UBC Theses and Dissertations

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UBC Theses and Dissertations

Linear transformations on algebras of matrices over the class of infinite fields Oishi, Tony Tsutomu 1967

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LINEAR TRANSFORMATIONS ON ALGEBRAS OF MATRICES OVER THE CLASS OF I N F I N I T E F I E L D S  t  by  TONY TSUTOMU O I S H I B.A.Sc., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 5 9  A THESIS SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Department of Mathematics  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e required standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA June, 1 9 6 7  In p r e s e n t i n g for  this  an a d v a n c e d  that  thesis  in p a r t i a l  shall  I further  make i t f r e e l y  agree that  for scholarly  p u r p o s e s may  publication  w i t h o u t my  of this  written  Department o f  M  thesis  eMA  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 Date  g~q*e  i 9  Columbia  £~?  I agree  f o r reference  and  copying of  this  by t h e Head o f my  I t i s understood  for financial  77C$  requirements  Columbia,  f o r extensive  be g r a n t e d  permission.  ATH  of B r i t i s h  available  permission  D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s , or  f u l f i l m e n t o f the  degree at the U n i v e r s i t y  the L i b r a r y  study.  thesis  gain  shall  that n o t be  copying allowed  (ii)  ABSTRACT The  problem of determining  the structure of  linear transformations  on t h e a l g e b r a o f  over  i s d i s c u s s e d b y M. M a r c u s a n d  t h e complex f i e l d  n-square  B. N. M o y l s i n t h e p a p e r ' ' L i n e a r T r a n s f o r m a t i o n s of Matrices".  The a u t h o r s  were a b l e  transformations which preserve properties of  matrices  on  Algebras  to characterize linear  one o r more o f t h e ' f o l l o w i n g  n-square m a t r i c e s ; rank,  determinant  and  eigenvalues. • The p r o b l e m o f o b t a i n i n g a s i m i l a r c h a r a c t e r i z a t i o n of transformations for a wider  class of fields  In p a r t i c u l a r , formations  their  holds  on d e t e r m i n a n t and  a s g i v e n b y M. M a r c u s a n d B. N. M o y l s b u t i s considered  i n this  thesis.  c h a r a c t e r i z a t i o n o f rank p r e s e r v i n g t r a n s -  f o r an a r b i t r a r y f i e l d .  One o f t h e r e s u l t s  preserving transformations•obtained  b y M.  B. N. M o y l s s t a t e s t h a t i f a l i n e a r - t r a n s f o r m a t i o n  maps u n i m p d u l a r m a t r i c e s  i n t o unimodular matrices,  preserves  Since  determinants.  hold f o ralgebras  of matrices  this over  infinite  fields.  i s limited  T  then  T  r e s u l t does n o t n e c e s s a r i l y finite  fields,  c u s s i o n on t h e c h a r a c t e r i z a t i o n o f d e t e r m i n a n t transformations  Marcus  to algebras  the dis-  preserving  of matrices  over  (iii)  TABLE OP CONTENTS Page CHAPTER ONE  CHAPTER  1.1  Introduction  1  1.2  Notation  1  1.3  Rank P r e s e r v e r s  2  TWO 2.1  Introduction  2.2  A Counter-example  '2-3 2.4  BIBLIOGRAPHY  5  1  f o rFinite  Determinant Preservers Comments a n d F u r t h e r for Investigation  Problems  Fields  4 5 15  15  (iv)  ACKNOWLEDGMENT I take t h i s o p p o r t u n i t y t o e x p r e s s ray a p p r e c i a t i o n t o Dr. Roy Westv/ick f o r t h e encouragement and t h e many h o u r s of v a l u a b l e a s s i s t a n c e he gave t o me d u r i n g t h e r e s e a r c h and w r i t i n g o f t h i s t h e s i s .  