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The division transformation for matric polynomials with special reference to the quartic case Niven, Ivan Morton 1936

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L £ 3 %J  ,Vi'  of-  THE DIVISION TRANSFORMATION FOR MATRIC POLYNOMIALS WITH SPECIAL REFERENCE TO THE QUARTIC CASE by Ivan Morton Niven  A Thesis submitted f o r the Degree of MASTER OF ARTS i h the Department of MATHEMATICS  The U n i v e r s i t y o f B r i t i s h Maroh, 1936  Columbia  1.  THE DIVISION TRANSFORMATION FOR MATRIO POLYNOMIALS WITH SPECIAL REFERENCE TO THE QUARTIC CASE  This t h e s i s i s based  on r e s u l t s which were obtained by  Dr. M. M. Flood of P r i n c e t o n U n i v e r s i t y and reported upon by him i n a paper presented to the American Mathematical at i t s New York meeting, February 2,5th  r  1933.  No. 39-3-92, The B u l l e t i n of the American Society).  Society  (Of. A b s t r a c t  Mathematical  P r o f e s s o r Flood t r e a t e d the problem f o r the l i n e a r ,  q u a d r a t i c and cubic cases.  The present paper extends the  theory to the q u a r t i c case. For the sake of c l a r i t y i t has been found d e s i r a b l e t o o u t l i n e P r o f e s s o r F l o o d ' s  treatment^"  f o r the e a r l i e r cases, although i n so doing, c e r t a i n m o d i f i c a t i o n s and e l a b o r a t i o n s are made. of Theorems (4.13), original.  For example, the p r o o f s  (5.21) and (6.27), as here g i v e n , are  The e n t i r e treatment  f o r the q u a r t i c case i s  original.  P e r m i s s i o n to do t h i s has-been v e r y k i n d l y granted by P r o f e s s o r F l o o d , who gave the author access to h i s manuscript.  I.  Introduction. A matrix a(A) whose elements are polynomials i n a s c a l a r  variable degree  A  , o f which at l e a s t  one element has the h i g h e s t  n, w i l l be c a l l e d a m a t r i c p o l y n o m i a l of degree  n,  and w i l l be expressed i n the form (1.1)  a(X)  y a. A X=o  =  a  ,  0,  where the  a . are constant m a t r i c e s . We s h a l l c o n s i d e r A. these m a t r i c e s square and of order r . I f now (1.2)  b(X)  »•  21b. A  i s a second such m a t r i c p o l y n o m i a l , i t i s w e l l known that there e x i s t unique m a t r i c polynomials q ( A ) , r ( A ) , q f A ) and (  r  (  (\) of which  and r ( A ) (  and q^ (A) are of degree  n^m  and r ( A )  are of degree l e s s than m; such t h a t  (1.3)  a(A)  =  q(A) b ( A ) +  =  b(A)  q, (A) +  rfX) r,  (A).  The scope of t h i s r e s u l t has been extended by J . H. M. Wedderburn-'- t o i n c l u d e the case i n which the l e a d i n g cient ^  b ,  coeffi-  of the d i v i s o r i s s i n g u l a r but n o t not z e r o .  I t w i l l be shown l a t e r , moreover,  that i n t h i s case the quo-  t i e n t and remainder are not of n e c e s s i t y each unique and  1  Gf. J . H. Ivf. Wedderburn, " l e c t u r e s on'Matrices," American Mathematical S o c i e t y , 1934.  3. that the degree of.the q u o t i e n t may (Theorem 3.1).  T h i s paper w i l l  he g r e a t e r than  consider t h i s l a t t e r  more f u l l y and w i l l give r e s u l t s concerning  n-m ease  the nature and  degree of the q u o t i e n t and remainder obtained under the stated conditions.  I I . D e f i n i t i o n s and n o t a t i o n . Small Roman l e t t e r s with a s u b s c r i p t denote constant m a t r i c e s , i . e . , m a t r i c e s v/ith elements i n a given The matrix of order  r,  except the one i n the 1, i s denoted Similarly  by  -v.  a l l of whose elements are zero  i*'  e.  row and the  1  1  column which i s  r e p r e s e n t s the u n i t v e c t o r whose  i d e n t i t y matrix of order p  i""  I t w i l l be c a l l e d a u n i t  nents are a l l zero except the  If  are p o s i t i v e  field.  r  matrix. r  I*** , which i s 1.  w i l l be denoted  by  compoThe  i .  