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UBC Theses and Dissertations

Pairs of matrices with property L. Chow, Jih-ou 1958

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PAIRS OF MATRICES WITH PROPERTY L BY JIH-OU OHOW  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  IN THE DEPARTMENT OF MATHEMATICS  We a c c e p t t h i s t h e s i s as conforming t o t h e s t a n d a r d r e q u i r e d from c a n d i d a t e s f o r t h e degree o f MASTER OF ARTS.  Members o f t h e DEPARTMENT OF MATHEMATICS. THE UNIVERSITY OF BRITISH COLUMBIA April,  1958.  ABSTRACT L e t A and B "be n-square complex m a t r i c e s w i t h eigenvalues respectively.  X^, • • •, jX^ a n d ^ / ^ y A , • • • ,jx^  The m a t r i c e s A and B3are s a i d t o have  p r o p e r t y L i f any l i n e a r c o m b i n a t i o n aA + bB> w i t h a, b complex, has as e i g e n v a l u e s t h e numbers a X + b/^, i . = 1,2, • • -,n. A theorem of D r . M. D. Marcus, w h i c h g i v e s a 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 such t h a t two m a t r i c e s A and B have p r o p e r t y L i n terms of t h e t r a c e s of v a r i o u s power-products of A and B, i s p r o v e d . T h i s theorem i s used t o i n v e s t i g a t e t h e c o n d i t i o n s on B f o r t h e s p e c i a l cases n = 2 , 3 > and 4,  when A i s i n J o r d a n c a n o n i c a l form. The f i n a l r e s u l t i s a theorem w h i c h g i v e s  a n e c e s s a r y c o n d i t i o n on B f o r A and B t o have p r o p e r t y L when A i s i n J o r d a n c a n o n i c a l f o r m .  In p r e s e n t i n g the  this thesis i n partial fulfilment of  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 i t 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 o f 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 .  I t i s understood  that copying or p u b l i c a t i o n of t h i s t h e s i s f o r 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  financial  permission.  J i h - o u Chow  Department o f  - Ma.thema.tins  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  A o r i l 16. 1QS8.  A CKNOWLEDGEMENT S  I am i n d e b t e d t o D r . B. N. M o y l s and D r . M. D. Marcus o f t h e Department of Mathematics a t t h e U n i v e r s i t y o f B r i t i s h Columbia f o r d i r e c t i n g and s u p e r v i s i n g my work. I would a l s o l i k e t o e x p r e s s my g r a t i t u d e t o 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 of Canada f o r considerable f i n a n c i a l  assistance.  T A B L E OF  CONTENTS  1.  INTRODUCTION  2.  A N E C E S S A R Y AND S U F F I C I E N T PROPERTY  •"  APPLICATIONS  4.  THE NECESSARY  1  C O N D I T I O N FOR  L  3.  2 4 C O N D I T I O N ON B S U C H THAT  . A AND B H A V E P R O P E R T Y L BIBLIOGRAPHY  ••  15 22  1.  Introduction, Definition.  L e t A and B: be- n-square complex  m a t r i c e s w i t h , e i g e n v a l u e s X, respectively.  • • •,^  n  andyx., u. , • • •, jx >>  i  n  The m a t r i c e s A and B-are s a i d t o have p r o p e r t y  L i f any l i n e a r c o m b i n a t i o n aA + bB, w i t h a, b complex, as e i g e n v a l u e s t h e numbers a\+ i n 1952  has  t>j*. i = 1,2, ''',n. lt  T. S. M o t z k i n and Olga Taussky ( l ] proved  the  f o l l o w i n g theorem: L e t the n-square m a t r i c e s A and B have p r o p e r t y L. L e t t be the number of d i f f e r e n t e i g e n v a l u e s of A and assume t h a t a l l the e i g e n v a l u e s A i of A a r e a r r a n g e d i n s e t s of L e t O K be the m u l t i p l i c i t y of t h e . e i g e n v a l u e s  e q u a l ones.  of A and assume t h a t t h e r e a r e m^ independent corresponding to e a c h ^ .  * v a l u e s of B.  Let  P  L  eigenvectors  be t h e c o r r e s p o n d i n g e i g e n -  *  -1  L e t B:=  X  -1  BP where A = P  AP i s i n J o r d a n  c a n o n i c a l form. Then B  *-  ' ll B  =  B  B  where  B  21  B  tl  B,. • • •  it  22  t2  i s an m^-square m a t r i x ( 1 = 1,2,  I xl  - B*| = .n | x l -  « »«,t  ) and  B |. n  b . , b . where the summation i s o v e r v  a l l i<k w i t h ( i , k ) o u t s i d e of every B ^  ( i = 1,2,  • • • , t ).  i n the above theorem A i s r e s t r i c t e d i n t h a t i t i s s i m i l a r to a diagonal m a t r i x , the s t r u c t u r e of B  i n t h e p r e s e n t paper we  f o r more g e n e r a l t y p e s of A.  We  consider  first  - 2 -  p r e s e n t a r e s u l t of P r o f e s s o r M. Marcus w h i c h g i v e s a  necessary  and s u f f i c i e n t c o n d i t i o n that A and B have p r o p e r t y L i n terms o f t h e t r a c e of v a r i o u s power-products o f A and B. We t h e n determine t h e r e s u l t i n g s p e c i a l c o n d i t i o n s on B f o r t h e cases n = 2,3 and 4. necessary 2.  we conclude w i t h a g e n e r a l  c o n d i t i o n t h a t £ and B have p r o p e r t y L.  A Necessary and S u f f i c i e n t C o n d i t i o n f o r P r o p e r t y L» Theorem 1.  L  A and B have p r o p e r t y L i f and only i f x y x y ^  A B " - A B ) = "~ x ^ . .. +x = P y . . . + y = k-p l ' 2»** ' y -o, j _ "'>yp X  1  r  p' . A' ), «•=•  r  r  1+  x  l*p*fc,l*k.-n,  9  r  i  0  x  ,  x  > 0  r  r  r  where X,, \, • •;  V  and yx, ,jx , • • •  a r e eigenvalues  i  of A  and B : r e s p e c t i v e l y . Proof.  I f A and B have p r o p e r t y L, t h e n  t r a c e ( aA + bB ) = t r a c e (aA + b B ) =  and  12  n £  j  a\+  b y A j  ( k = l , 2 , * " , n ).  k  Expanding b o t h s i d e s of t h e second i d e n t i t y , we have K-I y x y a t r a c e A + H a b ~ t r a c e ( 2__i A B " « A B ) + b t r a c e P=' x ^ . • .+x = p x  k  k  p  k  p  1  1  r  r  k  B  k  r  l * r ~ x-^0, .Xgj • • ,x >0  y  +  ,  +  y  =  k  p  r  k.( yr^o, yr_1,--..y >0 lc T -,k , V p-Js-p , L v ..p k-p -j£y k = a + L a ' b * ( ) 21 M C + • 1  B  V = l  &  p=l  P  The c o e f f i c i e n t s o f a b  t.