A l s o I would l i k e t o thank  Dr. B. N. Moyls f o r r e a d i n g t h e t h e s i s and o f f e r i n g h e l p f u l suggestions. The  f i n a n c i a l support  o f the U n i v e r s i t y o f  B r i t i s h Columbia and t h e N a t i o n a l R e s e a r c h C o u n c i l i s much appreciated.  CHAPTER  ONE  RANK PRESERVERS  1.1  INTRODUCTION The  purpose o f t h i s  chapter  i s to establish  n o t a t i o n t o be u s e d i n t h i s p a p e r a n d t o p r e s e n t  a character-  ization of linear  the rank  transformations  which preserve r  Although  n-square m a t r i c e s .  and  B. N. M o y l s i s c o n c e r n e d w i t h t h e a l g e b r a t h e i r proofs  theorem on r a n k p r e s e r v e r s istic proofs  the paper  i *  of  over t h e complex f i e l d ,  L1J  o f M.  of matrices  a r e n o t dependent on t h e c h a r a c t e r In fact, the  h o l d word f o r word f o r t h e a l g e b r a o f m a t r i c e s No p r o o f s  main r e s u l t o b t a i n e d  Marcus  o f t h e lemmas a n d t h e  nor the algebraic closure o f the f i e l d .  arbitrary field.  1.2  some  are given  i n this  chapter  o v e r an. but the  i n [ 1 ] i s s t a t e d b e l o w i n T h e o r e m 1.3-2.  NOTATION The  n o t a t i o n used i n [ l ] w i l l  denote a f i e l d . ive  be a d o p t e d .  Let  P  L e t t h e f o l l o w i n g symbols denote t h e r e s p e c t -  sets:  Numbers i n s q u a r e b r a c k e t s  refer t o the bibliography  2.  the a l g e b r a of U  the unimodular group of m a t r i c e s  "  n  n-square matrices  (i.e.,  matrices  in  RANK  1.3-1 A  .  into  M_ • n  of  l)  A .  PRESERVERS Definition  Let  a(A)  denote the rank of the  A linear transformation • T  preserver 1.5-2 —  in  F  w i t h determinant  denotes the transpose  1.5  over  i f  a  (T(A))  Theorem .  T  =  Let  T  be a l i n e a r  i s a rank p r e s e r v e r  T(A)  =  UAV.  T(A)  =  UAV  i s said  for a l l  o(A)  A  i f and  A  ,  .for a l l  A  .  a  rank  .  $'r.  only i f there  such t h a t  for a l l  s  e  t o be  transformation of  U and V  exist non-singular matrices  or  on  matrix  either:  M  n  3. CHAPTER  TWO  DETERMINANT PRESERVERS  2.1  INTRODUCTION The  main r e s u l t o f t h i s  Theorem 2 . 3 - 8 are  chapter i s given i n  which states that the f o l l o w i n g  conditions  equivalent: (1) v  '  (2)  T  maps  T  preserves determinant; i . e . , det T(A) = det A  for (3)  a l l  U  n  into U  A e  n  .  .  There e x i s t n o n - s i n g u l a r  matrices  w i t h d e t UV = 1 s u c h t h a t ' or  fora l l  A  T(A) = UA V  fora l l  A .  r  t h e major d i f f i c u l t y  results lies  either:  T ( A ) = UAV  When t h e b a s e f i e l d field,  U a n d "V  i n showing t h a t  F  i s an a r b i t r a r y  infinite  encountered i n o b t a i n i n g condition ( l ) implies  t h e above  condition  (2).  The s o l u t i o n o f t h i s p r o b l e m i s t o a l a r g e e x t e n t t h e  point  of this  (l)  thesis.  When  F  does n o t n e c e s s a r i l y i m p l y  w i l l be g i v e n  t o show t h i s .  i s a finite  field,  condition (2).  condition  A counterexample  4.  I  2.2  A COUNTEREXAMPLE FOR F I N I T E F I E L D S The  f o l l o w i n g example i s a l i n e a r  v/hich maps u n i m o d u l a r m a t r i c e s does n o t p r e s e r v e  Let  field  G-, a n d G^  T  T(G^)  of  = G^  2  2 x 2 matrices  two w i t h e l e m e n t s  { (o  0°)  ,  l) ,  (( I) ,  over {0,1} .  