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Matrices with linear and circular spectra Chang, Luang-Hung 1969

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MATRICES WITH LINEAR AND CIRCULAR SPECTRA by  i  LUANG-HUNG CHANG B.Sc.  N a t i o n a l Taiwan University,1958  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n the Department of Mathematics  We accept t h i s t h e s i s as conforming t o the " r e q u i r e d standard.  THE UNIVERSITY OF BRITISH COLUMBIA August, 1969  In p r e s e n t i n g an the  this  thesis  advanced degree at Library  I further for  shall  by  his  of  this  written  fulfilment of  University  of  make i t f r e e l y  agree tha  scholarly  the  in p a r t i a l  permission  p u r p o s e s may  representatives.  be  available  granted  gain  permission.  Mathematics  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, Canada  Columbia  Date  1969  September  17.  for  for extensive by  the  It i s understood  thesis for financial  Department of  British  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  (ii) !  i  ABSTRACT  Much i s known about the eigenvalues o f some types o f m a t r i c e s .  special  F o r example, the eigenvalues o f a  h e r m i t i a n o r skew-hermitian  matrix l i e on a l i n e while those  of a u n i t a r y matrix l i e on a c i r c l e ; t h e i r s p e c t r a are " l i n e a r " or " c i r c u l a r " .  T h i s suggests the q u e s t i o n :  have t h i s property ?  What matrices  Or, more g e n e r a l l y , what matrices have  t h e i r eigenvalues on plane curves o f a simple k i n d ?  Is i t  p o s s i b l e t o recognize such matrices by i n s p e c t i o n ?  In t h i s t h e s i s ,  we make a s m a l l s t a r t on these problems,  e x p l o r i n g some m a t r i c e s whose eigenvalues l i e on one o r more ••; l i n e s , o r on one o r more c i r c l e s .  .  /  (iii) TABLE  OF  CONTENTS Page  SECTION 1 :  Introduction  ..  1  SECTION 2 :  M a t r i c e s with Eigenvalues  on L i n e s  or C i r c l e s SECTIOI 3 :  6  The Expanded M a t r i x o f A Complex Matrix  11  SECTION 4 :  Nonnegative M a t r i c e s  17  SECTION 5 :  Tridiagonal Matrices  21  SECTION 6 :  Compound M a t r i c e s and Some Other Theorems  SECTION 7 :  25  Extended Polynomial-problem • Eigenvalues  on /  BIBLIOGRAPHY  29 38  /  (iv)  ACKNOWLEDGEMENTS  I am g r e a t l y indebted  t o P r o f e s s o r B. N. Moyls  f o r suggesting the t o p i c o f t h i s t h e s i s , f o r a l l o w i n g me a generous amount o f h i s time and f o r h i s many c o n s t r u c t i v e comments d u r i n g the p r e p a r a t i o n o f t h i s thesis.  I a l s o wish t o thank P r o f e s s o r Roy Westwick  f o r h i s c r i t i c i s m o f the d r a f t form o f t h i s work.  The f i n a n c i a l support Columbia i s g r a t e f u l l y  o f the U n i v e r s i t y o f B r i t i s h  acknowledged.  /  1.  Introduction Let a 12  11 a  A  21  \ n1 a  a  a  22  = ( a nn  n2  j  k  )  J  be any square matrix o f order n w i t h elements i n the complex field  C#.  A non-zero v e c t o r  complex e n t r i e s  x^ ,  ... »  s a i d t o be an e i g e n v e c t o r o f value X j  such  x  A  n  that  corresponding  n  AX = \ X to  )  T  with  is  the e i g e n -  X i s a r o o t o f the c h a r a c t e r i s t i c equation det(  of degree n. of A.  ... , x  X = ( x  A I - A ) = 0  Counting m u l t i p l i c i t i e s there are n eigenvalues  The s e t o f eigenvalues o f A i s c a l l e d the spectrum o f  A. There e x i s t s a l o t o f i n f o r m a t i o n about the eigenvalues of  some s p e c i a l types o f m a t r i c e s .  seen t h a t d i a g o n a l ( a ^ = 0  if  F o r example, i t i s e a s i l y  and t r i a n g u l a r ( a ^ = 0 i f j > k )  matrices e x h i b i t t h e i r eigenvalues on t h e i r main d i a g o n a l : =  ^ j j ' «5  =  1»2,...,n.  I t i s not q u i t e so t r i v i a l that the eigenvalues o f a T r e a l symmetric matrix A (A = A , where A of A) l i e on the r e a l a x i s . [8]  T denotes the transpose  T h i s was f i r s t proved by A. Cauchy  i n 1829» and many subsequent proofs have been g i v e n , by  2.  other eminent mathematicians,  i n c l u d i n g J a c o b i and S y l v e s t e r .  