= l  K-P  '  (p = 0,1,•••,k) i n b o t h s i d e s of  the i d e n t i t y must be e q u a l , so we have t h e f o l l o w i n g c o n d i t i o n s : k t r a c e A =T X , trace B = y x and n  n  k  J  lJ  k  -  L  x  3 -  y  x  y  x . ....++xX__== pP X„+. y +« • • +y = k-p 1 +  P  r  x  i/=l  r  l* 2'" * * r y -o, y^!.*** . y ^ o x  0  ,  ,  X  x  >  0  r  Conversely, i f t r a c e ( YLi A * ^ -  A* ^B  1  y rr  (^)(Lx?J5" ). P  ) =  l ^ P ^ l ^ n ,  t r a c e ( aA + bB )k  then  a A + L a ^ ' P L A % ^ . • .A*V K  = trace(  = a trace A k  7",  .*.  X r  /  k  K  + Zla b " trace(  k  p  y7'  k  k s  + 2_,a^b  = £ (  bu ) .  a\+  p  p-,k-p/kx  = fc ( a X , 1  r*  _ p  f( ) p  y A 1 1 B k-p , ^ + J  •A k  .  r  b B k  +  B k  )  k  )  ) + ITtrace B  v  k  L  t r a c e ( aA + bB ) = J L ( a\+ by**,)*, L=  1^-k^n.  i  For b r e v i t y o f n o t a t i o n we now l e t aA + bB = C  and  hjx^z^c^ ( i = 1, 2, • • •, n ) .  aX+  Suppose C- has e i g e n v a l u e s ( i 1,2, •••,n ), we want t o c prove t h a t t h e o^s a r e e x a c t l y t h e Y ^ s . S i n c e t h e r e e x i s t s a u n i t a r y m a t r i x U such t h a t =  1  1  r s.  U C u  and U C U = _ 1  _1  , k = 1,2,  k  <L- n.  O  <  0  r  a r e upper t r i a n g u l a r , t h e n n  n  L^i = G = t r a c e U*" C U = L^ = S , 1-k-n. By Newton's f o r m u l a f o r symmetric f u n c t i o n s , we have t  r  a  c  e  k  1  k  K  k  p  l =  V  S  Po = 5 (  s  i  S  Q  ),  p^ = r a t i o n a l f u n c t i o n o f S-^Sg, •••»S| , C  -4-  where p^., l ^ k ^ n , a r e t h e elementary v a r i a b l e s S,,  ^ ,and n  l^k-^n, so p^,  f u n c t i o n of the n  f ( x ) = x -p-^x ~' '+• • • + ( - l ) n  n  1  1-k-n, a r e a l s o t h e elementary  of V, , T^, . .., Y„ .  n  X  p  functions  Hence t h e s e a r e a l s o t h e n r o o t s o f  the c h a r a c t e r i s t i c e q u a t i o n  f ( x ) = 0 o f C.  Since  f ( x ) z 0 i s of degree n and has o n l y n r o o t s , t t h e o^'s a r e e x a c t l y the V s .  Therefore  =  aX^+byu^  n ) a r e t h e e i g e n v a l u e s o f G = aA + bB.  ( i =  1,2,  T h i s completes  the proof o f t h e theorem.  3•  Application. I n t h i s s e c t i o n we s h a l l d e t e r m i n e  necessary  explicitly  and s u f f i c i e n t c o n d i t i o n s on t h e c o e f f i c i e n t s  of an n-square complex m a t r i x B, when n = 2 , 3 and 4 , such t h a t A and B have p r o p e r t y L by u s i n g t h e Theorem 1. We'assume t h a t A i s i n J o r d a n c a n n o n i c a l form. Case 1. when n=2, suooose B=  has e i g e n v a l u e s y/.,andjx^  There a r e t h r e e cases t o be c o n s i d e r e d . A, o (i) LetA = , \jz\, then A and B have p r o p e r t y L o i f and o n l y i f °2.2 °21 ~ °* 1  proof. if  By Theorem 1, A andB have p r o p e r t y L i f and o n l y  the f o l l o w i n g r e l a t i o n s hold: (2)  t r a c e (B) = b]_]_ + b,  = 2  t r a c e (AB) = 0^,+ or  h  / i / a' A  +  u  22\ / t\ /zK' =  i  (/,-b ) + ( ^ - b ) = 0, 11  (/ r n)^.