Mg :  G ]  {\ (o o) , (o o) , (o o) , Co o) j .  2  be a l i n e a r t r a n s f o r m a t i o n o n  possible G.^ n G  M^  be t h e f o l l o w i n g s u b s p a c e s o f  =  Let  the algebra  of characteristic  «! " G  into unimodular matrices but  determinant.  Consider the f i n i t e  transformation  a n d k e r n e l T = G^ .  since  M  g  ='  M  g  such  that  Such a t r a n s f o r m a t i o n i s  i s the direct  sum o f  and d i m e n s i o n  M  2  G^ a n d G^  (that i s ,  = d i m e n s i o n G^ +  d i m e n s i o n G^) •  The  transformation  T  maps  U  into n  that i s ,  T(U ) C U n  exists a matrix  Be  . G^  itself: '  F o r each unimodular m a t r i x and a m a t r i x  C €  A  there  5-  such  that  Since T(B) all  T  A = B + C i s linear  ( + denotes the u s u a l matrix and  + T(C) = T(C) . o f whose members  modular,  and  B e Gg , t h e n  T(U ) G U n  n  Go) 0  +  Consider  But  determinant.  T(A)  T  (Q I /  *  Since  (o  DETERMINANT 2.3-1 said A e M  matrices,  therefore  2.3-2 root of  =  T  10  det T  1  ^) = d e t T  does n o t p r e s e r v e  Thus t h e t r a n s f o r m a t i o n matrices  T  maps  ^  ;= 1 .  zero  unimodular  b u t does n o t p r e s e r v e ^  PRESERVERS  Definition  A linear  t o be a d e t e r m i n a n t n  zero  l )  determinant.  2.3  o a )\0 0.  "  i s uni-  does n o t p r e s e r v e  T  i n t o unimodular  l  G  = (o ) •  0  det E = 0 ,  matrices  E =  (Ji ) •  preserves unimodular  Thus  T(C) e  .  On t h e o t h e r hand, determinant.  T ( A ) = T(B+C) =  But by the h y p o t h e s i s are unimodular.  addition).  transformation  preserver  T  on  M  is  n  i f . det T(A) = det A  for a l l  .  Lemma  If  F  i s an i n f i n i t e  o f an e l e m e n t of- F  elements o f  F .  exists  in  field, F  then  the  n-th  f o r an i n f i n i t e  number  6.  Proof:  Let  a-j  be a n o n - z e r o e l e m e n t o f  Consider  the equation  t h e r e a r e a t most  x  n  a non-zero element  bg F b^ .  Suppose  i = l  N  a^ € F  distinct  non-zero elements  such t h a t Therefore v/ith an 2.3.3  Since  a^  = b  + 1  N  +  F  ^^j. and  1  a  n  d  b  N  +  Notation  Let  ^N+l  ±  in  b ^ , b^,  n-th root. x  i s an i n f i n i t e  t h e r e a r e an i n f i n i t e n-th root i n  in  '  D  e  l  o  F  .  n  there  Then  there  = b. ,  n  i  and  ... , b ^  field, S  n  g to  there F  such  ^ b. , I = 1 , 2, . . . , N .  number o f e l e m e n t s i n  F  F . A = ( a . .) e M n  (1) (2)  (i,j) ^ (s,t) A(s,t)(x) A  and  i s the  • a ^ = x .  .)  (That i s ,  nxn m a t r i x obtained  by r e p l a c i n g t h e e n t r y  indeterminate x  and  But  F ,  n a^ = b^  such t h a t  elements  b ^ = a'  one s o l u t i o n  s o l u t i o n s of the equations  2, . . . , N .  }  exist  nN  is  a 1  number o f e l e m e n t s  have been f o u n d , e a c h o f w h i c h has an a r e a t most  and  s o l u t i o n s of the equation  s i n c e t h e r e a r e an i n f i n i t e exists  Now  = b  P  a. st  by t h e  from  7.  2.3-^  Lemma  f o r some  I f A = (a.. .) € M a, .A, . ^ 0 .  j ,  A, . ^ 0 IJ  f o rthis  Proof:  Suppose  a i + i  n  d e t A ••= that since  E  (-1) ~  det A ^ 0 .  Consequently,  .A, . = 0 Ij I J  fora l l j  a. A, . = 0 Therefore  which  f o r each  a.. . 5^ 0  and  0  1 < j -  < n . -  and  of  ,  Then  0  f o r some  F ,  then  3  j  . And  a., .A., . 7^ 0 10 l j  A., . r-'0 .  I f det A ^ 0  b e F  a, .  