i n t e g e r s suoh t h a t  s  r  (2.1)  =  £L  p  we d e f i n e S.by means of the r e l a t i o n Pi.  (2.2)  E. =  JLp+'p '  k»l  1  It follows that  i  + — + P  =  \  k  ( i = l , — - , s).  •  k= 1  Analogously, we d e f i n e  N• ;  Pi  (2.3)  N  = 1  Having  S -V P  k T l  + P + '  chosen the"  may c o n s i d e r the matrix A 3  + *  — ,  a).  so that (2.1) i s s a t i s f i e d , we i A<  whose elements  +P  v  as a matrix I  Q  S  o f order  are r e c t a n g u l a r m a t r i c e s with  s, p.  5.-  rows and^ columns such that  A  ]  and  A  ... 11  ±j  =  i *  o,  i s the i d e n t i t y matrix of order  l a r manner, the constant matrix •r - space may elements Aw  a  ,• i n s - space. p^  of terms i n  a..  rows and  u n i t v e c t o r s in°  Thus  p^  E  (2.6)  A^  e. and  and EL^, the  s - space.  The  following  proved:  E.  and  E. E . =  0  ,  i # j.  H.  s  K.  and  E. B.  0  ,  i * j.  k  K. i  =r  i <  s  ^-  * = 1  H  *  with  'V. , the u n i t m a t r i c e s and"  -  i  . i n  columns appearing as a b l o c k  E.  i  A  a<  i s a rectangular  r - space, correspond to  r e l a t i o n s are e a s i l y  (2.5)  In a s i m i -  w i t h elements  Then A  u n i t m a t r i c e s and u n i t v e c t o r s i n  3.  p. . x  he considered as a constant matrix  matrix with  (2.4)  i,  I D  A^ " . k «* i  *  r  Ill.  E x i s t e n c e of Quotient and Remainder. (3*1) Theorem:-  I f a(A) and b(\) are the polynomials (1.3) except that lb 1=0, but h  d e f i n e d by (1.1) and  then there e x i s t polynomials q(A) , r ( A ) , q (A)  a(A) =  q(X) b(A)  s  0,  and r, (A) o f  (  which r(A) and r,(A) are of degree leHS than  ^  m,  such t h a t  r(A)  b(A) q (A) 4-  r, (A).  (  Proof:Let the rank of b  be  p <. r .  U s i n g the development of  paragraph 2 v/ith  s •= 2. we have  m a t r i x theory  can be expressed i n the form  b  (3.2)  B  ^  m  P  1  p  Sj_ Q  1  r - p^.  =  ,  where P^ and Q, are n o n - s i n g u l a r m a t r i c e s . -1  h^A)  (3.3) we  see t h a t  b^A),  =  Writing  (S A +  d e f i n e d by  b (A)  (3.4)  By elementary  hj(A) b(A) ,  «  1  i s a m a t r i c p o l y n o m i a l whose degree i s not h i g h e r than the HI +• 1 "I degree of b(A) s i n c e the c o e f f i c i e n t of \ , E P , vanishes. Now (3.5)  |b (A)| 1  =|h (A)|1  |b(A)|  =r  ( A  P 2  )  P~ | X  b(A)  so that the degree of J TD-^ CA) | exceeds the degree o f by  p  g  .  I f the c o e f f i c i e n t of  \  m  i n b (A)  i s singular,  t h i s p r o c e s s may be repeated g i v i n g  bg(A), bg(A),  where the degree of each j b^ ( A ) j  exceeds that of  ( b ^  ( A. ) | •  B  u  t  t  n  e  degree of each  b^A)  |b(A)  ,  i s l e s s than  or equal to  m  and the degree of the determinant  p o l y n o m i a l of the  m  degree cannot exceed  p r o c e s s must terminate; y i e l d i n g  r m.  a polynomial  h i g h e s t term has a n o n - s i n g u l a r c o e f f i c i e n t ; law  of f o r m a t i o n (3.4) we  b (A) = h(A)b(A)  of a Hence the  b . ( A )» whose and from  the  have  where  h(A)= h.(A)h (X) 0  h.(X).  A p p l y i n g the d i v i s i o n t r a n s f o r m a t i o n (1.3) to a(A)  and b  (A) J  we  have (3.6)  a(X)  = =  = Since P^and and r(A) chosen,  s(X) b.(X) + s(A)  r(A)  h(A) b(A)+ r ( \ )  q(X) b(x) + r(X). i n (3.2) are not unique,  depend upon the p a r t i c u l a r  and  since  polynomial h(A)  these l a t t e r polynomials are not unique.  which i s  The  method of p r o o f i s o b v i o u s l y a p p l i c a b l e t o the dextrolateral  case.  q(A)  above  8. IV. A s s o c i a t i o n "by a L i n e a r P o l y n o m i a l . We  s h a l l say that the polynomial h(\) i s a s s o c i a t e d w i t h  the polynomial b(\) i f the degree of the polynomial h(A)b(X) equals the degree of b(\) and i f the c o e f f i c i e n t l e a d i n g term of h(\) b(A)  of the  i s non-singular.  As i n s e c t i o n 3, l e t m  b(A) •=  (4.1)  b^A*  be a m a t r i c polynomial f o r which b and  suppose h(A)  i s s i n g u l a r but not  zero,  i n (3.6) i s l i n e a r and a s s o c i a t e d w i t h  b(A),  ffi  so t h a t  *  h(A)  (4.2)  h \ +  h  Q  and (4.3) Since we h(A)  h b = 0. 