T h i s theorem was g e n e r a l i z e d by Hermite [%2] i n 1 8 5 5 t o matrices for  which  A = A  (A  denotes the transpose conjugate  of A),  and r e s u l t e d i n such matrices b e i n g named a f t e r him.  A well-  known simple elegent o n e - l i n e proof o f t h i s r e s u l t ' i s the following : For a u n i t eigenvector  X  corresponding t o X ,  X = XX*X = X* X = X*AX = ( X*AX ) * = A. A  In a s i m i l a r way the eigenvalues o f a r e a l skew-symmetric matrix (A = -A ) l i e on the imaginary a x i s . proved by A. Clebsch [ 9 ] ,  and l a t e r by W e i e r s t r a s s [ 1 9 ] .  That the same i s t r u e f o r a skew-hermitian was shown by G. S c o r z a j t 8 J notes t h a t  iA  T h i s was f i r s t  i n 1921.  matrix  (A = -A )  T h i s i s immediate i f one  i s hermitian. T  The  eigenvalues o f an orthogonal matrix A (AA  = I , where  I i s the i d e n t i t y matrix o f o r d e r n) have absolute v a l u e 1 and occurc  i n reciprocal pairs.  T h i s was proved by F. B r i o s c h i  [7] i n 1 8 5 4 , and a g a i n by F. Rahusen £17} i n 1 8 9 4 . *  v a l u e s o f a u n i t a r y matrix A (AA value 1 .  The e i g e n -  it-  = A A = I ) a l s o have absolute  The proof was f i r s t g i v e n by H. Aramata [3] i n 1 9 2 7 ,  and a s h o r t proof was g i v e n by R. Brauer r e s u l t i s obvious i f we observe  [ 5 ] i n 1928.  The  that f o r a u n i t eigenvector X  corresponding t o A,, 1 = X*X = X*A*AX = ( XX*){XX)  .', • = XA.  3. As  f a r as the eigenvalues  nothing  specific  obviously  o f a general matrix a r e concerned  c a n be.said about t h e i r  location;  l i e anywhere i n t h e complex p l a n e .  A great  theorems have been proved about t h e l o c a l i z a t i o n many h a v e b e e n s u m m a r i z e d b y  M. M a r c u s  and  they can many  o f eigenvalues;  H. M i n e  [15] a n d  M. P a r o d i [ 1 6 ] .  The in or,  three  common.  types  o f matrices  Their eigenvalues  i n other words, t h e i r  suggests t h e question more g e n e r a l l y , curves  each l i e on a l i n e  spectra arelinear  sort  by i n s p e c t i o n  ? ?  have t h e i r  I s i tpossible  t o answer.  exploring  some m a t r i c e s w h o s e e i g e n v a l u e s  A matrix  In this  thesis,  eigenvalues t o recognize  we m a k e a s m a l l  eigenvalues  l i e on a l i n e .  rotation  of matrices  translation.  f o ra matrix  Or,  on plane such  start,  called  t o b e HORT a n d UOT.  I t s  w a y we d i s c u s s "UOT"  from u n i t a r y matrices  We o b t a i n n e c e s s a r y  "HORT"  from an hermi-  and/or t r a n s l a t i o n .  In a similar  m a t r i c e s , which c a n be obtained suitable  ?  l i e o n one o r more  t o b e HORT i f i t c a n b e o b t a i n e d  t i a n matrix by a suitable  This  '  s e c t i o n 2 we i n t r o d u c e a c l a s s i ssaid  circle;  property  o r o n one o r more c i r c l e s .  In  tions  o r on a  These seem t o b e r a t h e r d i f f i c u l t \  questions  lines,  something  or circular.  What m a t r i c e s h a v e t h i s  what m a t r i c e s  o f a simple  matrices  :  mentioned above have  by a  and s u f f i c i e n t  condi-  4. With each a n a t u r a l way A.  n x n a  complex matrix A there i s a s s o c i a t e d i n  2n x 2n  r e a l matrix c a l l e d the Expansion o f  In s e c t i o n 3 we d i s c u s s some simple r e l a t i o n s between  A  and i t s expansion. In s e c t i o n 4 we examine matrices.  n x n  nonnegative  indecomposable  We show that i f such a matrix has a l l i t s eigenvalues  on a l i n e , i t has r e a l eigenvalues, t i v i t y i s a l most 2 (Theorem 4.2).  and i t s index o f imprimiI t s eigenvalues l i e on a  c i r c l e i f and only i f i t s index o f i m p r i m i t i v i t y i s n (Theorem 4.3). In s e c t i o n 5 we introduce two f u r t h e r c l a s s e s o f matrices called  "almost" h e r m i t i a n ( D e f i n i t i o n 5.2) and  (Definition 5 . 3 ) ,  and o b t a i n necessary'and  "almost" HORT  s u f f i c i e n t conditions  f o r a matrix t o be "almost" HORT. In s e c t i o n 6 we d i s c u s s the r e l a t i o n s h i p between the e i g e n values o f the r  compound matrix  F i n a l l y , i n s e c t i o n 7, -problem on eigenvalues ; roots  C ( A ) o f A and those o f A. r  we looked a t the extended  polynomial  t h a t i s , the d e t e r m i n a t i o n o f the  X of det( A A r  + 7i. ~ A r  0  where the A^  are n x n  the degree  is 1  r  or  1  l  +...