contradicts the hypothesis  a, .A .  n  Lemma  then  .  a, . and A . a r e elements lj l j .  implies that  2.3.5  j  det A ^ 0 ,  and  n  j  and  such  that:  t h e n f o r some f i x e d  3  a e F  t h e r e e x i s t s an element  det A ( l , j ) ( a ) = b . B y Lemma 2 . 3 . ^  Proof: that  A, . £ 0 .  S (-1) j=l,jA For any  Let this value of  a , .A . = c , ±  t h i s root by  x  be  1 _< j _< n ,  the polynomial  a .  d e t A ( l , k ) ( x ) = xA,, + c . xA-ij^. + c - b  namely  Thus, f o r any  such  k . I f  l  has a r o o t ,  such that  then  j  j ,  J  b e F ,  determinate  a e F  J  there e x i s t s a  i nthe i n -  x = (b-c)A~^ .  b e F  det A(l,k)(a) = b .  k  Denote  t h e r e e x i s t s an  8.  2.?.6  Lemma  Let F  then f o r any  Proof:  an  A e  F  and d e t A ^ 0 .  such t h a t i f b e B  n-th root  i n F .  b y Lemma 2 . 3 - 5  each  there  Let  S  i s an  be t h e s u b s e t o f  corresponds a  b e B  n  ,  a e F  be a n I n f i n i t e  b ^ 0  then  }  there  F  I f T(U ) C U  Let B  B y Lemma 2 . 3 . 2  det A ^ 0 b e B  field.  det A ^ 0 , det T(A) = d e t A .  such t h a t  Let A e M  subset o f  be a n i n f i n i t e  and  has  such a s e t e x i s t s .  exists a  k  f o r which  Since  such that f o r  det A(l,k)(a) = b .  such t h a t f o r each  f o r which  b  a e S  b = det A(l,k)(a)  .  there I n other l/n  words, f o r each e F .  Since  det A(l,k)(a) ^ 0  a e S  B' c o n t a i n s  an i n f i n i t e  and  number  e l e m e n t s t h e n s o does S . F o r e a c h (det A ( l k ) ( a ) ) ] = 1 . And s i n c e  [det A ( I , k ) ( a ) ]  of distinct  a e S , det[A(l,k)(a)/ T(U ) C U - f o r each  1 / / n  3  n  3  a e S, d e t T ( A ( l , k ) ( a ) ) = d e t A ( l , k ) ( a ) . Denote by p o s i t i o n contains  E. . t h e n x n m a t r i x  t h e element  det  T ( A ( l , k ) ( x ) ) = d e t T(xE.,  det  (xT(E,, ) + 1 K  1  is a polynomial Lemma 2 . 3 . 5 some det  s a (i,j)^(l,k) in x  1  Since  T(A(l,k)(x)) ,  s  T ( E .)) ~  f o r each xA  e l s e w h e r e . ' Then  a . , E , .) =  = p(x) ,  o f d e g r e e _< n .  then  (i,o)-th  where  p(x)  J  i t was shown t h a t  c e F .  and i s zero  +  J  whose  l k  In the proof of  det A(l,k)(x) = x A x e S  l k  + C for  det A(l,k)(x) =  + c = p(x)  f o r each  x € S .  9.  But  S  c o n t a i n s an i n f i n i t e number o f d i s t i n c t  therefore any  xA  + c = p(x)  p  x e F ,  identically  A(l,k)(a ) = A ,  then  2.3.7  Let F  1 k  Corollary.  )C U  Proof: A(u)  in x .  ,  ;  l k  .  In parti-  and s i n c e  d e t T(A) = d e t A .  be an i n f i n i t e  t h e n f o r any  Let A  A e M  n  ,  field.  be any m a t r i x b e l o n g i n g t o  f o r i = 1, 2, 3,  If  d e t T(A) = d e t A .  be t h e m a t r i x o b t a i n e d by r e p l a c i n g  a. . + u ,  Thus f o r  det T(A(l,k)(x)) = det A ( l , k ) ( x )  c u l a r , the e q u a l i t y holds f o r x = a  T(U  elements,  , n ,  M  ,  n  a ^  and l e t  in A  where  u  by  i s 'an i n -  determinate. Det A ( u ) = u polynomial i n u  n  4- p ( u ) = p ( u ) ,  o f degree _< n-1 .  n det ( s ( u + a ) T(E..) + S a . T ( E i=l . i ^ j ± i  = q ( u ) where Since u  if  But  has a t most  n  )) ,  thus  t h e n by Lemma 2.3-6  isa  det T(A(u))  o f degree _<• n .  s o l u t i o n s , then  f o r i n f i n i t e l y many v a l u e s o f  det A(u) ^ 0 ,  P-j^u)  det T(A(u)) =  q(u) i s a p o l y n o m i a l i n u  p(u) = 0  = P ( ) "r 0  where  1  u  det A(u)  i n F . But  det A(u) = det T(A(u  and c o n s e q u e n t l y  p ( u ) = q ( u ) f o r i n f i n i t e l y many v a l u e s o f  u  p(u) = q(u) i d e n t i c a l l y i n u  in F .  