1 m r e q u i r e the c o e f f i c i e n t  b(A)  i n the product  m  set t h i s  coefficient  i , without l o s s of g e n e r a l i t y ^ that i s , r  (4.4)  h b Q  + h b _  m  (3.2) we  H  \  to be n o n - s i n g u l a r , we may  equal to  Using  of  1  m  get from  P  = 0  (4.5)  G  Q  =  ^  =  1  i  r  .  (4.3) or d e f i n i n g  H  Q  P  2  ,  G  x  G  2  G =  Q  Q l  and R  ±  G  1  by  ^  we have (4.6)  " G '  Similarly, (4*7)  i f we  i*? = •xi  ==  0  ,  K  x  =  0.  define P~'  B o/' Jk 1  (k =: 0, 1, —  ,  m-1),  then (3.2) and H  (4.4) give  h\h  0  whence (4.8)  G  Q  E  +  L  G-L  1-  h  ! i-i \ =s  E B  B^j  «  r  I  • M u l t i p l y i n g on the r i g h t by (4.9)  & 1  B  £)  H  2  =  G,H  g  »  . , we o b t a i n  £  b£{  =  2Z  H  E  ,  from which i t f o l l o w s t h a t (4.10)  (1) I BI _ _ m-  0.  i a 2  I f we set (4.11)  G  G we  i  i -  K  Q H L  °«  they are  K  2  - \ ^il-L^  1  O  Q  (4.4) are s a t i s f i e d i d e n t i c a l l y .  i n (3.2) are not of n e c e s s i t y unique,  m-lsa  i s independent  of the manner i n which  chosen.  (4.12.) Theorem:f i n e d by  i  ^ - V^-LE^VIEI ;G  =  see t h a t (4.3) and  Although P^ and (1) the rank of B  G  I f b(X)  i s the m a t r i c p o l y n o m i a l  de-  (4.1), the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t  there e x i s t a l i n e a r polynomial a s s o c i a t e d w i t h i t i s t h a t the equation (4.10) be s a t i s f i e d . d e f i n e d by equations (4.13) Theorem:f i n e d by  (4.1),  One  such polynomial i s  (4.11). I f b (A ) i s the m a t r i c polynomial  de-  the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n  that a l l polynomials a s s o c i a t e d with b(X)  be l i n e a r i s that  v  10. (4.10) be s a t i s f i e d . lows from  That the c o n d i t i o n i s n e c e s s a r y  fol-  (4.12,).  Assume now that h ( A ) , as i n (4.2), and Ji  (A)  =  SL k= 0  i  A  i  k  U o ^  £  , (t >  are each a s s o c i a t e d w i t h b(A) and that equation satisfied.  1)  (4.10) i s  Then we have 0  =  L. B t m  1, B  =  t  . 4m-1  L, .B-, . t-1 m  Setting R  k  =  Q  lk L  P  l  =  0, 1, —  t)  we have 0  R,  =  E  =  l  R  t  B^>  4-  R _ t  1  S  l  whence R.  1  t  Since  B^|_ ra-122  AL giving  1  t  L  = . 0 and - .  R. B ^ E = R B ^ = 0. t m-1 2 t m-122  i s non-singular;. ' &  t *1  =  R  t  "  0  •=• 0, which c o n t r a d i c t s the h y p o t h e s i s ,  11. V.  A s s o c i a t i o n by a Quadratic As  Polynomial.  i n s e c t i o n 4,. l e t m  (5.1)  •  ^  21  bOt) =  b, \  be a matric polynomial f o r v/hich zero, and  suppose h(A)  b  i s quadratic  i s s i n g u l a r but and  not  associated* v/ith  bf A) •, so t h a t (5.2)  hQ)  (5.3)  h b  (5.4)  h b , 2 m-1 h b _  (5.5)  2  =  h  =  m  m  A  2  0 +  2  +  \  +  2  A +• b  Q  ,  ,  . h, b 1 m h b _ 1  m  =0, +  1  and h b Q  m  =  i  .  r  Since l i ( A ) i s not l i n e a r , i t f o l l o w s from Theorem (4.13) that (5.6)  B  ] _ . I =. m-122 1 In t h i s s e c t i o n , we which e a r l i e r we B ^ m-122  is  r  =  used  p ^ . ^ P  f2) l 2 +  and  -> 7  B  m  () 2  s h a l l use  p>  and  Hence we (2) * 3  P  In .this new (B  0.  f l  P  S  l  E  i n the  w  h  same sense i n  suppose that the rank of  have  (2) . „  P  n o t a t i o n we "  p^  e  r  e  may  (2)  ?! - P  •  8 1 1 4 x  choose  P=  ?2  2  P  2  and  Qg  + P  so  that  \  such that i f (2) (5.8)  B  fc  = P  g  B  k  Q  2  (k = 0, 1,  then the f o l l o w i n g r e l a t i o n s are s a t i s f i e d : -  , m-1  (2) 3  )  (2)  12.  