+ l  b  z  22  ^ - 2 2 ^ a b  +  =  °*  +  - 5 -  Since  1  1  X,  X  2  22 Now  u u „ = b- _, b„^ - b b =u ^1T2 11 22 12 21 l n  r  r  U 2  - b 12  b  21  . .  b b = o . 12 21 Conversely i f b b = o, t h e n u = b ,/A = b and (2) x^ ^x J_ IX 2 22 holds. ; t h e n A and B have p r o p e r t y L i f and ( i i ) L e t A= f A, 1 ° A, only i f b =o. 21 Proof. ( t r a c e (B) = b., + b = JU. + JU 11 22 O ^2 U r a c e * (AB) =A(b + b } + b„ = A.(u +u ) 1 11 22 21 / l 2 n  7  . . b = 2 1  1  o....  T h i s i s c l e a r l y n e c e s s a r y and ( i i i ) I f A =Al, Proof.  sufficient,  t h e n , f o r any B, A and B have p r o p e r t y L.  The c h a r a c t e r i s t i c  e q u a t i o n of a A + ^B i s  (3)  ^|<xA+(aB-<Tl| = o ,  or  'iJo{Al+^B-<Tl|=o.  I f (3 = o, t h e n the e i g e n v a l u e s of  A are b o t h  <* X .  I f (3 ^ o , (3) becomes  I B -( £  ) I |= o  I f B has e i g e n v a l u e sy/C^ andyx^ , t h e n each and  <5- . = o( X + ^ ^ A .  ,  i=  1,2.  —  ——  - 6 -  Case I I . n = 3. We l e t B = ( b j j ) ,  ( i j = 1,2,3) have e i g e n v a l u e s  There a r e s i x cases t o he c o n s i d e r e d ,  but we s h a l l  give  the p r o o f o n l y f o r t h e f i r s t case and s t a t e t h e 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 s f o r the r e s t w i t h o u t g i v i n g t h e i r  proofs.  ( i ) L e t A =|o Xi o [o o X j  then  , where A, ^A^and A  A and B have p r o p e r t y (4).u,=b„ ^l ll-> =D  U,„b  L i f and o n l y i f  kx „= b UL _=b M 22*> - 3 33  .  =  D  M  ;  2  ; b, b,, i b _ b _ = ( X -X ) : (X^-Xj) : (\-A ),  on 12 21 • " 3 ln " l 3 • ~23~32 P r o o f I f A and B have p r o p e r t y b  trace B = b  b  1 1 +  t r a c e AB = b (.5)  are d i f f e r e n t ,  3  £ 2  {  z  3  L, t h e n f r o m (1),  + b ^ =^  .  A, + b X + b A~„ =M X, +u. X + u 11 1^ 22 | 33 3 ^ 1 1 ^ 2 L ~3  trace A B = b^X* + b^A** b ^ A ^ ^ A * = £  2  £  i=)  trace AB = ( 2  b . ^ . = £  k«l -3  -3  ) A +( L  lk  (  L  2 b  3k\  ) 3  A  +  3  / b  b 2  k=l  3 +  2  1=1  b b  k=i  ^  ) k  2  V  ;  3^1 1 ^2V/ X  k  +  X 3  3'  For example, t h e t h i r d e q u a t i o n o f (5) i s d e r i v e d f r o m (1) when k=3, p=2; i n t h i s case (1) becomes.  (i)lX/-;= t r a c e  ( A B +ABA + B A ) 2  2  = t r a c e A^B + t r a c e ABA + t r a c e BA =3trace A B , . 2  s i n c e , f o r any m a t r i c e s M,N, t r a c e MN = t r a c e  5  +^ X^ ^ A* .  2  trace B  3  E M.  - 7 -  Prom t h e f i r s t t h r e e e q u a t i o n s o f (5) we have - b  [LK. v /  l  11  ) + ( M. - b ) + ( M - b ) = o. ^2 22 ^ 3 33  V^l  11  ) +A  (u  2- J  7  - b )+A„(u - b )=o. 22 3 / 3 33  2  ) + A , (/A - b ) = o \\ ( M. - b ) + A (JA, - b \ 1 11 22 22 3 33 r  Since  (6)  r  5  J  1 1 1  Aj Aj. A  3  Ai A  3  =b  M.  ^1  A  2  11  = ( \ - \ )  =b  , X A  ^ 2  ( ( A  , JJ<  22  ^ 3  3  = b  - X J )  jt o  33  Prom the l a s t two e q u a t i o n s o f (5) and ( 6 ) , we have • 12 21 W ^ l = o , (b  b  +  b, r,b„„ + b„  12 21  or  31  ^b  O I 7  23  b  b  32  b  +b b  23  b  b 31  13  b  +  3 1  : b 32  X  : b 31  b  23 32 2 b  +b  2  b b 12 21  b 13  +  U  +  (  V l 3  +  = o -  23 32  +b  13  b  + b b  13  b A 32 1  ,b  j 23  < 21 12  +  b A. 21 3  12  = o  =o. b  12  = ( X - X ) : (A -X ): {X -\ ) 21 2 3 3 1 1 2  Hence a n e c e s s a r y c o n d i t i o n i s (4)  1  J \b_  23  = b "11 b  ,/X  : b 32  = b  2  b 13  = b  22 ; b  31  b 12  33 '  = (A - X ) i (A - A ) : (A - X ). 21 2 3 3 1 1 2  C o n v e r s e l y , i f (4) h o l d , then  t r ac e A B = A  1  b  11  +A  b  +A „b  =Au+Aju.+A  M  «  2 22 3 33 I I 2 2 3^3 2rs 2 . ^2 ^ . 2 =AA 2 M. .+ . A 2K + . A 2 U trace A B = X b + A b + A b 1 11 ^2 22 3 33 1 1' 2^2 3^3' r  2  N  - 8 -  (4)  Cont'd t r a c e AB = A ( b + b b + b b ) + A (b b + b + 1 11 12 21 1 3 31 2 21 12 22 2  2  2  2  + b ,, b,J + A„ (b„ b + b b .+ b ) 23 32! 3 31 13 32 2 3 3 3 0  "* \ = A  2 h _ 1 11  (A  *\ 2  b  2  2 b  +A  nn  22  3  +b  12  3 3  b  21  ( A + A )+ b b 1 2 223 3  3 2  + A ) + b b (A +A) 3 31 13 3 1  2  = X u +A u + A u + k (A l l 2 / 2 3 ^ 3 + k ( A - A ) (X + A ) 3 1 3 1 2  2  - A ) +  2  r  = X ^l  + X  lJ  2  .  /  2  A ^  +  1  2  K(A  - A ) (A 2  3  +A )  2  .  2  Hence by theorem 1 , A and B haize p r o p e r t y L. A4i o  (ii)  o  o At o ^1 0 o A 3J  Let A =  where A  ^X 1  , t h e n A and B hatee 3  p r o p e r t y L i f and o n l y if ^  1  ^  2  b.,, b 1 3  (iii)  1 1  + b 3 1  2 2  b 2 3  ^  3  3 3  = o . 3 2  Let A =  t  A  ^A 1  , t h e n A and B have p r o p e r t y L 3  if and o n l y if  ^1^2 " ll V b  b  +  >S  =  b  33 '  21= ° ' (b  b  + b b ) 13 31 23 32  (X  1  - A. 3  ) = b b 23 31  3  \  (iv) Let A=  x  9  -  1 o  then A and B have property L  o  ^  i  O  O  .Xj  i f and only i f j  b  21  \  = o , b  \ (v)  Let  1  o  then A and B have property L  A=  O  0  \  1J  i f and only i f b  b  31  32  b  rt  21  (vi) Let  = o, = "  b  2 l >.  (b  - b  11  A=  \  33  o  )  =  = X I o  have property X L . o  c  O  3  , then, f o r any B , A and  -10-  Case i n , n = 4 : in this case we let B = ( ij)> (i»J 1>2,3,4), have eigenvalues/^y^'T^ andpf.4. . There are fourteen cases to be considered, but we shall only state the necessary and sufficient condition for these cases without giving their proofs. b  A,  o  O  (i) Let A =  O  o  o  O  D  A3  O  where A A , A 3 and A^ are (/  O  2  O O O  different, then A and B have property L if and only if .j^i =b , i = 1,2,3,4, ij ji( i+\j) = o, j 2. 2. < V . i j"b j i (A,i+ A A j+ A j) = 0, xl  b  b  X  b  ±  i,j=i  ^H  (  i _ , i J J ^ k i " i i ^ i ~ °* b  b  b  A,  (ii) Let A =  b  A  O O O  o  A., o  o  o  o  A3  o  O  O  ^ O  where A1 \$ and  are different.  then A and B have property L if and only if ,M± =b , i = 3,4, 13 31+ l4 4l+ 23 32 24 42 34 43 = °> ( 31 13 32 23 34 43) (A -X J (b4 b ^bz .2 24+ 43 34) = 0. 