+ A ) = 0, r  complex m a t r i c e s . 2,  In the case where  we g i v e some c o n d i t i o n s f o r the r o o t s  to l i e on a l i n e o r on a c i r c l e .  In  this  t h e s i s we  C#  s h a l l use the f o l l o w i n g n o t a t i o n :  t h e complex number  H  hermitIan  S  skew-hermitian  field.  matrix. I  matrix.  i  U  unitary matrix.  ;  (p><l)  a p o i n t i n the complex  Exp A  expanded m a t r i x  Q,  — -  totality k  of  plane.  A.  of s t r i c t l y  i n c r e a s i n g sequences of  integers chosen from  1,2,...,n.  th C (A) r  Ajxjy]  r  compound m a t r i x submatrix  y.  of integers. determinant  A.  o f A u s i n g rows numbered  columns numbered  det(A)  of  of  A.  /  x  and  Here x and y a r e sequences  2.  Matrices with eigenvalues  Definition its  2.1  A matrix  eigenvalues  if  for a l l  that  case,  (m-circular)  One  should  Those m a t r i c e s  just  m f o r which A i s m-linear.  It  the  1 - l i n e a r and  1-circular matrices.  Every  s h a l l be  We  b u t we  those  as i n D e f i n i t i o n  one d i s t i n c t  O-circular,  are  could  content  1 - l i n e a r and  eigenvalue  eigenvalues 0,  These m a t r i c e s 1-circular.  parametric  a = r+si  equations  for  x = r + pf\ y = s + qt j Let Now  them  a=r+si,  if  there  exist  B  has r e a l  B =  spec-  O-linear among  are not the  Such  matrices  two p o i n t s .  i f pj^O,  ( A - al)e~^  and  I f the l i n e  w i t h angle  1  A  t i s a real  0 = n/2  has r e a l  i s 1-linear.  of  of inclina-  roots.  parameter.  i f p=0. Conversely,  a c o m p l e x number a and a r e a l number 0 s u c h  roots, then  L  are :  where  q/p = t a n 0  the matrix  2.1.  i s 1 - l i n e a r and 1 - c i r c u l a r .  of A passes through  then  (the  t o i n c l u d e them  Suppose t h a t t h e m a t r i x A i s 1 - l i n e a r .  tion  call  f r o w h i c h t h e s p e c t r u m c o n s i s t s o f a t most  2x2 - m a t r i x  pesky when  and  ones t h a t a r e b o t h  In  discussion only  trum i s a point) are r e t h e r s p e c i a l .  only  m-linear  b u t n o t more t h a n m l i n e s .  t o us to define m-linear with  be  s o m e t i m e s b e f a c e d w i t h some  are i n t e r e s t e d i n the least  appears reasonable  plane.  i t i s also  could argue that A should  l i e on m l i n e s  h o w e v e r , we  i f a l l  ( c i r c l e s ) i n t h e complex  combinatorial considerations, which merit we  circles.  implies that i fA i s m-linear,  p^m.  i t seigenvalues  or  i s m-linear  l i e on m l i n e s  This definition p-linear  A  on l i n e s  Hence  that  Theorem 2.1  A i s 1 - l i n e a r i f and only i f there e x i s t a complex  number a and a r e a l number 0,  0^0 <"r such t h a t (A - a l ) e " " ^ i  has  real roots. One might be tempted t o say that the problem o f r e c o g n i z i n g 1 - l i n e a r matrices r e a l l y amounts t o r e c o g n i z i n g matrices with r e a l eigenvalues. Theorem 2.1  But i t i s not t h i s simple.  The c r i t e r i o n o f  may not have too much value i n r e c o g n i z i n g 1 - l i n e a r  matrices from those w i t h r e a l r o o t s .  However, matrices r e l a t e d  to h e r m i t i a n matrices can be r e c o g n i z e d . D e f i n i t i o n 2.2  L e t A = ( a ^ ) be an n-square matrix with  elements i n C#.  We say that  and/or t r a n s l a t i o n "  A  i s HORT "hermitian on r o t a t i o n  i f (A-aI)~ ^  i s h e r m i t i a n f o r some complex  i  number a a n d - r e a l number 0. I f A i s HORT and H=(A-aI)e~ ^  i s h e r m i t i a n , the eigenvalues  i  of A are  A.(A) =  ( H j e ^ + a,  J  the l i n e x = r  y = t a n 0-(x-r) + s  i f 0 ^  ir/2 and on the l i n e  i f 0 = ir/2.  (S - 0 - I ) e ~  i 7 r > / 2  Theorem 2.2 v  i s HORT  and w  a  jj  with  = " jj T  kj =  matrix  S  i s HORT  since  = - i S i s hermitian.  A  a  and  and they l i e on  J  Note that a skew-hermitian  numbers  j=1,2,...,n  w a  a  jk  +  (  i f and only i f there a r e complex i wI = 1  V  -  W  )  such t h a t  ' i  o  v  j ,k=1 ,2,... ,n.  8. Proof.  If A  number a  i s HORT, there e x i s t , by d e f i n i t i o n , a complex  and  hermitian.  