t h e m a t r i x  a  In  diagonal  Thus  S  E,. + E,..  that  located positions.  ( j , i ) t o some  entries.  with  7T^ .  entries  l o c a t e d p o s i t i o n s and then  fixed  D  on t h e d i a g o n a l .  i t i s necessary  l ' s at symmetrically  symmetrically or  S preserve  will  that the  31  i j  In  ( c , ,) 1 j  diagonal  = ^-^^^  located  of the diagonal  the effect  S  only,  i t i s necessary  s and r e m a i n i n g  a permutation  Consider  Eg  of S  n  1.  Eg then  entries  the e f f e c t  S preserve  with  holds  located  on M  that  E^, Eg.  of symmetrically  symmetrically  Mote  to characterize the trans-  entries  T h e same  l * s and t h e r e m a i n i n g  matrix  proceed  of diagonal  Consider  that  ( c , ,) i n M . ij' n  of the set of symmetrically  interchanges  order  v  transformation  If S preserves  a permutation  Proof.  fixed  a l l diagonal  is  two  a n d some  C and S which  fixed  18 -  other either  s  ( .jj E  +  Thus pair  ^j ±~) S  of  interchanges  -  We following  can w r i t e  -  19  any m a t r i x  (a,.) = A i n M as t h e 1 j n  sum A = D + y  P, . 3  <_- a. i  where  <  D = diag(a  j  ...,a  1 1 #  ) nn  ^ 1 1  P. . =  lj  ( a . .E, , + a . ,E . . )  Then  <-—^ ij.  SA = SD  < 3  i  sp.;. i  The  effect  <  o f S on P.,  j  i s known.  We  have  either  S ( a . , E . . + a,.E..) = a,.E.-, + a . . E . ij i j ji ji i j kZ j I -ik f  or S (a . . E . . + a..E.,) = a , . E, „ + a, J . , ij i j j i] i j i kl 13 -Zk We  can c o n s i d e r t h e e f f e c t  following  order.  ^ i  S takes  o r n o t , d e p e n d i n g on S. \  < J  S P  ij  a  s  ' l' 2^ w  r  ^ i j ) P  i < j  to take  the p a i r  (k,-?) , (-2, k) and t h e n  to a^,  First  of S on P,.  place  of e n t r i e s  interchanges  Then we  a^  i n the at and  can w r i t e  where T  2  is a  permutation  -  on  the s e t of p a i r s interchanges  depending  20 -  of symmetrically  or leaves  on t h e e f f e c t  fixed  located entries  th.e e n t r i e s  o f S on P ^ j -  Then,  effect  o f S on t h e d i a g o n a l  effect  o f S on t h e o f f - d i a g o n a l e n t r i e s  in  entries  at  (i,j),(j,i)  since the  i s independent we  and  have  of the  for a l lA  M n S(A)  Lemma Then  4.2.  c.. = c,. ij jl  c . , = — 1, ii Proof. A  Suppose  =  V^^W^K  C:(a^)  and e i t h e r  1  =>(c^ja^^) p r e s e r v e s c , . = 1,  ,  1  i=l,...,n  1  Eg. or  i=l,....n. ' L e t i , j be f i x e d  i n t e g e r s between  1 and n.  Define  as f o l l o w s : a..=x.. n i l  a . . = x., i j i j  a . , = x . , J J JJ a. . = 0 kl Then  elsewhere,  E _ ( A ) = x..x,. - x..x.. and E „ ( c ( A ) ) 2 i iJJ i jj i 2 V  = c..c.,x,.x.. n JJ n JJ  V  c..c..x..x... ij j ii j j i expression  i i  gives  S e t t i n g these  identically  (c..c,,  This  a..=x,. Ji J i  JJ  -  zero  1)x,,x..  us c , . c , , i i JJ  i i  JJ  equal  f o r any n  we  have  the following  x,.,x,.,x..,x.,. j j i j j i  + ( l - c..c..)x,.x,, i J  J i  = 1 f o r any i f i .  