1±  b  b  b  b  b  b  b  +b  b  +b  b  +b  b  +b  b  b  3  \  (iii) Let A =  0  0 0  b  0 0 A, 0 0 0 A3 °  ]  +  ]  1  b  b  (  0  0  0 A3  where A, ^ A3 , then A and  -11-  B have p r o p e r t y L i f and o n l y i f yU /*2 = bll+D22 , 2. 2. * y*L+/«2 = lk kl+ 1+  t )  At^? b  (iv)  = t  r* L 2k k2 > .  b  I  b  b  lk ki il+ L  b  b  ^  b  13 31 l4 4l 23 32 24 42 b  + b  b  + b  Let A =  b  + b  A,  o  o  o  0  A, o  o  O  o  O  O O  A ,  b  =  b  2k ki' 12 ' b  b  °'  , where A,  A^, t h e n A and B  0  A4  have p r o p e r t y L i f and o n l y i f ^ 4  = bA4 ,  3  1L  "°kA.k  = 0,  K= 1  A,  (V)  Let A -  l o  o  o  A,  I  o  °  o  A,  I  o  o  o X,.  , then A and B have p r o p e r t y L L  i f and o n l y i f *>4i = 0 , b  31  b  21 H  42 =  + b  + b  32 D43 = 0 , f  ( 2k kl+ 3k k2+ 4kbk3) = 0 , b  b  b  •(b  '  0  i r  b  4 4  b  b  b  2k ki il+ 3k k£ i2+ 4kbkibi3) = o,  )b  D  3 1  b  b  b  b  - b 4 3 b £= 0 . 2  b  -12f X,  (vi)  Let A =  o 1  1 0  o  o  X,  i  o  o  X, o  O  O  O  , then A and B have p r o p e r t y  L  X|  i f and o n l y i f °  = 0 ,  3 1  21  b  + b  32 -  '  0  t, ( 2 k k l 3 k k 2 ) = b  b  + b  b  IL t L ( 2 k k i i l b  K-=.|  b  b  3k ki i2) = b  b  0  '  " « . - I  I  o  o  o  X| o  o  o  o  X,  |  o  o  o  X,  'X,  (vii)  + b  '  0  Let A =  t h e n A and B have p r o p e r t y  L  i f and o n l y i f 21 43 [ ~2J b  + b  =  »  0  < 2k kl 4k k3) = b  b  (b  + b  2 k  b  b  k i  b  ' 23 4r 43 21= b  b  (viii)  b  b  Let A =  i l +  b  4 k  b  L  A,  I  o  o  2k kl =  . II L  b  0  A,  o  o  o  o  X,  o  6  O  O  X,  b  >  0  2k ki il= 0 . b  b  b  i 3  ) = 0  "j  0 , b  k i  0  i f and o n l y i f '21  '  0  then A and B have p r o p e r t y ,  L  -13-  (ix)  A, I o o A, o  Let A =  o  o  A  O  o  o  D o  , t h e n A and B have p r o p e r t y L  o  3  A^.,  i f and o n l y i f ^3 =  33  b  hi =  '/^  =  0 ,  2  2 3(L  44  b  2  v*  2  ^ £ 3 ^ 3 3 ) ^ 3 - ^ 1 )  + 3(L,  -2(b h +b b )(A -A ) 13  31  32  23  L to b i+  +8\  2k  1  b  b  b  K-I  i.= I  b  b  l4  b  +  b  A  b  _ b  4l  b  2  t 3k ki i3- 33)( l- 3)'  ' ( L  4k k4  2  2  4 4 ) (^4-^1)  _  2(h b +b b )(A -A4f+  -  3  b  (^3 23 31 V 4 4l)  4  k  2  A  24  =  1  °>  t; 4k ki i4- 44)(\-^4)-  < L  +  Zf2  b  b  b  b  2 k k i i l = °« b  b  I o o o A, o 0 O O A/3 o  A,  (x)  Let A =  ,where X, ^ A 3 ,  t h e n A and B  f> A3 .  ,00  have p r o p e r t y L i f and only i f b  21 =  '  0  +^4 = 3 3 + 4 4 ' b  L  <  ^3  b  b  2k kl=  ( l3 3i+ i4 4l  b  b  V*4 = 33+ b  2 b  b  b  b  + b  23 32 b  + b  24 42)(\"^3) ' b  34 43+ 44 > b  b  L ' & k i i l = (-|^ 3k ki i3 t 4k ki i4- 33- 33 34 44+ t.it=i + b b44+b3 b44-b4iji -4b34b44b4 ) ( X - A ) . b  b  b  b  33  Let A =  b  3  D  +  b  b  b  3  A|  (xi)  b  I o  o  O  O  A,  O  O  _ O  o  A3 I  , where X. £ A,,  t> A , 3  have p r o p e r t y L i f and o n l y i f  b  1  4 b  b  3  t h e n A and B  b  -14-  bg! = 0 , b  = 0 ,  4 3  ( L  b  3k k3 b  L  +  4  -( 13 31 * b  b  ( .L_ +  b  b b b b H 2 k k l + f_i 4 k k 3 =  +  k.