a r e a l number 0  such that  H = (A-aI)e ^ i s _ i  That i s . (A - a D e " " ^ = (A* - a l ) e ^  or  A* = e " ^ ( A - a l ) '+ a l ,  thus  a, . = e "  2 i  a_j = ~  ^ 3k  2 i  e  Put  v = a  and  2 l  ^a  + ( a - ae"" ^) 2i  i i  a  f  o  J^ » 0,k=1,2,...,n.  r  k  w = e"~ ^. 2i  The converse i s immediate by r e v e r s i n g the order o f the ab ove i; argument. In p r a c t i c e the r e c o g n i t i o n o f an HORT matrix i s even simpler than the c r i t e r i o n o f Theorem 2.2. can be assumed r e a l . imaginary,  If 0=0,  I f 0 ^ 0 , a(=v)  a can be assumed t o be pure  a(=v) = b i where b i s r e a l ,  0 = 0 , and w = 1.  Thus we have Theorem 2 . 5  A i s HORT  i f and only i f e i t h e r (1) o r (2)  holds. (1)  / &^  =  w  a  jj  + v(l-w),  • •  aj.j = w a , Jk  j,k=1 ,2,... ,n  f o r some r e a l v and complex w such that | w I = 1. (2)  a ^ j = a^. - 2 b i ,  f o r some r e a l number b.  = a^,  j ^ k , j ,k=1 ,2,.., ,n  HORT matrices are t - l i n e a r , but o f course they a r e by no means the only 1 - l i n e a r m a t r i c e s , j u s t as h e r m i t i a n m a t r i c e s are not the only ones with r e a l eigenvalues. I n ' f a c t the example A =  P  a  lo  |  \  shows t h a t a matrix w i t h r e a l elements and r e a l  2)  eigenvalues can be j u s t about as "unhermitian" as one can conceive. Suppose that the matrix  A  i s 1-circular.  The equation o f  the c i r c l e o f eigenvalues must be o f the form (x - r )  + (y - s ) = b ,  2  2  where b i s a non-negative  2  r e a l number.  Analogous t o the case o f HORT  matrices,  we make the  following definition. Definition  2.3  elements i n C#. (A - a l ) b ~ ^ number  L e t A = ( a..^ ) A  be an n-square matrix with  i s UOT " u n i t a r y on t r a n s l a t i o n "  i s u n i t a r y f o r some complex number  and  real  b^O. /  If  a  i f  A  eigenvalues  i s UOT  1  and U = (A - a l ) b " '  o f A are \ j ( A ) = X j ( U ) b + a,  i s u n i t a r y , the j=1,2,...,n  and they l i e on the c i r c l e (x - r )  2  + (y - s ) = b 2  2  i f a = r+si.  to Theorem 2.4 number v  A  and  i s UOT  i f and only i f there e x i s t a comple  r e a l number w^O  such that  n 11 _ y a_.„a..„ m=1  - va^ . - v a ^ + | v|^ -  n _ V jm km ~ m=1 a  Proof.  If  number a  a  A  and  i s unitary.  v a  kj "  _ j k = °'  v a  3» =1»2,...,n. k  i s UOT there a r e , by d e f i n i t i o n , a complex a r e a l number bj£0  such that  U = (A - al)b"~^  That i s UU* = (A - a l ) b ~ ( A *  - aDb"  1  or  = 0  AA  - aA  1  = I  - aA + |a| I = b I,  n thus  _ I 0 m j m - **U m=1 a  a  n y ~ ~  _ Z j n  3  - ^  "  a a  "  a  a  jj  _ _ icj ~ j k a a  +  =  |  ° '  a  |  2  =  b  2  d»k=1 ,2,.. .',n.  m=1 Put  v = a  and  w = b.  ;  The converse i s immediate by r e v e r s i n g the order o f the above argument.  ^  11 3.  The expanded m a t r i x In  there  looking at 1-linear,  are really  matrices real  A  note,  there  A",  matrix  o r more g e n e r a l l y m - l i n e a r ,  two m a j o r a v e n u e s o f i n v e s t i g a t i o n  and r e a l  matrices, We  o f a complex  matrices.  We  but explore  both  denoted by  real  Exp  A = (  a  problems.  matrix  t o each complex  called  matrix  t h e "expanded m a t r i x  of  and defined f o r  A,  jk  complex  s h a l l not confine ourselves to  however, t h a t corresponding  i s a related  :  matrices  = (  }  V  *  c  d  k  i  b  ),  b  jk'  c  j k  r  e  a  1  '  by °11 /  ~ 11 C  ;" 2 i b  •  Exp  A  u - 21 C  b  11  c  21  b  1960,  E. G o t t det(  A one-line proof in  12  c  b  2 2  ~ 22 c  °n1 n1  -°n2  [10]  Exp A  b  1n  2 n  c  2n  ~ 2n  b  2n  c  c  22  b  2 2 . '* °  ...  • •  b c  • •  c  n2  nn  b  n2  nn  • •  °nn b^^ nn  |det(A)| . 2  t h e o r e m was g i v e n b y J . L. B r e n n e r [ 6 ]  Re d e t ( A ) Exp A  1n  ~ tn  12.  1961 b a s e d o n a n i n t e r e s t i n g  det(  c  proved/that  ) =  of this  1n  b  b  • •  •  b  °12  " 12  •  •  In  2 1  b  theorem o f Im  S. N. A f r i a t  det(A)  ) = det  = -Im d e t ( A )  [1]  Re d e t ( A ) j  |det(A)|  2  .  12. B r e n n e r a l s o showed t h a t Exp  A  consists of the eigenvalues This  roots.  i s particularly  The r o o t s  certainly We  give  and  if  results  A = ( a..^ )  complex elements. A  interesting  Exp A  A i s skew-hermitian E i r s t we  A is,normal.  