'Ciis (5.9)J B  +  B  (2)  mm-xai  ~  m-18 2  =  0  A aa  (2) m-132 *  B  B  i ~  (2) — m-123  f 2 )  m-133  =  B  =  0  T h i s f o l l o w s , s i n c e , as i n (3.3)^ we such that tha  B  = m  If, a l s o , P"  B  1  1  E, x  x  Q"  m-1  A  n  and Q  A  81  A  ia i3 A  1  A  /A . A then, since/ \ i s of rank Hz 3 3 2  1  x  \L  2 2  can f i n d P  3  p  hz  A S 3  31 32 A  , we  A  33  can f i n d  non-singu-  A  Xc 1  l a r matrices  and  each of order p x  A  aa  A Now,  Y  32  A  a3^  A  Y  33 j  =  (3)  (A  3 3 0  0  0  1  _(2) p" 'such that ' 3  let and  Y  0  •11  = 0 •  then  X P:  B  1  m  , Q"  1  Y  / C / U  has the form  \ °81  G 12  0  G \ 13 1  0  If /  0  1-  n  A  0 ' A22  \0 then  -°ia  0  •ii  0  OA  33  \aad  G= 2  -G 0  21  0  0  A  0  o  A 33  \  13.  x  G - G G .0 11 12 21  °31 Note that X, Y, 0^ and G since  P  l  a  o  X E  =  _ _ 1i  e  P-  then  P  Y  E  2  E^ = E^ G  _ - l1  .  C X P" =  %  X-  G"  1  =  g  ..-1  and  a  x  °  °  /  are each n o n - s i n g u l a r , and t h a t  2  * G  _ _  =  • 0 13  X  ^  1 E]  _ G"  1  1^  = Y-  , then  _ - 1l  Q"  =P  1 Q  l  Y ^ E  X  ,  x  Q  x  =  B  m  and P  2 m-i  V l  ^2 -  B  (2)  as d e s i r e d i n (5,7) and (5.8) where  B _-]_  has a form such  m  that c o n d i t i o n s (5.9) are s a t i s f i e d . Using  (5.7), we get from (5.3) H  2 2 P  E  (5.10)  l%  °  S  G  k  G  a  °  = Q  H  2  r  k  d  e  P  f  i  n  i  n  S  G  * l' G  0  G  2  b  y  (k = 0, 1, 2)  2  we have (5.11) Using  E  1  = 0  ,  G  z  N  =  x  0 .  (5.8) and (5.10), we get from (5.4) (5.12)  G. B'W  ,4. G  2  x  E  X  M u l t i p l i c a t i o n on the r i g h t by (2) m-1 % = G  B  =  0 N  9  gives '  a  0  2  whioh y i e l d s on a p p l i c a t i o n of (2.6) and (5.9) (5.13)  G  Multiplication  a  N  2  s  0  of (5.12) on the r i g h t by  N  gives  14, which g i v e s , as a consequence o f (2.6), (5,14) Using  &  1  (5.7),  =  l  - G  2  H  (S) B _  3  m  (2) m-2  +  (2) lm-1 0  G  B  +  G  l '  S  2 m-2 %  +  (2.6),  (5,15)  s gives  f2)  ,(2) B  Applying  J  1_ o  M u l t i p l i c a t i o n on the r i g h t hy  G  1 3 r  (5.8) and (5.10), we get from (5,5)  B  o  G  B  (5.9) and (5,11),  G  lm-1 %  G  (5.9), F  2  * . S ff  B  *£l 233  [  3  (5.11),  (5.13) and (5.14),  we get  m-131'^m-113  Setting (5.16)  V  . 33 "  fa) V-233  (a)  (a)  ~ ^-131^-113  we have (5.17)  G  F  2  V  3  *  3 3  F  3  It n e c e s s a r i l y follows that (5.18)  ,(2) (a) (2) ^m-aSS - ^m-131 m-113  33  +  B  0.  I f we s e t a r b i t r a r i l y  G  G  l  G  l  (5.19)  G  l  K  W  K  a  " %  3  0 %  a  G  -  K  o  l ~  N  0  « V  a  %  G  a  N  3  -l (a) 33 V l 3 1 J » V  G  33  o %  =  = 3 ^3 K  l%  G  v  = 2" % K  T  0  " (2) (2) (2) V-2.31 ^m-131 V l l l p  _! (2) 3 3 m-E3E B  we  see that  (5.3), (5.4) and  Again, although P rank of  and Q  (5.5) are s a t i s f i e d  identically.  are not of n e c e s s i t y unique,  the  V__ i s independent of the manner i n which they are 33  chosen. (5,2,0) Theorem:f i n e d hy  (5.1), the n e c e s s a r y and s u f f i c i e n t  there e x i s t that  I f b(A) i s the m a t r i c polynomial  c o n d i t i o n that  a q u a d r a t i c polynomial a s s o c i a t e d w i t h i t i s  (5.18) he s a t i s f i e d .  hy equations  One  such p o l y n o m i a l i s d e f i n e d  (5.19).  (5.21) Theorem:f i n e d hy  de-  I f b(A) i s the m a t r i c polynomial  de-  (5.1), the necessary and s u f f i c i e n t c o n d i t i o n s that  a l l polynomials a s s o c i a t e d w i t h i t he q u a d r a t i c i s that (5,18) he  satisfied.  That t h i s c o n d i t i o n i s n e c e s s a r y f o l l o w s from that h(\),  Assume now  (5.20).  as i n (5.2), and  j  t Jt(X)  =  ^  X  q  A , k  k  i^o,  are each a s s o c i a t e d w i t h b(A).  (t > 2)  Then we  have  0 = 1, B = L B +• L B = L B -«-L, B , +• 1, B t m t m-1 t-1 m t m-2; t-1 m-1 t-2 m 0  n  Setting R we  k  = Q  have  R  t  ^  x  0  2  L  k  P  2  (k = 0, l i —  t)  16, R R  Kg  t  B  - -0 .  (2)  R  B  m-233  I t follows  t-1 %  f2)  B  t  -  (2)  m-131  0  m-113  3  whence  that  %  h  L  =  *  p 2  R  t  =  Hence  which c o n t r a d i c t s the  0, hypothesis.  m-l.il  0 .  R  t  K  3  =  0.  17. TI.  