= l  y^3 1^4 ) ( ^ 3 - ^ 1 ) 4  ( L ^ A l - L  V  4 k k 4 ) (^5-^1)  K^l  K = l  =  b  £b4  b  i  + b  4 k  b  23 32 b  3k ki i3 b  b  b  k  + b  k 3 ) ( l X  :i i3 =  3 )  A  +  2 b  l4 4l+ 24 42) 4 b  b  H  +  b  23 4lb  (^1-^3)  b  b  =  0  >  4  4k ki i4)( 3 b  - A  A  ( y ^ + X ) ^  b  k  b  >  A  X  ) +  b  2k ki il b  ) •  A, I o o LetA = O  (xii)  o  o  A j 0  » X , 96 X , t h e n A and B have 4  o  X, o  p r o p e r t y L i f and o n l y i f b 21 = 0 , yW-4 = b 4 4  ,  (f_i 4k k4" 44)(X4 - \ b  b  b  ) + I l 2 k k l = 0, b  b  ( l l 4 k k 4 - 4 4 )(^4 + ^ l ) + 2 4 4 l = 0 , b  b  b  b  4 (  ^  4  . L L 4 k k i i 4 - 4 4 ) ( ^4-A-L b  b  b  \  (xiii)  b  Let A = o o  b  b k k i i i = 0. b  +  b  2  y K=H  o o o X  o  o X  0 0 0 have p r o p e r t y L.  )  0  * X  = A l » then, f o r any B, A and B_.  b  +  -15-  A, (xiv)  I o ° o A, I o  Let A =  o  o  X,  o  *  o  , A,£  o  A^,  t h e n A and B have  \  p r o p e r t y L i f and o n l y i f  b  2l  32 =  + b  >*4 =  >  0  ,  ( Il 4k k4- 44)(*4- l)+ b  <  b  K=l  b  b  b  K=  K= l  ( L  b  4k k4~ 44) (^4-\)--% b  b  b34b4i =  < (  b  0  >  b  b  b  b  +  2  0,  k i  k  =  b  t  ( 24 4l+ 34 4 ) (\~\)  b4 b bi4-b44)(A4_A ) + + /  4.  IL 2k kl+ L 3 k k 2  A  b  b kb  1  3k ki i2 b  b  =  2  k l  b  1 1  +  0  The N e c e s s a r y C o n d i t i o n on B such t h a t A and B have  p r o p e r t y L. Theorem 2. L e t A = d i a g ( A-,, Ag? /  A  1  =1  A 0 0  1 1  0 • • • 0 A 0 0  A  r  (u^ 1  N  12  A ) .'where  and  A j =  0  o" '0 ] s  u. 1*• ' 0  i  V0  i s .1 /  0  O'^'U,  Here A i s of o r d e r n, A, i s of o r d e r a., and A . i s of o r d e r i  my ±  where  m  1]L  + m  and u-^jUg, • • •, u  p  i2  + • • •+ m _ is  =  ,  n +n +-..+ n a  2  a r e d i f f e r e n t e i g e n v a l u e s of A.  r  = n ,  -16-  And l e t B  B  B.=  l l 21  B  B  lr) B 2r  12 22  B  B, r l r2 rr, be an n-square m a t r i x w i t h e i g e n v a l u e s B  .where B  v  v - ^ V g , '">^ > n  = ( b ^ ^ ) i s of o r d e r a^. X  i i  If A  r  B  and B have p r o p e r t y L, t h e n  _r  b  r bR ,  y  1  v  -  r b(D .  +  r  r  b^)  v  -  r ^  r  ., . . H . r b<> x  +  = 0, ( k = 1,2, • • -,^-1  ) , Vr)  ;_7l.i  J J"  f-H. 2 1  k  m  + J  '  - 0, ( k = 1,2, • ",a  m  21  ^—' J.K*. o - 2o o 2 +J» s,  + J _ k  n  m  n  o " 2o o 2 +s,J " in  -1 ) ,  + • • •+  r  b  ( r )  jsk*-i  r  r  rs  = 0, ( k = 1,2, *' ' . n ^ - l ).  Proof:  There &  orders of m  s  no l o s s of g e n e r a l i t y i n assuming t h a t the  ( j = 1, 2, • • •, il  i  m  i2-  S s .  ) ore so a r r a n g e d such t h a t ( i = 1,2, • •'  ,r )  i  -17-  One can show e a s i l y by i n d u c t i o n t h a t  A 0  •  •  / Is i  <  ,f  ^  k-2  1  k v . . k - l  ( (T)  u  l  *  u  l^k<.m. .-1  • •  u. ui / ,k-m. .+ 1 ^  k  •• •  I.  .-  0  k  \  m. .-1.  