conjugates. A with  those o f A counted  relating  A  and  real  twice,  Exp A  he a n n - s q u a r e m a t r i x  i s normal i fand o n l y  i s symmetric i fA i s h e r m i t i a n ,  Proof.  f o rmatrices  of  1-linear.  some s i m p l e Let  of the eigenvalues  o f A and t h e i r  o f Exp A a r e j u s t  real  T h e o r e m 5.1  Exp  the collection  Exp A  i n  with  i fA i s normal. i s skew-symmetric  and  Exp A  i s u n i t a r y i fA i s u n i t a r y .  show t h a t  Exp A  i s normal i f and o n l y i f  Suppose  a ^ =  + c ^ i ,  and l e t  Since a  *  n  m=1 if  and o n l y  m=1 /  i f . n 7~  n a  ,im km a  Y~  =  m=1 (Exp  *  a  ,im km  j ,k=1,2,... , n .  a  m=1  A)(Exp A ) * -  ( ( f^^Xm  )  j k  )  )  j k  )  m=1 =  ( (  2_  m=1  A  jm km A  = ^  A  )  ^  A  >  13. if  and only i f *  M  n =  (  _  _  n  TZ jm -n m=1 a  (  a  Thus A i s n o r m a l If  A  )  j k  =  21  jm km m=1 a  a  Exp  A  has r e a l roots.  Exp  A  normal.  of  Exp A  Hence  elements a r e r e a l , Similarly,  Exp A  A  Exp A  Assume  that  A  .  Exp A  '  a r e r e a l , and, since  o f A and t h e i r  conjugates,  being hermitian also implies  |a-bi|  eigenvalues  must  sincei t s  i sagain  i f one pure  .  be a n n-square m a t r i x w i t h  i s HORT;  i . e . , (A - a l ) e " " ^  a=r+si  and  complex  i s  r e a l number  0  i s 1 - l i n e a r i f ( 1 ) 0=TT/2, o r ( 2 ) s = 0 of A i sa point.  otherwise.  Exp A  i s 2-linear  /  I f 0=n/2jthe e i g e n v a l u e s of  A  i s normal.  o f a pure imaginary  |a+bi |= 1 =  0=0, o r ( 3 ) s p e c t r u m  those  A  t h e l a s t two statements,  h e r m i t i a n f o r some c o m p l e x n u m b e r  Proof.  =  i s symmetric.  L e t A = ( a^j. )  0 = 0<T  )  i sh e r m i t i a n , and indeed,  one c a n prove  and  T h e o r e m 5.2  a r e those  However  that the conjugate  imaginary  and  j k  Now a n o r m a l m a t r i x w i t h r e a l  be h e r m i t i a n .  with  )  i shermitian, i t s eigenvalues  eigenvalues  elements.  ( (  *  i f and o n l y i f Exp A  the  notes  )  _  E x p A.  I f  A  lie  on t h e l i n e w i t h  x = r  s = 0 a n d 0=0, t h e e i g e n v a l u e s  a r e a l l r e a l and l i e on y=0. Exp  o f A l i e on  i sobviously 1-linear.  I f t h e spectrum Otherwise,  equation  y = tan0*(x - r ) + s .  a n d s o do  o f A and  Exp A  of A i sa point,  the eigenvalues  of A  14 Those o f  Exp  A  l i e on  y = tan0«(x - r ) + s and This  y = -tan0*(x - r ) - s . proves our  Theorem 5.5  theorem.  Let  complex e n t r i e s . complex number (A - a l ) b the  A =  )  a=r+si  and  i s unitary.  Proof.  By  the  A l i e on  the  be  Assume t h a t  spectrum of A  A  Exp  A  on  two  of  the  2  Exp  +  (y - s )  2  A  = b  above e q u a t i o n 2  Exp  + y  A  i n pairs. circles  U and  UOT;  with  i . e . , there b^O  such  i s 1-circular  2  =  b  that  i f s=0,  i s 2-circular the  are  or i f  otherwise.  eigenvalues  may  be  to the  a l l l i e on I f s^O,  .  2  written  2  which i s symmetric w i t h respect  lie  is  r e m a r k s s h o w n i n S e c t i o n 2,  (x - r )  conjugate  n-square matrix  circle  i f s=0,  eigenvalues  an  r e a l number  i s a point.  (x - r ) Now,  (  the  this  /  - r )  2  +  (y - s )  2  =  b  (x - r )  2  +  (y +  2  =  b .  2  2  axis.  circle  eigenvalues  :  s )  real  Then  f o r they of  Exp  A  the are will  of  t5. More g e n e r a l l y , Exp is  A  one c a n s e e t h a t  i seither 1-linear or 2-linear.  1-linear,  then  A  then  A  i s 1-linear;  c a n he e i t h e r 1 - l i n e a r  however,  that  i f  Exp A  Exp(Exp A) = Exp (A) eigenvalues is  A lines  i s  i salso  a r e added.  A "blown up"  applies  to  This  more i n t e r e s t i n g  B  (2)  B  = B  Q  k  k  m  i f Exp A  i f Exp A  i s 2-linear, I t i s obvious,  c( =1 , 2 , 3 , • •. , t h e n  °< - l i n e a r .  No n e w l i n e s o f  i snot s u r p r i s i n g since duplicated.  Exp (A)  A similar  discussion  matrices. question  i s t o a s k f o r t h e number o f  of  B  l i e , where  m  k  + b I);  1  k  a  k  , b  k  complex,  e a c h t i m e we make a n E x p a n s i o n ,  number o f l i n e s 2 .  Conversely,  then  i s 1-linear,  = Exp(a B _  In general,  i s 1-linear  ©(-linear,  w i t h each root  <*-circular  A  or 2-linear.  