A s s o c i a t i o n "by a Cubic P o l y n o m i a l . As p r e v i o u s l y , l e t m  b(A) =  (6.1)  A*  b oL*  0  *  be a matric polynomial f o r which, zero, and suppose so  h(A)  b^  i s s i n g u l a r but not  i s cubic and a s s o c i a t e d w i t h  that (6.£.)  o  h b =0 3 m h b . +• h b 3 m-1 £ m h b _ h b _  (6.4) (6.5) f 6  *  6 )  h„ A° 4 h  h(\) =  (6.3)  3  h  m  2  +  2  m  A  +• h  A +  A  1  c.  0  =0 1  h b  +  3 m-3 + W E * b  1  h  ^  m  l V l  /  +  0 Vm  = \  Since h(A) i s n e i t h e r l i n e a r nor q u a d r a t i c Theorems (4.13) and (5.21) that n e i t h e r can be  * we see by  (4,10) n o r (5.18)  satisfied. (3)  In t h i s s e c t i o n , we s h a l l use p sense as, m  s e c t i o n 5, we used p  (3) and p  and p (3)  and suppose that the rank of V f*)  -  P  l  (3) 3  +  +  ( )  (3)  Z  P  £  + 3  (3) 4  P  ( ) 3  t*\  + 4 P  w  h  e  r  e  ?!  =  .  3 (2) ?! a n  i n the same respectively,  Z  is p 33  r  b(/\),  d  Hence we have  P2  (3) = P£  (2) a  n  d  (2) 3  In t h i s new n o t a t i o n we may choose P^ and Q_ so that 3 3 (6.7) B -  18, and  such that i f  (6.8)  = P B 5  (3)  (k =0, 1^ -- , m-1).  ' Qg  K  then the f o l l o w i n g r e l a t i o n s are s a t i s f i e d : >(3)  (3) _ -lEl  B  = B  m  (3) Bm-14k  =  (6.9) B  B  =  0  0  B 2,  (k  3, 4)  22  m-122 (3) m-233  3  =  v  \  /  (3) ( 3 ) m-1.23 m-124  (3)  -  B (3)  -  B  (3)  -  B  = o  B  = 0  V-113 F S }  m-114  m-141 (3)  =  BJ I ; ^ f 3 -)  B  0  F 3 )  m-114  'm-141  'm-244 (3) Conditions  (3)  B,(3)  m-2.43  33  m-113  'm-131  m-234 (6.10)  B (3)  m-131  (6.9) correspond t o c o n d i t i o n s  (5.9) and a r e  d e r i v e d by methods which are i d e n t i c a l with those f o l l o w i n g (5.9).  In order to d e r i v e v  33 * ^m-233  which i s of rank p  (3)  (6.10),  ~ V-131  consider  PJ2)  m-113  (°) (3) (3) ' and of order p* = p l +- p! . r  Y/hen we pass t o the n o t a t i o n of t h i s s e c t i o n , J5) m-233 B  (3) m-243  \  (3) \ m-234 f B  3  )  m-244  m-141  B  (3) m-113 0  becomes (3) m-114  B  0  \  /B' )  - B' >  3  m-233 or  B  V  B'  3  m-131  f3) m-243  B  3 )  m-113  f3) ^ f3) (3) m-234 m-131 m-114 B  (3) B m-244  ^(3) ( 3 ) m-141 m-113 R  (3) m-141  We oan f i n d n o n - s i n g u l a r m a t r i c e s H-^ and (3) (3) P + n 3 ~4 r  H  l  V  33  l=  K  0  33 0  0  whence c o n d i t i o n s (6.10) a r i s e . 0  "11 s  of order  such t h a t  r  A  H  (3) m-114  I f now, we w r i t e / A  0 \  0  A 22  0  0  0  H  and  K  0  -  V  1/  i t w i l l he noted  0  u  oV 0  22 0  0  that the H and K can be absorbed  and Q , as were the X, Y,  and Gg i n the l a s t  thus l e a v i n g the forms of B ' m-1  and  l  B m  by the P, 3  section,  = E, u n a l t e r e d . 1  U s i n g (6.7), we get from (6.3) H  3 3 P  E  (6.11)  l%  * °  Oj, = %  ° H  r d e f i l l i n  P  k  S  G 0  '  G  l'2 i 3 G  G  b  y  (k =. O, 1, 2, 3)  3  we have (6.12) Using  G  3  E  ±  =  0 ,  G  3  1^ = 0 .  (6.7), (6.8) and (6.11), the r e l a t i o n  (6.13)  S  3  B^f» +. 8  S]  _  = 0  (6.4) g i v e s  M u l t i p l i c a t i o n on the r i g h t hy G„ B ] 3 m-1  K  f 3  =  2  Q  G  Multiplication  G  s  Kg  3  givers  0  which g i v e s , on a p p l i c a t i o n (6.14)  Kg  of (2.6) and (6.9),  0 .  o f (6.13) on the r i g h t hy 3  m-1  B  l  K  +"  G  2 %  •=  N  yields  0  whence  (6.15) . G  K  1  - G„ 1„ B  =  3 Using  3  (3)  '  (3) B ' 4 m-141  + Grr  m-131  3  (6.7), (6.8) and (6.11), we get from (6,5) (6.16)  G„ a  + G m-2  B<3) m-1  0  2  G  M u l t i p l i c a t i o n on the r i g h t by G._ B J, 3 m-2  K  f2  which, on a p p l i c a t i o n  3  S 1  =  0  .  1  yields  +- G  2 0  of (2.6),  B  ] m-1 f 3  H„ 3  (6.9),  =  0  (6.15) and (6,10),  gives (6.17)  G  Multiplication  G  F  3  3  =  of (6.16) on the r i g h t by F  3 m-2  ^2  B  which, on a p p l i c a t i o n (6.14) and (6.10), (6,18)  0 .  Gg Hg  +  G  2 m-1  of (2.6),  B  K  2  - G  3  1  4  =  (6.9), (6.15),  gives =  yields  B^  2 4 2  .  