Thus  det A ( u ) = d e t T ( A ( u ) ) f o r a l l v a l u e s o f particular,  u  the e q u a l i t y holds f o r u = 0 .  and  i n F . In But s i n c e  A(0)  10.  then  d e t A = d e t T(A) and  therefore  The  .  A  was a n a r b i t r a r y e l e m e n t i n  d e t A = d e t T ( A ) f o r a l l A e M^ .  p r o o f s o f Lemma 2.3-8 a n d Lemma 2.3-9 a r e t h o s e  g i v e n b y M. M a r c u s a n d B. N. M o y l s i n t h e p a p e r [ l ] .  The  p r o o f s a r e i n c l u d e d i n t h i s paper f o r t h e sake o f  completeness.  2.5.3  T  Lemma  If T  s i n g u l a r and hence Proof:  Suppose  preserves determinant,  A ,  Since  denoted  by  There e x i s t n o n - s i n g u l a r m a t r i c e s I  r  + 0 where n-r  0  the  sum.  F o r any  r = a(A) x  (n-r)x(n-r) n  1  3  ,  -1  X = 0^ + I  det X = 0 T  n  _  unless  .  r  is'l e s s  M and N  such  than  that  n . MAN =  t h e r x r u n i t m a t r i x and +  denotes t h e d i r e c t 1  - 1  XN  d e t M~"*"XN~~ = d e t X d e t M !!" Set  a(A) ,  [ d e t ( M A N + X ) ] [ d e t M" !*" ] =  = det( A+M  1  I r  det A = det T(A) = 0 ,  z e r o m a t r i x and  X e M_ ,  d e t M" (MAN-tX)N"  i s non-  onto. T(A)' = 0 .  then the rank o f  then  Then  r = 0 .  1  - 1  .  1  ) = det T(M~ XN~ ) = 1  Therefore  1  det(MAN-tX) = d e t X .  det(MAN+X) = d e t 1 = 1 . But  r = 0  implies  But  A = 0 .  Thus  i snon-singular.  2.5-9  Lemma  determinant,  Let F then  T  be a n i n f i n i t e preserves  rank.  field.  I f T  preserves  11.  P-<"oof:  Let A e M n  be a n a r b i t r a r y m a t r i x .  non-singular matrices M_AW = Y ]  1  r = a(A)  = I  x  r  M  + 0 _ n  1  , N  1  M  3  and N  g  and M T(A)N  r  2  = Y  2  There e x i s t such t h a t  g  * i  g  g  + 0 _ n  where  g  a n d s = <j(T(A)) . 0 : M —•* M  Define The m a p p i n g  0  n  0(X) = M ^ M ^ X N " ) N  by  n  1  2  .  has the f o l l o w i n g p r o p e r t i e s :  (1)  0  (2)  d e t 0(X) = k d e t X  i s linear  since  T  is linear, where  J  k =  det(M M~%~ N ) ; 1  2  2  d e t 0(X) =  This r e s u l t s from  d e t M - d e t T ( M ~ X N ~ ) • d e t Ng = 1  1  2  det. M - d e t M ^ X N "  • det< N" =  1  2  2  d e t ( M M ^ N ^ N ) • det. X 1  1  2  (3)  0(Y ) 1  p  = Y  2  = M T(A)N . = Y 2  Set  y  = \  I _ n  .  r  p  r  ,  + 0(Y^)) = p(\) ,  therefore  2  1  1  1  2  1  1  .  Por each s c a l a r  1  o f d e g r e e _< s .  = k\  2  a n d d e t 0(\Y _ +  T  det(\Y X  = 0 r  0 ( Y ) = M T(M^ (M AN )N^ )N  since  \  i  det(\Y  + YJ)  1  ) = d e t ( x 0 ( Y ) + 0{Y^)) = . 1  where  p(\) i s a polynomial i n  B u t ' d e t 0( \Y  ±  p(\) = k\  r  + Y^) = k d e t ( \Y  ±  + Y^)  f o r any \ e F . .Since  2  12.  k ^ 0 , and  p(\) = k V < G(T(A))  a(A)  identically  all  B e M  .  n  is,  T  T  Therefore  T  The  and s i n c e  preserves  - 1  = a(A)  ( A ) < a( T( A ) )  preserves  exists;  - 1  1  _ 1  a  Therefore  d e t B = d'et(TT" (B)) = d e t T  a ( T ( A ) ) _< a ( T T ( A ) ) shown t h a t  X .  r _< s  .  By Lemma 2 . 3 . 8 , serves determinant,  in  ,  T _ 1  pre-  (B)  determinant.  for Thus  and s i n c e i t . has p r e v i o u s l y been then  a  ( A ) = a(T(A))  .  That  rank.  statement  o f the f o l l o w i n g theorem d i f f e r s  from  t h a t g i v e n b y M. M a r c u s a n d B. N. M o y l s o n l y i n p a r t ( 3 ) particular,  when  F  i s the f i e l d  were a b l e t o f i n d u n i m o d u l a r conditions 2.3-10  linear  i n part (3)  Theorem  Let  matrices  o f Theorem F  o f complex numbers, U and V  M  n  .  