on which t h e eigenvalues  (1)  is  i f  o f eigenvalues,  But o f course  B  k=1,2,...,m. we d o u b l e t h e  h e n c e t h e maximum  c a n be  linearity  2 ^ - l i n e a r f o r each q,  m 0 ^= q ^ m, f o r s u i t a b l e c h o i c e s B  could  J u s t when  a l s o be B  ffi  i s  <*Z-linear, c< - l i n e a r  o f B , a^. a n d b .  Conceivably,  k  where  << i s n o t a p o w e r o f 2 .  f o r any integer  °C = 2  m  seems  t o be a complex c o m b i n a t o r i a l p r o b l e m .  To i l l u s t r a t e t h e  possibilities  we g i v e  B^ a n d B  is  skew h e r m i t i a n  hermitian,  thelinearity  of  and u n i t a r y .(Tables  2  when  B  1 and 2 ) .  Q  16. Let Table  1.  B = Q  Conditions P ,q ,r ,s 1  q  1  1  0  =  (p  1  & 0  1 =  a =P2 <l2 ^  a.j=p,|+q.j i ^ O , b ^ = r ^ + s ^ i ^ O ,  +  2  1  0  =  (q  1 =  on  Exp  B  1  Conditions r  s ^ O  & r ^ O & r  1  1-linear(y=0)  q =0  &  2  1  =  (q  0 1 =  & 0  1-linear(x=0)  0 )  =  s ^ O  &  s  1  on  1  (p  =  0 1 =  & r  1  =  0)  =  0  & T^Q)  0  2  1- l i n e a r ( p e r p .  otherwise  2- l i n e a r  2  P =0 &  s =0  1- l i n e a r ( y = 0 )  q =0  s ?^0  t-linear(perp.  2  &  1-linear ( p e r p . t o y=0)  2  2  otherwise  2- l i n e a r  q =0  1- l i n e a r ( p e r p .  2  ' P =O  2-linear (paral.to  2- l i n e a r ( p a r a l . t o  y=0)  2-linear(paral.to  y=0)  q =0  s ?^0  4-linear(paral.to  y=0)  &  2  2  2  2-linear(perp.  q =0  &  s =0  2-linear  q =0  & s ^0  4-linear  P =0  &  s =0  2-linear  P =0  & S J^O  4-linear  2  2  2  2  2  1  2  2  B = IJ 0  1  1  s ^ O  otherwise  Exp  t o y=0)  s =0  2  Conditions on P ,q ,r ,s  t o y=0)  q =0 & 2  2-linear  otherwise  t o y=0)  y=0)  2  2.  j  2  1-linear(y=0)  s =0  P =0  Table  B  s  2  q  Exp  p =0 & s ^ 0  2  P  2  P2»^2' 2 * 2  1  & 8^=0)  0  and b2=r2+s i^0.  H (S)  2  P  i  B  1  Conditions  on  Exp  B  P2» 3.2' 2'^2  1  (  1-circular  2-circular  r  s =0  1- c i r c u l a r  otherwise  2- c i r c u l a r  s =0  2-circular  otherwise  4-circular  2  2  2  t o y=0)  17. 4.  Nonnegative  Definition called We  matrices.  4.1  A  real  nonnegative,  write  A  Definition  =  i f  4.2  A to  matrix  P  that  as  such  matrices.  i s the  follows  Theorem  of  A)  A.  has  a  which  I f  f o r  =  ( a^  j ,k  =  1,2,...,n.  n-square  decomposable  PAP  =  T  Otherwise  theorem  matrix  A  B  0  C  D  (  I  where  then  they  called (3)  A  i s  on  indecomposable  6  -  e  l 2 l T / n  ( a ^  )  permutation  and  C  are  indecomposable.  nonnegative  [J5],  which  we  state  A  be  an  n-square  nonnegative  indecomposable  real i s a  positive  eigenvalue r  simple root  i s any  of  the  (the maximal  characteristic  e i g e n v a l u e o f A,  has  are  the  h  equation  |/V:(A)I  then  e i g e n v a l u e s of modulus h  index of  distinct  then  roots  imprimitivity  X - j , '/^2» • * *'  ,  eigenvalue  =  of  r.  J  A  the I f  =  B  J I f  i s  :  7V.(A).  (2)  )  k  i f there exists  Perron-Probenius theorem  Let  Then  A  0  A  :  4•1  matrix. (1)  be  fundamental  matrices  =  k  nonnegative  i s said  The  a^  matrix  0.  (n > 1 )  square  n-square  7^9,  a  r  e  a  ^  of of  r  ?\.  : /Vj=r, T^* -  r  =  0;  • • •» ^ - ^ t h  A.  "kke e i g e n v a l u e s o f A  "X^O,..., 7^6  i s  are  A , 1  T^,...,  and  X  n  18. in  some  (4)  order.  , I f  h>1,  then  there  exists  a permutation  matrix  P  such  that  PAP,T  where  the zero  Suppose  blocks  that  the nonnegative  1-linear.  Let  of  suppose  A  and  9^ •JL  r .  Since  i s also — 7\.  7\_and  and  9^  Theorem A  H e  and  roots  of  placed  were  d e t ( A I  with  I f an n-square then  - A)  then  are a l l real.  det(A.I = 0 x  2  + y  Exp A  2  square.  matrix  real)  eigenvalue  - A)  = 0  |xl= = r  2  to the x-axis.  i f A.= A  .  nonnegative  Hence  r .  and  are real.  i s nonnegative  is  also  1-linear;  real,  Then are  Thus we  say  are  r ,  have  indecomposable  A  A i s  eigenvalue,  and  a l l i t s eigenvalues  that i f  are  a complex  of  respect  be on a l i n e  i s t r i v i a l  1-linear,  there  (positive  on o r i n t h e c i r c l e  can only  4.2  that  diagonal  indecomposable  be t h e maximal  the coefficients  i s 1-linear,  It  r  a root  symmetrically  down t h e m a i n  A  :  matrix  Furthermore,  indecomposable i n fact, i t s  19. P a r t ( 3 ) o f Theorem 4.1  i n d i c a t e s that A i s a l most  n/h-circular. If satisfy  A  i s 1 - c i r c u l a r , then a l l the eigenvalues o f  |A.