IJ  J i  = 0  -  -  As  n >  2 we  have  the  21  -  equalities  c . , c , , = 1 i i 33  for  any  and  that  either  now  c, , c . , = k k 33  1  c.. ii  product  I  This  ir(i)  s i n c e by  l*s  at the p o s i t i o n s s i n c e row  L e t N be  (i,7r(i))  j consists  Let  N have  off  diagonal positions.  at  (i,i)(j,j)  f  s at  and  I  X  by  I  7T^, ir^,ir^ exhibiting,  non-  matrices  l e m m a 3-7  a  by  direct  ( j , j )to position  (i,j)  and On  s u c h t h a t 7T ( j ) a permutation  of at  (j,i)  (i^,ij).  = j  and  matrix  with  T h e n ir^U' i s  for i=l,...,n.  S u p p o s e ir^ m a p s e n t r i e s l  at  7T o f l , . . . , n  i for a l l i £ j.  singular  i=l,...,n.  preserves non-singularity.  i s a permutation ^  1  i s done  S u p p o s e TT^ m a p s t h e e n t r y There  o r c . , = +1,  I  a r e mapped t o s i n g u l a r  i s sufficient  preserving  I  c , , = c , . = c,,  the transformations  N which  This  ,7r  imply  i=l,...,n,  products.  singular matrices  iri» 2 3'  These  = -1,  show t h a t  not d i r e c t  7r  =1  i ,j ,k distinct.  We are  c . . c. , n kk  zeros. (i,j) and  (j,i)  zeros  the diagonal  l * s elsewhere.  Then  to  (k,J?)  i n the  remaining  l e t N have ^  s  C^,k).  zeros  singular  and  N i s non-singular. S u p p o s e 7T leaves  fixed  1  those  interchanges at  (k,^),  entries  C^,k).  at  (i,j),  (j,i)  and  I t i s not d i f f i c u l t  to  -  find  a non-singular  N  22 -  s u c h t h a t 7T.. N i s s i n g u l a r . 4 1 7T^ i n t e r c h a n g e s t h e e n t r i e s a t (.1,2) ( 2 , l ) i n M„  or M  3  For  example,  and  holds  fixed  those  E  + E„„  + E, „ 4J  i s non-singular  £j 0  0  We  suppose  proceed  to  at  (4,3)  (3,4).  construct  7T. E 1  and an  Then  N  in M  Then,  none  set  mutation 2  and  P  equal  rows  of  IT o f  following equalities = k  i =  1  j  = k  j  1.  + E.. ^ + E ^  l,...,n leaves  such  identity  Similarly  we  can  find  We  now  show  that  the 7T»  1  where We  R  have  = PQ, that  maps tr  %  such  fixed  4 x 4  non-singular  singular.  the  find  E  fixed.  derived  21  per-  1 and j , Then,  by  setting  permuting  t o 7T we  V  +  a  will  the  have  +  that  = E  1  2  +  E  2  1  • E  3  •  4  E ^ .  transformation : A — ?  R7T, ( A ) R '  1  a non-singular  interchanges  holds  +  E  can  7T i n t e r c h a n g e s  according  QPir^P.Q.  We  k  numbers  " 12  a Q  =  E^, .  matrix  matrix  ^ i V  +  that  remaining  to the permutation the  occur:  i  = E ^  i , and  of  the  + E„. + <• 1  f o r n > 4.  n Suppose  = E., 14 singular.  is  E  entries  at  matrix  It follows  the  (3/4) C  such  matrix  to  entries (4,3)• that  a  at  singular (1,2)  Then  we  7T* d i a g ( C , I  t h a t 7T^diag(C, I _ ^ ) n  ( 2 , l ) and can  n  matrix.  ^)  find  a  i s non-  i s singular. '  -  If find,  any o f t h e e q u a l i t i e s  as b e f o r e ,  an R s u c h IT  interchanges entries  above  to f i n d  that  above  the modified  hold  we  can  transformation  1  at (1,2)  the entries at  listed  : A—>R77\ ( A ) R »  1  1  the  23 -  (3»l)  (l»3)-  We  a non-singular  ( 2 , l ) and h o l d s  and  can proceed  N such  fixed  e x a c t l y as  7r'N i s s i n g u l a r .  