0  •  0  u  k i  Now t r a c e (A B)  — trace  'A* 0 o ki 0  B  i l  12  lr  21 22  2r  l l  B  B  B  B  r  A  r  B  rl  B  r2  Br r  /  k ) + t r a c e ( A B ) + t r a c e (A^B-^) + k  T  where  0 0  0  = t r a c e ( Akj B +...+  i  ^ o  <*-i> i  0  A j  s  0 ••• 0  A. . = 0  and  0  2  0  1  0  o  A^ =  0  k  - (  U  where  •  •  •• • A  0  and  '4«  0 N 0  0 k A  k  i : L  T  2 2  trace (A B^ ) r  r  ( k  = 1,2,...n-l )  = ( b [ ^ ) ( i = 1,2, " T  ) i s of o r d e r  n. ±  -18-  Furthermore, trace ( A p  1 1  A  =  ) = l l  0  trace  A  0  (i)  b  .  In.  I  k  0  2  0  b  21  b  b } -vV  22  ( 1  1  ft,* )  i.  S  ...  1  = trace ( A  b(i)  0  U  >2n.  •••  b^ )... i 1  n  2  a  ± l< a  A  1}  1 mi l  11  )  +  m. -, m., il i l  m + trace ( A  l,  i l +  .  m  1  +  l  m. l 1 +  m  il+  m.  2  12 b  U)  ...  . il m  +  f  f  i  l  i l  m 2  +  b m  1  (i) il+ml2  m  il  +  m  i2  + . .. +  r (D  ...b^)  b  n  + trace ( A  i" l8 m  + 1  1  ' l-  IS:  n i ni-ai. is  J  W ; ,  n  J^i  > « . - ,  +1  ,  n  l8+  1  ...  " i ^ i s . *  n. n.  |=k  N n 1  n  i  -19-  H ^ n + J m.-. + j - l ^ l ^ i  1  +  +  Lm, +j n  m, j 1+  +  + (r  b  jtt  /r  (  l  •  )  ^ i s .  .)u + k  +^ i ^ i s . +J '  1  v^(i)  w  • frl " i- i8 n  i  m  +  1  - ( i t,* ) v u 1  1  J»  " i ^ i s * - ^  c r b^) + F b ^ )  k +  1  i  s  i  1i  ... + (r * -\K + r {  i  s  b(i)  +  +L b where n-1 ,  / . h i j^K+i  1  ? .  .  ( i )  .  ,  ,  t + J ' t*.j-k i = 1, 2,« • • r . +  k , k-1  •  ,,+•••  ..) , 0, when k = m, , and k = 1,2,' i j  =  S i n c e A and B have p r o p e r t y L , by Theorem 1 we have the f o l l o w i n g c o n d i t i o n : ^ + L.*\V  Trace (B) = L ^ V  J - l  r  M r  J  + ...  +  j=l  X  Trace ( Ak -B-) = ) = L> L Trace - — ( •A,B,, **  (CV ) 1  )u  k +  (Fb  ( i )  i +  +  L  -20-  I  0  1  i -  n  m  i B  +  '  n  = A L v + u< L v  i-*lS  +  ) ...  k  +  J ~  k  „*( 2 >  +  )  ( k = 1, 2,» • • n-1 ) . The d e t e r m i n a n t o f t h e o o e f f c i e n t s o f t h e above n s i m u l t a n e o u s l i n e a r e q u a t i o n i s as f o l l o \ ^ s :  1  u t  D =  x  o  0... 1  0  (J)  .. 0  ..  uf (^)u (2)  (  • • •0 ' '*  - J  •  •  •  n-1  ' n  .n-1 u-  2  •  l 2, n  n n 1  3  •  •  • • »0 ••  0  •.  0  n  r  -1  <4 -1>"  1  n-1 u.  n-1 n  0  .. 0  .• o n,-l  1  • • ( u ., -u )r - l r r-1 r n  v  n  ;  * 0 .  Hence t h e s o l u t i o n s i s  f The v a l u e o f D was found by L. S c h e n d a l I n 1891. See t h e book " The Theory of D e t e r m i n a n t s i n t h e H i s t o r i c a l Order o f Development ", V o l . 4 , p . l 7 8 - l 8 o .  -21-  = 0 , ( kz  7  b(. \ 2  +  1,2, • - ; n l - l  r  ) ,  +.  b< ) 2  +  r  b^) 2  2s,  = 0 , ( k = 1,2, • • • , n - l ) , 2  I>  (r) n  = 0 , ( k = 1,2, The  ...,n  -1  theorem i s p r o v e d .  ) .  r" rs a i  +  J  n  ~ rs m  +  BIBLIOGRAPHY  1.  T. S. M o t z k i n and 0.- Taussky, P a i r s of m a t r i c e s w i t h p r o p e r t y L, Trans. Amer. S o c , 114.  v o l . 73,  1952,  pp.  108-  

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