(A)| = r ,  j = 1,2,...,n,  r e a l maximal eigenvalue of A. x  n  of  - r jJ  1  n  = 0, - r  and = 0  n  h = n.  Theorem 4.3  where r i s the p o s i t i v e  Thus  allA  Conversely,  (A)  are r o o t s of  i f h = n,  the r o o t s  c o i n c i d e w i t h the eigenvalues of A,  has 1 - c i r c u l a r eigenvalues.  A  and A  Hence  An n-square nonnegative indecomposable matrix A  is 1-circular  i f and only i f n = h.  An example o f the matrices appearing i n Theorem 4.3 the permutation  matrix /  p  0  1  0  ...  0  0  1  0  0  ...  0  1  1  0  ...  0  0  =  A  The eigenvalue's o f P are the n A.(P) for  = e  i ( 2  */  n )  /roots of u n i t y :  J,  j =  1,2,...,n,  P  i s the companion matrix of the polynomial  P  i s a s p e c i a l case o f the more g e n e r a l matrix  B  is  a  b  0  a  b  0  ...  0  a  b^ n  ...  0  0  1  0 2  ...  0  ...  0  =  b . n—l  A  n  - 1 .  20. I Here,  det(Al  - B) = ( A - a)  + (-1 )  || b, . k=t  The e i g e n -  K  values o f B are : A.CB)  =  i ( 2 7 r r e  / ^  + ,  n  j = 1,2,...,n, n even  a  J  A,(B) = r e  or  i ( l r / n  +  27r  J/ > + ,  j=T,2,...,n, n odd,  n  a  J  where  r =  n / n / I I b^ w k 1 =  The eigenvalues o f hence  Exp B  Exp B are j u s t those o f B counted  twice;  i s again 1 - c i r c u l a r .  B i s i n t e r e s t i n g i n that i t i s 1 - c i r c u l a r even i f the nonzero e n t r i e s are n o n - r e a l .  XJ(B) =  y-prr  c =  || b, = |c|e * . k=t *  Exp B ' i s 1 - c i r c u l a r i f wise .  i  -•  (  e  ,  1-1,2....,».  j r i-  n  where  I t s eigenvalues are  •  q = 0  l e t a = p+qi, and  Exp B  p,q a r e r e a l ,  then  i s 2 - c i r c u l a r other-  21 5.  Tridiagonal  Definition or  5.1  Jacobi  matrices.  A  matrix  matrix i f  A  a..^ =  =  ( a^^  0  )  i s called  whenever  j j  -  a  k J =  tridiagonal 2,  Thus  L  n  ,0  1  c  1  d  2  b  2  °2  d  3  b  0  (5.1)  0  b  =  3  0 c  3  0  0  d  0  is A  a  general  i s similar  in  n-square to a  is  complex  Jacobi  matrix  which  L  matrix. [14] ,  d  Since we  i s m-linear;  c b  n  any  [4]  Theorem  proved  by  F.  n  are interested  particularly  when '  f o l l o w i n g t h e o r e m was  n-1  matrix  1-linear.  The 1961  0  Jacobi  conditions under  „ b j n-1 n-1  M.  \  Arscott i n  :  5.1  ' I fa l l the entries' of the Jacobi  matrix  L  are n  real are  and  c..d..  real  and  We  shall  +1  > 0,  j  =  1,2,...,n-1,  then  a l l i t s eigenvalues  simple. g e n e r a l i z e Theorem  5.1  as f o l l o w s  :  22. Theorem  5.2  Let.  ————————• and  complex  the  eigenvalues  Proof. the  c^,  The  ML  j2  d^.  i n  these  + 1  of  such  minor  always  i n d e t ( X l  same  coefficients  are  real  L^,  i f  &2>  d  the  case  of  the  following  c a l l  L  are  n  j  (  i  j  5.2 * —  +  Theorem  b  are  n  >  Let  0,  5.2,  -  M  a  )  1  =  n-1  ..  n.  3  =  product  when  a l l  5.1  the  of a  Thus  c.d.  -  n,  Then  as  versa.  =  sum  occurs  detCAl  k  i s the  c.  vice  given  real,  c^ ,  numbers  ..  J  J  M  ) =  c^  It  0  and  eigenvalues  L  and  n  are  discussed  a  Jacobi  > • >,  such  i t s eigenvalues  "almost"  1  as  Theorem  and  complex  then  n  c  as  *  equation  roots  By  a n (  =  b, ,  of  simple.  ; _.| that  together  and  definitions  n  »  real  1  V/herever  does  1-linear matrices  Definition YJe  minors.  0.  5.1  3=1,2,...,n-1,  c  k  characteristic  b^ , b 2 » * . . ,  d^,...,  of  occur  the  Theorem  ^»  A  has  By  J^J+-J ^  and,simple.  the  and  II  real  that  real  with  are  so  c d . 3 3+1  L^  that  follows  and  matrix  such  principal  a  elements  are  the  coefficient  (n-k)-square  factor  be  that  c n  matrix  _-|  a n  Cjd^ ^  >  +  a l l real.  ^ 0,  Analogous  i n Section  2,  we  to  make  :  be  a  Jacobi •  hermitian d2,  3=1,2, —  i f  d^,..., ,n-1  .  matrix  with  b> , b , . . . , 1 2 0  d  n  are  complex b, n  complex  are  entries. real,  numbers  such  -  23. Note  that  i f L  3=1,2,...,n-1,  then  Definition "almost" some  5.3  HORT  complex  n  i s hermitian.  l e t L  n  be a J a c o b i  (L  number  5.3  L  complex  numbers  hermitian  L  i f  Theorem  i s "almost"  n  a  i s  n  - a D e "  v  and  ^  1  real  "almost"  and  i s  w  with  d  j + i  "almost"  hermitian, f o r  such  i fand only  |w| =  1  such  L  c. , J  We  0  c a l l  =  matrix.  