that  1 Thus and  we  have  which A  direct  linear  transformations  are not d i r e c t  C  the matrix  Then  d e t A = A."  product, preserves :  = ^  with  find  l  sense  E  12  +  E  2  3  +  A E  A. £  1  3-7/  lemmas  will  linear  in  i s defined  as  31 C  i s not a  i f a direct  the absolute  S which  3»1«  ^* ^  =  where  t o lemma  an E ^ - p r e s e r v i n g  of theorem  and i s not a  ( a . .) e M  d e t C ( A ) = 1.  i t preserves  transformations  E^  elsewhere.  n  f o r according E^  °21  = 1  The f o l l o w i n g t h r e e  cannot the  1  ^*  diag(A,I _^) A  preserves  5> ( c . a . . ) =  E^  as f o l l o w s  : ( a . .)  c.. C maps  C which  is-given  c^2  preserve  products.  transformation  product  S which  value  be u s e d  direct product  of the  t o show t h a t  transformation  i s not a d i r e c t  determinant  product  amongst in the  we  -  4.3.  Lemma  If A  elements  are  r  A  Since  determinant  M  zero, E  Proof.  e  then  =  (  r  r  We  also  sum  l's  r  r >  i s of  rank  k >  and  r  E (a  r  <  r  (A)  < —  1  =  I.  of  A  has  integer  (det(X )) d  over  2  r  A  2  a l l r  x  r  subdeterminants.  j|C (A)||  =  tr C  2  r  (AA*)  »  t r C (A)  =  E  r we  a ) 2  =  have  0  C (A*)  r  =  r  (a ,...,a ) r 1 n 2  E^(A)  2  <  (  ).  n  So  we  continue:  2  )  The  i f A  J  Z.—i.r  =  k <  r  (  1  only  r  take  remaining  have 2  A  the  submatrix  (X )  extends  j>Jd (A)  If  and  have  E  the  n  i f and  x  det  where  -  for n >  )  n  every  we  has  24  ( *  inequality  of  =  E (a  ,a )  2  <  2  r  )k" (tr(AA*)) X  r  =  (l) is justified  ( \ by  ( \  )k~ E^(a ,...,a ) r  2  )k- ||A|| r  theorem  2  2 r  5.2  of  [5]«  Since  l|A|| we  = n  <  ( *  r  have  (2)  E (A) r  We (i)  2 r  k of  c o n s i d e r two = n. A  are  A of  )k"  r  n  r  cases:  i s a permutation modulus  one.  matrix  and  a l l eigenvalues  L e t Tv., , . . . ,X,  1'  ' n  be  the  - 25 -  eigenvalues  o f A.  We E  which  states  modulus is  one  one.  the  r  have  that  (A,, , . . . ,A. ) = 1' ' n  (  r  that  a sum  of  ( ^  is ( ^  )•  We  show t h i s  Set m  =  summands.  ( ^  Since  )  n  ) complex  numbers  implies  j  would  each  summand  ) a n d l e t a_. + i B . . ,  j=l,...,m  a,  a.  <  1.  j=l,...,m;  J some  each of  <  be  1 for  jo  imply m  —i /  „ a, <  n.  * J j=l Thus  a . = 1 a n d B.  J Now A. , c  E  r  = 0,  j=l,...,m.  J we  show  A.^ b e a p a i r  a l l the eigenvalues  of eigenvalues.  (A., , . . . ,A, ) we l ' n  will  have  ^c^i  two  In t h e e x p a n s i o n  Let of  terms  "" ' i X  1  1  are equal.  i*  r - l  2  '  3=1/'.-/r-l  J  A, ,A,. • • - A,, d l I r-l N  Since  the eigenvalues  equal  t o o n e we  have A, c  Also, is  A  i s normal,  diagonal.  by  A  a n d t h e two t e r m s a r e  cancellation  = A. , d s o we  have  a unitary  L e t A, b e t h e e i g e n v a l u e UAU"  Since  are not zero  consists only  1  =  \l  of l  T  A s a n d O's  U such o f A.  = A.I  that Then  UAU""  1  -  (ii) If  k <  If  k  k >  r = 1.  jJ(A)  =  =  I f E^ (A)  so,  from  )k"  Lemma 4.4.  