0  (6,17),  (6.12),  M u l t i p l i c a t i o n o f (6,16) on the r i g h t by  gives; i n a  s i m i l a r manner (6,19)  &  F  1  ^  x  G  H  3  4  [ B ^ l  4  B ^  1  (3) m-241  B  11 ~  "* 2 3 m-131 *" 2 4 m-141 G  Using  F  B  G  H  ( 6 . 7 ) , (6.8) and (6.11), we get from (6.6)  (6.80)  «  3  B ^ ' 1- 8  + G,  2  +  M u l t i p l i c a t i o n on the r i g h t by tion  B  (6.21)  8 G 11 V > m-o44 3 •- 4  - B  ( 3  S  Q  ^  =  I.  g i v e s , upon s i m p l i f i c a »  B' ' 3  m-141  m-214  m_242 m - 2 2 4 m-141 m - l l l ra-114  B  B  B  +B  B  B  m-241 m-114 B  Setting (6,22)  V  4 4  ,(2) =B ^  *  (3) - B _ t  4  4  -(3) m-141  m  u l 4 1  (3). V-lll  we have (6.23)  G^ K V 4 * 3 4 44 I t n e c e s s a r i l y f o l l o w s that (6.24)  44  f  0 .  I f we s e t a r b i t r a r i l y  (3) B : m  2 1 4  ,(3) - B _ m  a 4 a  _(3) B _ m  2 2 4  (3). .(3) (3) ^m-114 " ^m-241 V-114  (6.25)  G  1^ ^ G  3  2  G U  2  G  l  F  1"  -F  3" 3  % *  K  F  1  2  - F  G  F  F  Q  X  44  m-141  3  F  1 4 N  V  4  T  -  G  B  3 L  f 3  >  f 3  (  3  0 3 K  B  F  B^ )  f  B  F  3  3  B  )  B  )  m-111  G  0 4  =  N  B  m-211  43 V - 1 3 1  )  -  B  B  B^)  3  m-141  + V  m-141  B  -  (3) L "m-342 B  3) (3) m-131 ra-111 m  B  m-341  '43  1  4  4 3 3  B f ' m-232  0  44  B  3  (3) m-131  ( 3  s  i [  Y  H  K  B  F  3  (  3  >  (3) f3) m-242 m-221 B  )  v  + B  )  m-111  0  (3) m-231  V  B  m-111 " 4 3  W  h  e  F  3  )  m-241  r  f  3  )  m-231 B  (  3  )  m-111  e  (3) (3) ( 3 ) (3) _(3) m-141 m-213 rn-242 m-223 m-343  B  B  +  B  B  (3) (3) - B' ) m-241 m-113 m-141 B  3  B  B  (3) m-111  (3) m«113  B  4 4  m-242  B  +  G  B! !  44  (3) 7 - m-241j n-2<'~  1  ( 3 )  V  4  0  W  » B -B » B » V m-141 m-212 - m-242 ra-222.  f 3  N  ¥jj  4  0  =  4  K„  2  2  1  4  K —  C  K  3  .G . 4=  V V 44 4 43 J  K  G  2  1 (3) B ' „ - B\ V, , 3 . m-232 4 44 4  - H  2  +  +  1  0 2. -  G  B  + , AA  G  3  -1 V V 44 43  N 4  F ,4. T  =  G Ng  Y  -  1 3  0  =  2  4 4 4 [^-141 m ? l l l  -B  G  K  4  +  +  F W  G  2  .S3.we see that identically.  (6.3), (6.4), (6.5) and (6.6) are s a t i s f i e d Again, although P  unique, the rank of V  and ^ are not of n e c e s s i t y . . 3  i s independent of the manner i n  which they are chosen. (6.26)  Theorem:-  I f b(A) i s the m a t r i c polynomial  d e f i n e d by (6.1), the n e c e s s a r y and s u f f i c i e n t  condition  that there e x i s t a c u b i c polynomial a s s o c i a t e d w i t h i t i s that equation (6,24) be s a t i s f i e d . d e f i n e d by equations (6.27)  One such polynomial i s  (6.25).  Theorem:-  I f b(X) i s the m a t r i c polynomial  d e f i n e d by (6.1), the. n e c e s s a r y and s u f f i c i e n t  condition  that a l l p o l y n o m i a l s a s s o c i a t e d with i t be c u b i c i s that the equation (6,24) be s a t i s f i e d . analogous to the p r o o f s of Theorems so w i l l not be g i v e n .  The p r o o f of t h i s i s (5.21) and (4,13), and  24, VII.  A s s o c i a t i o n by a Q u a r t i c Polynomial As p r e v i o u s l y , l e t  (7.1)  m <•  b(A) =  . <*"  b_  A  be a matric polynomial f o r which zero, and suppose  h (A)  b  i s s i n g u l a r but not  m  i s q u a r t i c and a s s o c i a t e d with b ( A ) ,  so that (7.2.)  h(A)  (7.3)  h  f 7  -  4 )  \  ( 7  '  5 )  f 7  -  6 )  h  (7.7)  »' h  b  4  4  ,  m  A  V-£  W  a  +  h  V =  +. h  3  h  3 V &  +  h  h  2  2  \  i s not l i n e a r ,  V l +  (5.18) and (6.24) can be  im  h  + T  *  b  h b 1  quadratic  Theorems (4.13), (5.21) and  + h b m-l' 0 m  = i  m  T  r  .  