they satisfy  2.3-10.'  be a n i n f i n i t e  t r a n s f o r m a t i o n on  which  field  and  T  a  The f o l l o w i n g c o n d i t i o n s  are e q u i v a l e n t : (1)  T  maps  (2)  T  preserves  (3)  There e x i s t with  D"  U  n  .  determinant. non-singular matrices  d e t UV = I T(A)  or  into  n  = UAV  T ( A ) = UA°V  such t h a t  either  f o r a l l 'A fora l l A .  In  U and V  13-  Proof: (3)  ( l ) h o l d s i f and o n l y i f ( 2 )  implies  holds by C o r o l l a r y  2.3-7.  ( 2 ) since  d e t T ( A ) = d e t UAV = d e t A • d e t UV  (3) -  T  = det A . (2)  implies  all  such t h a t e i t h e r  A e M  -  there exist  T ( A ) = UAV  Let T(I) = det 1 = 1 ,  field  F  infinite  i s sufficient  elements these  n-th root.  elements  F  field  F  condition.  < n  but the  I n o r d e r t o show  are i d e n t i c a l l y  equal, distinct  and t o p o s s e s s  an  contains  n(n+l)  non-zero  elements,  n + 1  distinct  non-zero  elements  n - t h r o o t ( s e e t h e p r o o f o f Lemma 2 - 3 - 2 ) .  f o r a given  o f Theorem 2 - 3 - 1 0  elements.  number o f e l e m e n t s ,  t o be n o n - z e r o  at least  each p o s s e s s i n g an Therefore,  requires the  The p r o o f o f Lemma 2 . 3 - 6 r e q u i r e s  I f the f i e l d  then there e x i s t  1 = det T ( l )  that the e q u a l i t y holds f o r n + 1  of the f i e l d .  n + 1  for  t  INVESTIGATION  i s not a necessary  t h a t two p o l y n o m i a l s o f d e g r e e it  T(A) = UA V  therefore  h y p o t h e s i s o f Theorem 2 . 3 - 1 0  t o c o n t a i n an i n f i n i t e field  non-singular matrices  J  COMMENTS AND FURTHER PROBLEMS FOR The  or  then by  T h a t i s d e t UV = 1 -  = d e t U I V = d e t UV -  2.4  preserves determinant,  and Theorem 1 . 3 - 2 ,  Lemma 2 . 3 - 9  U and V  I f  Again this  n ,  i ti s sufficient  contains at least  n(n+l)  i s not a necessary  s i n c e f o r some g i v e n  t h a t each element o f t h e f i e l d  n  non-zero  condition  there exist  has a  that the f i e l d  on t h e  fields  n-th root;  such  f o r example,  14.  for  n = 3  take  The  F = Z  c  = Z/5Z  paper [ l ] g i v e s a c h a r a c t e r i z a t i o n of  transformations which preserve i n the a l g e b r a of  nxn  eigenvalues  matrices  A f u r t h e r p r o b l e m w o u l d be holds  .  over  t o see  f o r the a l g e b r a of m a t r i c e s  i f the over  A n o t h e r p r o b l e m w h i c h may following.  In order t o o b t a i n the  preservers given  the  be  for a l l matrices complex numbers.  same c h a r a c t e r i z a t i o n  a larger class of considered  transformations preserve  ranks  to  find  c o n d i t i o n s on a l i n e a r  T  such t h a t  a(T(A))  other  sufficient T  = CT(A)  i s a rank p r e s e r v e r .  n  .  For  f o r a l l symmetric m a t r i c e s ,  algebra of matrices preserver?  2 and  t h a t the  over  a field  F ,  then  rank  linear  I t may  be  possible  transformation .  example, i f A , is  fields.  i s the  c h a r a c t e r i z a t i o n of  in [ l ] i t is sufficient 1.  linear  i n the T  a  rank  15-  BIBLIOGRAPHY  [l]  M a r v i n Marcus and. B. N. M o y l s , L i n e a r  Transformations  on A l g e b r a s o f M a t r i c e s , Canadian J o u r n a l o f Mathematics,  Volume  11.(1959).  

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