of  be  described  or  <  on  the  as  left  that  ( *  ( »  for a l l A e M  S(A)  of  e n t r i e s at  ( i , j ) and  = ( * that  Thus we  by  P and so  P  T  that  e l e m e n t s of  S by  consider  N  The  where n > — any  4,  pair  ( j , i ) are  either  ) we  must  e n t r i e s on can  have S ( l ) = I the  main  m o d i f y S by  diagonal  pre-  and  r e s p e c t i v e l y , where P i s a the  transformation  ——?-PS(A)P»  diagonal  modification  n  position  implies  diagonal.  A the  )  )  E^S(l)  permutation m a t r i x chosen  We  <  fixed.  This  post-multiplying  leaves  r  follows.  located  In o r d e r  lemma 4-3«  remain  n  0  symmetrically  Proof.  r  If E„S(A) = E ( A ) J i  interchanged  by  contra-  (2) r  can  A = I which  2  E (A)  S  = n,  1.  ( I and  -  n.  k = 1 then  dicts  26  fixed.  We  denote  vr.  the 0  A  e f f e c t of T  on  the  « diacf(0 _ , J ) n  2  wh e r e J  =  1°  matrix:  such  a  -  We  wish  27 -  t o show 7TN  = N o  We  assume 7TN  ( i ) 7TN  Either  ( k , p  and or  o  *  1  a t some  o  has a  1  (k,.|) s u c h  (k,,f)  a t some  (i)letD £ M _ n  elsewhere  such  2  S i n c e 7r l e a v e s d i a g o n a l rows o f 7 r d i a g ( D , j )  we  can c o n c l u d e  fixed  can c o n c l u d e  the e n t r i e s Now  and  zeros  entries  fixed,  we  at (n-l;n)  We  call 1  o f non-  = 0 ( i ) cannot  interchanges  occur. (ii). or leaves  (n,n-l).  consider a matrix  i s a P such  (k,k),  Therefore  can e l i m i n a t e a l t e r n a t i v e  t h a t ir e i t h e r  elsewhere.  a 1 at  t h e number  i s a t most 2.  TN  There  and  d i a g ( D , j ) = -1  3  that alternative  In a s i m i l a r manner we Thus we  (k < / )  Then  E^ 7T d i a g ( D , j ) and  that  that k >  be d i a g o n a l w i t h  on t h e d i a g o n a l . E  zero  o  (n,n-l). In  n-3 O's  has a  £ N  t (n-l,n)  ( i i ) irN  (kj)  o  o  t N  with this  1  that PN P» = N., o 1  l  T  s a t ( i , j ) and ( j , i )  matrix  N^ and assume  -  28  -  Then TT(PN  or  o  P»)  PN  P»7r(PN P»)P o  v  This  £  inequality  with »7rN » follows: o  alternatives The  t  o  N  If there  o  "P»7r(PN P )P«».  irN^ -  Lemma 4.5. —  £ N  to the  P»7r(PN P')P o or  P»  '  l e a d s us  r e p l a c e d by  o  =  (i) ( i i ) conclusion  w o  1  exist  U,V '  in M  such  n  that  S(A)  = UAV  for a l l A  in M  S(A)  = UA»V  for a l l A  in M  either  n or  n then  U,V  are permutation  Proof.  matrices.  C o n s i d e r i n g members of M  described  i n the  i n t r o d u c t i o n we  representation  o f S as  direct  of V  product  f  and  U must be  an  Then e v e r y  e n t r y of  element, which, V i s the one  row  same. or  Thus U = A,P and and  i n t u r n , i m p l i e s every  Q contains  two  from the  Thus  one  direct  matrix. or  non-zero  0 s T  and  l  T  s .  product  Since  zero.  the  this  element  of  elements i n of V  and  1  V = A. "*"Q where P i s a p e r m u t a t i o n only  the  i n v e r s e of  non-zero  as  matrix  a permutation  U must be  column so w o u l d t h e  column v e c t o r s  matrix.  