or cubic we see by  satisfied. (4)  , p  (4)  ( 4 )  and — * "p  in (3) same sense as, i n s e c t i o n 6, we used p f ^ , p,( ^ and p 3 (4) 2  3  Hence we have (4) (4) r*P + P + x  1 ' 3)  ,  Q  >  0  r e s p e c t i v e l y , and suppose that the rank of V  IP  A + h  (6.27) that none of (4.10),  In t h i s s e c t i o n , we s h a l l use p  the  4- h^  2  ° '  =  h b +rhb , h b 4 m-4 3 m-3 + 2 m-2  Since h(A)  A  2  ° •  V l + +  X  3  0 j  V l ^ 3  4  4  2  P  2  5 ) >  P  3  3 )  P  3  ;  3  is p  4 4  4  (4) (4) (4) (4) (4) (4) +- P + % where p ^p , p v  3  1  4  > ^Vectirely,  and p<4>+  In t h i s new n o t a t i o n , we may  choose  g  p  P  U) 4  g  f > 3  =  p  and Q  4  . so that  £5. (7.8)  B  =  m  ]  and  p  <k  4 %  suoh t h a t i f  \=  (k = 0, 1,  %  h\  , m-1)  then the f o l l o w i n g r e l a t i o n s are s a t i s f i e d : (7.10)  (4 /  m-112  R  B  -  m-1 E l (4) x»(4) m-13k ' m-14k  >J4) _ \ m-122 " B  B' * = 0 m-llk  =  0  except when  j = k = 3 , i n which case  J4)  (4) m-113  m-131  3  3 (4) B m-aak  _U)  _(4). . J 4. ) B - B B ' m-llk m-ajl m-llk  (U except when  A  "  L  f 4  (4) (4) _ J. B B m-ljl m-111  _  (4) (4) B ^,,_ - B'*:.„ "m-alk m-a^a  (7.12) / B ^ ) - B ] m-ljl 7 m-3jk +  ( j , k = 3 , 4, 5)  4  n  T,(4) m-S33 "  (4)  EE  f 4 )  0  = B  ( 4 )  -J4) ^W-lSk  (7.11) / B - B .. m-Ejk m-ljl U )  B = B m-1 as  = 'JO  k = 4, 5)  j =t k = 4, i n which case the  \^ above e x p r e s s i o n equals  A  ^  .  C o n d i t i o n s (7.10) and (7.11) correspond  to c o n d i t i o n s  (6.9) and (6,10) and are d e r i v e d by methods which are i d e n t i c a l w i t h those used i n the l a s t  section.  In order t o o b t a i n  (7.12),. c o n s i d e r 'V^, which i s of rank p ^ ^ and of order 4  (4) (4) . p + ; i n the same manner as \r was o o n s i 4 4 5 33 dered i n the p r e v i o u s s e c t i o n , remembering, however, that p  (3)  =  V„. reduces to 44 A  4 4  0  0  0  upon m u l t i p l i c a t i o n r i g h t and l e f t by s u i t a b l e n o n - s i n g u l a r matrices. Due to the complexity of the above and f o l l o w i n g r e l a t i o n s , i t i s expedient  to introduce f u r t h e r n o t a t i o n . (4)  Henceforth, the matrix Thus the r e l a t i o n s  B  m  i  w i l l  be designated by f i j k ) .  (7.10), (7.11) and (7.12,) are w r i t t e n  (7.13) /(112) ~ (121) =• (123) =. (124) » (125) = 0 . (13k) =  (14k) »  (12.2) »  A.  2  (15k) = 0  (k = 2, 3, 4, 5) .  .  2  (2jk) - f l j D ( l l k ) = 0  ( j , k = 3, 4,.&)  except when j » k * 3, i n which case =  (233)-(131)(113)  33 '  A  (3jk)-(ljl)(21k)-(2j2)(22k)+(ljl)(lll)(llk)-(2Jl)(Ilk) ,-  0  except when j = k = V s i on equals Using H  4 4 P  E  S  4  4  .  (7*8) we g e t , from  l ^4 *  (7.14) .  A  4, i n which case the above  °»  = Q  4  o  (7.3),  cLe^ining G , G , G , Gg, G , by  r  Q  H  K  P  4  x  2  4  (k = 0, 1, 2, 3, 4) ,  expres-  27. we have (7.15)  G  E  4  Similarly,  (7.16  -  ±  0  ,  G  M  4  =  1  0 .  (7.4) g i v e s (4) B ' -t- G, K  G 4  m-1  Multiplication  o  =  0 .  1  on. the r i g h t hy  H  yields  (4) 4 m-1 2 which g i v e s , on a p p l i c a t i o n of (2,6) and (7.17) G 3% = 0 .  (7.13),  4  Multiplication (7.18) Again,  G  of (7.16) on the r i g h t hy [&  3  3ST (131) +• G  4  3  "•  9  )  8  l-1  4  Multiplication  +• G  K (151)J  4  5  .  l-1  B  +• 3 G  (7.15),  (7.2.0)  0  G E 4 3  Multiplication G  3  F  E  +  S  2  E  on the r i g h t hy  c a t i o n of (7.13),  (7.21)  4  yields  (7.5) g i v e s B  1  K (141)  4  1^  =  l  "  K , g  (7.17) and  ° • t o g e t h e r w i t h an a p p l i (7.18) g i v e s -  of (7.19) on the r i g h t by =  - £s .'M (242) 4  4  l  £  yields  + G^ N (252) J . g  M u l t i p l i c a t i o n of (7.19) on the r i g h t by If produces (7.22) G % = - G % ( 1 3 1 ) - G I%(141) - G N '(151) 2  3  G  4  3  K  4  3  5  38. -  G 4  Again (7.6) transforms  (4)  d-M)  \  [f~  >  ~ fl51)(lll)J  into (4)  mi  B  %  5 1  % V a  +  (4)  +  G  aV i  +  Multiplication  on the r i g h t by F  (7.34)  =  G K 4  4  G K 2  g  = G  4  K  5  K  x  yields  3  (352)]  N (242) - G Kg(35a) . 4  g  o f (7.33) on the r i g h t by H  Multiplication 1  s i m p l i f i e s this, i n t o  [_(151) (313)+ (353) (222) -  3  G  0  0 .  - G Kg(232) - G  ('7.36)  E  of (7.33) on the r i g h t by F  Multiplication (7.35)  4  i i= •  G  gives  = G K gj(351) (111) + (252) (221)+- (151) (311) 4  +G :  (151) (111) (111) - (351 )J  Kg [(131) (111) - (331)]  - Gg Kg(131)  + G K [(141) f i l l ) - (241)]  - Gg K (141) 4  3  + G  3  Kg [(151) (111) - (251)]  - G K (151)  of (7.33) on the r i g h t by F  Multiplication  g  .  