entry d i f f e r e n t  If U contained  2  must have t h e  a permutation  Suppose V c o n t a i n e d non-zero  n  as n  U.  matrix  direct  product  -  of  V  no  rows o r columns o f z e r o s .  or  column  29 -  and U i s n o n - s i n g u l a r ,  T  Q i s non-singular  I f Q h a d two l * s i n any row  so w o u l d t h e d i r e c t  Q i s a permutation  and c o n t a i n s  product  of V  a n d U.  T  matrix.  If E^S(A) = E ^ A ) f o r a l l A i n M  Theorem 4.1.  Thus  n + 2  then  either (i) or  S ( A ) = PAQ  ( i i ) S(A) = PA»Q  where P, Q a r e p e r m u t a t i o n  Proof. n=l , a By  By i n d u c t i o n on t h e s t a t e m e n t  E^(A) f o r A i n  linear  transformation  corollary  which preserves  2.1 we have a U, V i n  or  If  Thus S becomes  the  determinant.  such that  either  = UAV  S ( A ) = UA»V. f o l l o w s from As  lemma 4«5  that  t h e members  we can w r i t e ,  of M  diag  E (TTC)  Since  fixed  f o r any C i n ^ 7T  3  U, V a r e p e r m u t a t i o n  matrices.  i n lemma 4«4 we u s e t h e m o d i f i c a t i o n , 7T, o f S i n  showing t h e i n d u c t i v e s t e p . of  o f t h e lemma.  i s the determinant.  S(A)  It  matrices.  and p r e s e r v e s n  +  2  diagonal  (0,C) = diag  3  = E (diag(0,C)) =  (0,TTC)  = E^tT E (c) 3  elements  E „ , by lemma  ,  = E (diag(0,irC))  3  1T l e a v e s  diag(0,C))  4«4  - 30 -  By  the induction  hypotheses, 7TC  or  = PCQ  7TC = P C Q .  However, s i n c e v l e a v e s  diagonal  we have 7TC = C o r 7TC = C  and  elements f i x e d  .  t  Similarly  P = Q = I  we can w r i t e  IT d i a g ( D , 0 ) = d i a g ( 7 r D,0) and  c o n c l u d e 7rD = D o r 7rD = D .  for  notational  NOW s e t m = n+2 and,  T  convenience only,  such that  a - . = a. = 0. ml lm follows:  as C  :  C  Ds  12 d  3  =  2  °  C  =0  i j d  d..  = a. .  d., ii  = a.. i i  =  2  a  3  l e t A be any member  Define  i +l  j+1  - a  3  C = (c,,) 1 j  of M  m  and D = ( d . . ) Ij  i,j=l,...,m  (i,j)£(l,2)  3  j=l,...,m i=l,...,m ' '  d.. = 0 elsewhere, i j  Now (0, C ) + diag  A = diag  (D, 0)  and 7TA = IT d i a g ( 0 , C ) = diag By  lemma 4«3  leaves  e  (0,7T C ) + d i a g  C  = D* .  ~  (TT D , 0 ) .  e i t h e r 7T i n t e r c h a n g e s  know t h a t  them f i x e d .  °12' 21' 7TD  w  + TT d i a g ( D , 0 )  In t h e f o r m e r a n c  *  s  l  n  c  e  case,  a  s i n c e TT  TT i n t e r c h a n g e s d  2  3  3 2  ,a  2 3  or  interchanges  and  31  -  -  Therefore 7TA = d i a g ( 0 , C » ) + d i a g ( D » , 0 ) = A» in  t h e same way t h e s e c o n d a l t e r n a t i v e  TA  = A. With the exception  (l,m) case  the e f f e c t with  of the e n t r i e s  of w i s determined.  the test matrix  E  1  0  12 I f 7T does n o t i n t e r c h a n g e and  interchange  a change  interchanges interchange  the entries  • We  eliminate  this  ., ., + E , „. m+1,1 m+1,2  n  a t (1,2) and ( 2 , l )  a t ( l , m + l ) and ( m + l , l ) we  1 t o 0.  the entries  conclude  a t ( m , l ) and  + E„ + E 2,m+l  the entries  the e n t r i e s  i n E^ f r o m  l e a d s us t o  A similar  change o c c u r s  have i f ir  a t (1,2) and ( 2 , 1 ) and does n o t a t ( l , m + l ) and ( m + l , l ) .  