yields  (7.27) GgKg ~ G KJj- (353) + (252) (223)+ (151) (313) +• (251) (113) 4  Designating G  3  F  3  the c o e f f i c i e n t of G Fg above by ¥ g 4  = G F 4  Finally,  (7.7)  5  Y  <  4  B  m - 4  G  (4) G  V  (4)  3 m-3 + 3 m-a G  B  B  +  (4) G  lVl  on the r i g h t by F g  4 5 [ 5 a ] + 3% [ K  we have  gives  -  Multiplication  4  .  5 4  (4) G  (151)(111)(113)] .  G  V  53] = ^5  +  V l» \ '  yields  29. in  whioh  V  52  =  (455) - (151)(315) - (252)(325) -  (251)(215)  - (151) (111) (215) + (151) (212) (225) i- (252) ( 2 2 2 ) (225) - (352)(225)+  (251)(111)(115) +  (252)(221)(115)  +(151)(211)(115)-(151)(111)(111)(115)  -  (351)(115)  and  V '(335)-(l31)(215)-(233)(225)+•(131)(111)(115)-(231)(115). 53  Substituting  in V  f o r GgNg  from  ( 7 . 2 7 ) , we  obtain  which  55- 52 V  * W 5 3  (455)-(252) (325)-(l51) (315)-(352) (225) + (252) (222) (22,5)+ (151) (212) (225)+(151) ( i l l ) (215) -(151)(111)(111)(I15)+(a51)(111)(115)+(252)(221)(115) + (151)(211)(115)-(351)(115) +- (151)(213)(335) - (151)(213)(232)(225)-(l51)(213)(131)(215) (151)(213)(231)(115) + (151)(213)(131)(111)(115)-(151)(111)(113)(335) + (151)(111)(113)(232)(225)+(151)(ill)(113)(131)(215) +(151)(111)(113)(231)(115)-(151)(111)(113)(131)(ill)(115) +•(252) (223) (335)-(252) (223) (232) (225}-(252) (223) (131) (215) -(252) (2.23) (231) (115) f  (252.) (223) (131) (111) (115)  + (251) (113) (335)-(251)'(ll3) (232) (225)-(25l) (113) (131) (215) -(251)(113)(231)(115)+(251)(113)(131)(ill)(115)-(353)(335) +(353)(232)(225)+(353)(131)(215)+(353)(231)(115) -^353)(131)(111)(115)  .  I t n e c e s s a r i l y f o l l o w s from (7.2,9) that "•  |f  |55  3 0 )  V  ° •  I f we set a r b i t r a r i l y (7.31)  G  = G  A  I  4  2  G  =  =G  A  =• 0  A  SB = H Y . 4 5 5 55 _ 1  3 1 =~ 5 55 '  G  H  F  G N  V  « I V~J Y C  3 3  5 55  1 5 1  >  . V  2 2 M  =  |^  5 55  K  Y  V& V \ = -  G  G  1 ! K  V  i 3 E  B  B  Y  A  G  51 *  1 5 1  3 4  l %  G  H  A  =  N  l "  K  5 55 V  4  5  1  )  t  "+(S&2)(22£)-(35S)]  -  ° ' ~ 5 55 K  8  f  0 .  V  V  50  0  1  V l  >  « *  =  »  3 5  f 2 5 S )  [ ( 2 5 1 ) - ( 1 5 1 ) (111)1 5 50 L J  2 1 2  V*5 G  -V55  . G^N. ^ G^K  C<1  5 4 .  G M. ^ -H V (131)2 1 5 55 54  G  =  2  M  5 55  1  ^151) (311)  Y  -+ N v " (252) ( 3 2 1 ) - H.V~* Y „ ( 3 3 1 ) 5 55 5 5b 54 1  C  - K V 5 5 [(151) ( 2 1 2 ) + ( 2 5 2 ) (222)-(352) J  (22.1)  + N V ^ £ Y ( 1 3 1 ) ¥ ( 2 5 1 ) ~ ( 1 5 1 ) (111) J G  " 5 55 51 K  T  (211)  5 4  Y  f l l l  >  -[ 2- % N  ¥  5 5 50] V  f l 2 1  >  >  31. GI 02  G E 03  =r  = G I 04  =  G 1 05  =  0,  i n which V  n  5l  = (251) (Ili)+(252){221)+(151)(211)-(151) f i l l ) (111)-(351) V  +  [" ( 1 3 1 ) ( l l l ) - ( 2 3 1 ) J  .  and V" „ = 50 c  (452)-(l51)(312)-(252) (322) + T  5 4  we  (222)  (7.5), (7.6) and  A g a i n , although ?  c e s s i t y unique, the rank of Y i n which they are  I f bQO  and Q  4  by equations  satis-  are not of ne-  i s independent  i s  of the manner  "the m a t r i c polynomial  (7.1), the necessary and. s u f f i c i e n t  c o n d i t i o n that  a q u a r t i c polynomial a s s o c i a t e d w i t h i t i s that  equation (7.30) he s a t i s f i e d .  by  4  (7.7) are  chosen.  Theorem:-  there e x i s t  fined  (332)  + (252)(222)-(352) J  fied identically.  (7.33)  54  (212)  see t h a t (7.3), (7.4)>  d e f i n e d hy  V  (131) (212) +. £(251)-(151) (111) J [(151)(212)  (7.32)  +  One  such polynomial i s d e f i n e d  (7.31). Theorem:-  I f b(Ji') i s the m a t r i c polynomial  (7.1)j the n e c e s s a r y and s u f f i c i e n t  de-  condition that  a l l polynomials a s s o c i a t e d with i t be q u a r t i c i s that equation  i(7.30) be s a t i s f i e d .  The p r o o f of t h i s i s analogous  the p r o o f s of Theorems (5.21) and given.  to  (4.13) and so w i l l not be  

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