Thus we can c o n c l u d e  that  either  1TA = A 7TA = A'  or and  from  t h e d e f i n i t i o n o f TT t h e t h e o r e m  We now s t a t e Theorem 4.2. =s are  and p r o v e  I f E3 _ C ( A ) = E3( A ) f o r a l l A i n Mn t h e n 0  d i a g o n a l U,V i n M C(A)  We w i l l integers  i s proven.  so t h a t = UAV  for a l l A in M  u s e Q^n t o d e n o t e t h e s e t o f 3 ~ " t P l s u  w = (i^  there  i ^ ) where  1 < i ^< i  2  < i ^< n  e  n of p o s i t i v e  32  -  is to  t o be  subspace  zero the e n t r i e s of t h e 3  positions w =  the  (i.,  J-  i _i ),  Lemma 4.6.  3  3 0  exist  h  : A  w  .  i n d u c e d on  L  n  .  The  2.1 R  L  w  by  t o M„ • J>  such  that  3n  there  = det  hC(A)  0  that  >  h" (U  h(A)V ) W W  1  have d e t h ( A )  transformation  : h(A)  >  (h)C(A)  and p r e s e r v e s d e t e r m i n a n t .  there exist  U , V W W *r U  : h(A)  such  Thus,  that  h(A)V W  which  equal  to the  Then f o r a l l w £ Q  W  i s linear  corollary  from  a Hadamard t r a n s f o r m a t i o n  E  R  setting  submatrix determined  S i n c e C p r e s e r v e s E^ we  a l l A in L  from  complementary  isomorphism  diagonal U V i n M. s u c h ^ w w 3  Proof.  by  the  f o r a l l A in M  C  for  A in  3 principal  x  L e t C be  E_C(A) = E A  resulting  of e a c h  and  0  j>  of  -  W  implies C(A)  = h  _ 1  (U  h(A)V W  We  have y e t t o p r o v e  to  do  this  following  we  that  restrict  our  ).  W  U , V  are d i a g o n a l .  attention  t o R and  In o r d e r employ  the  notation R U  : h(A) = U. W W  = (a..) V =  7>(r..a..) V.  i,j=l,2,3.  -  We h a v e , where U^"^  33 -  i s t h e i - t h column  o f U,  Since R  : E. .  >• r . . E . . 1 1 1 1  11  we  have v..U ii  - r..e n  ( i )  v..U > Ij  ( i )  Now r . . £ 0 as C i s n o n - s i n g u l a r ii  holds  u, . = 0 hi  h £ i  v.j  j * i .  f o r any  i  = 0  j * i  and so  y  This  =0,  (i  so U, V a r e d i a g o n a l .  We a r e now i n a p o s i t i o n t o p r o v e t h e o r e m 4*2. It  i s sufficient  t o show ( c . . )  i s o f r a n k 1.  I j  every row.  F o r , then '  row o f ( c ^ ) i s a m u l t i p l e  o f some row, s a y t h e i Q  I f we r e - l a b e l t h e e n t r i e s o f t h e i - t h row a s o d, = c, , j =1,...,n o J  J  then c . . = c. d. U l j  i /= i o  where c. i £ i i s t h e m u l t i p l i e r . 1 o I f we  choose U = diag(c ,c 1  2 <  ...,c,_  V = diag(d ,d ,...,d ) 1  it  follows  that  2  for a l l A in M  C(A)  = UAV.  n  n  l l  l,c,  + 1  ,..,,c ) n  - 34 -  In 2 x 2  o r d e r t o show ( c . . ) i s o f r a n k  submatrix  matrix  i s singular.  ;  2  can  a 3 x 3 principal  and  ( ^Pj)«  y  a  four  2  or l e s s .  as a p a r t .  b  s e t {a^ a , B ^ ., B ] c o n t a i n s e i t h e r  integers find  We d e s i g n a t e t h e 2 x 2 s u b -  rows a^,a,> and columns P ^ r P ^  involving The  1 we show e v e r y  distinct  I f t h e l a t t e r h o l d s we a r e a s s u r e d we  y  Suppose  columns o f t h i s  submatrix which i n c l u d e s  = (y^  y^ y^)  principal  (cx^p\)  i s t h e s e t o f rows  submatrix.  By t h e p r e c e d i n g lemma we know C where  : Jy  *h" [U 1  Y  h(J )V ] 3  y  i s the n x n matrix with l * s at p o s i t i o n s  principal  submatrix  by y  determined  and z e r o s  of the  elsewhere;  and Uy  From t h i s  =  diag(u  =  diag(v  i t follows c  i Q  a P 2  and t h u s If  l  p  =  sets  v ,v^).  x /  2  f o r some i ^ i  Z  U  ,  J  V  ,  x  2  1  a  l  2  c ^ i 2  j^ j  2  =u.v. l 2  D  P  C 3  X  =  D  a B 2  J  U .  2  V ,  i  2  j  2  (cx.^p\) i s s i n g u l a r . fa^ a ,B^,P } #  2  2 #  contains four  we c o n s i d e r t h e two p r i n c i p a l two  2  =u.v, l l  ~ a a  C  that  u ,u^)  1 #  of d i s t i n c t  distinct  integers  submatrices derived  i n t e g e r s p, = [ a ^ , a , B ^ } , 2  from t h e  V = [a.^,a B } 2/  2  - 35 -  By t h e p r e c e d i n g  lemma, we C  C where  J  :  V  d  i  a  ^  (  v  l  1  v  2  = u ,  n  x  C  =  n  2*V  and f o r some n^ n ^ c  m  a B  2  f o r some  Jv.  l l  U ,  a  V ,  2  X  c  l  a  »J # 2  1  i  1  1  = Xu .  i i  =  2  t t  v.  l l  a  C  J  = u' 1  v' m  n j L  o  a B  that  l  x  U .  V ,  2  l  X  X  2  D  1  C  R  v* v,» ) .  Vl a  3  V  i t follows c  this  > h~ [U . h ( J ) V ]  u» u» ) V  this  ) y  3  = diag(v»  From  3  u )  = diag(u» =  — • _ . - ! [ h ( J  : J •  = diag(u^ Uy  From  3  have  2  = u* n  c  2  2  s e t of e q u a l i t i e s  v' m  a_a  c  2  a a 2  = u 1  n ; L  = 1  v'  T  U  T  n  V* g  i t i s easy t o e s t a b l i s h  that  V i and t h u s (o.^Bj) i s s i n g u l a r . of  Theorem 4 « 2 .  This  completes the proof  -  Theorems 4»1 and 4«2, 3.7  f  a l l o w us t o c o n c l u d e  C and S w h i c h p r e s e r v e 3«1.  The q u e s t i o n  which p r e s e r v e  36 -  i n conjunction with  that  the only  are direct  remains:  E^ d i r e c t  lemma  transformations  products  Are a l l l i n e a r  as i n Theorem transformations  products?  C o n c e r n i n g Eg p r e s e r v e r s , we were a b l e t o e x h i b i t types  of l i n e a r  were n o t d i r e c t these  transformations products.  transformations  exhaust  the class  Eg and w h i c h  I t i s n o t known w h e t h e r o r n o t  and d i r e c t  of l i n e a r  which p r e s e r v e d  two  product  transformations  transformations  which p r e s e r v e E_.  BIBLIOGRAPHY  1.  M. M a r c u s  A l l linear operators leaving the u n i t a r y group i n v a r i a n t , t o a p p e a r i n Duke Math. J .  2.  M. M a r c u s and J . L . McGregor  E x t r e m a l p r o p e r t i e s of Hermitian matrices. C a n a d i a n J . Math., v o l . 8,  1956, p p . 524-531.  3«  M. M a r c u s and B.N. M o y l s  L i n e a r t r a n s f o r m a t i o n s on a l g e b r a s of m a t r i c e s . C a n a d i a n J . Math., v o l . 11,  1959, P P . 61-66.  4«  J . Williamson  5-  G.H. H a r d y , J.E. Littlewood, and G. P o l y a  M a t r i c e s whose s - t h compounds are equal. B u l l e t i n of t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y , v o l . 39, •1933, P P . 108-111.  I n e g u a l i t i e s . 2nd e d . , Cambridge U n i v e r s i t y P r e s s ,  1952.  

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