number  HORT  and  that  i f  !  Q  0 = 0 < TT  there are  that  (b.- v)w  J is  real,  j=1,2,...,n  Proof. a  I f  complex  such  (bj  i s  n  number  that  - a ) e  L  -  i  and  and -  j  d  j  "almost"  a  ( L  ^  i sreal  +  - |  w  a real  and  ^ °»  HORT,  a.I)e~^  B =  n  c  2  3=1,2,...,n-1.  by d e f i n i t i o n ,  number  i s  c^.d^  0  with  "almost"  + 1  e~  2  i  there  exist  0 = 0 < IT  hermitian; i . e . ,  ^ > 0,  j = 1 , 2 , . . . ,n-1 .  * ch Put  v =  a  The  converse  above  i.e.,  and  5.4 ( L  number A  w = e~  .  i s immediate  by reversing  argument.  Theorem  Exp  and  n  i s Exp A  - a l ) e  L^  be a J a c o b i  ^  i s  and  real  _  i  1-linear i s  o fthe  /  L e t  a=r+si  the order  "almost" number  i f(1)  2-linear  matrix  that  hermitian 0  0 = T/2  otherwise.  with  i s  "almost"  f o r some  complex  0 = 0 < TT .  o r (2)  s = 0  HORT;  Then  and  0=  0,  Proof. of  We  Theorem  omit  the  proof here  since  3.2.  /  i t i s similar  to  that  25: 6.  Compound  Definition  1  6.1  =• r = n .  (r)  (ri  X  For  m  y 6  and  matrices  Let  The  r^*  a t r i x  Q  In  particular,  6.1  C (A),  1 = r = n,  y  and  radius  the  point  C  (A),  r =  and  ,2] )  C ^ ( A )= A  Theorem  .L e t  jp+qij  A =  be an n-square  are  r = 2  When  n  1,2,...,n,  A,  det(A[l  I f  C ( A )=  x  and  p+qi  )  det(A[l det(A[l  r  n  y.  j4 0 .  with  1,  a r eon t h eunit  | p + q i |=1 6-1  , 2 i 2,3] ) ,312,3] )  d e t ( A [2,3| 2,3]  )  The eigenvalues o f  center  a t the origin  r = n, theeigenvalues  Fig.  x €" Q  det(A).  n  Jp+qi | =  i s the  r  ),  and  then  d e t ( A [ 2 ^ 3 i 1,3] and  matrix  C (A),  i n  ,2|1,3-] ) ,3|1 ,3J)  det(A[l  (p+qi)U,  (p+qi) .  o f  det( A[xjy]  l i eon a c i r c l e .  theorems.  lexicographically  ,2|1 ,2] ) ,311,2])  ^det(A[2,3M  )  compound m a t r i x  1  i f n = 3  det(A[l  =  other  ( a ^  whose e n t r i e s  /det(A[l 2  A =  arranged  example,  C (A)  a n d some  coalesce a t  a l l theeigenvalues of circle  ( F i g . 6-1).  |p+qi|<1  26. Proof. As  Since  the eigenvalues  ( A ) A . ( A ) . . .A-  since-the  A.(A)  (A),  of  where  a r e on  C (A)  are the  r  1 =  j . <  j  the circle  with  center  0  <  products  ... < j  =  at the  n,  and  origin  J and  radius  circle  with  Theorem real  i f  r  center  at the origin  The  axis  the  i f last  i f  A  =  The  vice  Theorem  this  radius  on  S  r  1 =  r  =  n,  l i e on.the  are pure i f  r  the  |p+qi| .  the imaginary  0^(11),  of  l i e on  r  l i e on axis  real  axis.  imaginaries,  i s even  and  pure  i s odd. follows  numbers  i s a  1-linear  f o l l o w i n g two matrix  from  immediate  from  the fact  that  f i e l d .  the set  {  i f  c r  ( A )  A  =  theorems  : r=1,2,...,nj  i s 2-linear  H.  t e l l  us  1-linear matrix  that by  we  can  certain  obtain  transformation,  versa.  6.5  .  a unitary matrix;  values  C (A)  r  are real  r  that  and  1-circular and  C (S)  statement  notes S  of r  set of real  One  of  of  C (S),  and  the eigenvalues  eigenvalues  The  those  and  of  r,. i s e v e n  while  Since  imaginary  the eigenvalues  eigenvalues  i f  i s odd,  Proof.  is  ,  6.2  the  the  jp+qij  l i e on  a unit  transformation  A  =  ( I  -  S  i . e . ,  we  circle  from  ( P i g . 6-2).  )( I +  S  can obtain a  )~  1  a matrix  skew-hermitian  whose matrix  eigenby  27.  a  Fig. Proof. of  S  "First are  note  pure  that  I  +  6-2  S  imaginaries.  i s nonsingular  Also,  ((I+sr  1  AA*  =  (I-S)(I+S)""  1  (I+S)(I-S)  =  the  roots  (I-S)(I+S).  )*(I-S)  =  (I-S)(I+S)~ (I+S*)" (I-S*)  =  (I-S)(I+S)~'(I-S)" (I+S)  1  since  1  (since  ,  S  =  -S)  =(I-S)((I-Sr (I+S)r (I+S) 1  = Theorem  I.  6.4  I f  det(I+U) A  is  skew-hermitian;  values  1  l i e on-the  transformation  =  0,  ( I  i . e . , imaginary  (Fig.  £  -  we  then U  )(  can  I  +  obtain  axis/from  a  U  )~ a  matrix  0  Fig.  6-3  whose  unitary matrix  6-3).  ->  1  eigenby  this  We  need  only  t o show  that  (u ur )*(i-u)* 1  +  (I+U  )" (I-U  )  1  (i+u~ r (i-u"" ) 1  1  1  (u- (u+i)r (u'" 1  l  1  (u-D)  (U+I)" UU~ (U-I) 1  1  (u+i)- (u-i) 1  (U+I)~ (U-I)(U+I)(U+I)"  1  (U+I)~ (U+I)(U-I)(U+I)"  1  1  1  -(I-U)(I+U)~ -A  1  .  /  29. 7.  Let  Extended  polynomial-problem  Let  A;j,...,A  X =  A , Q  r+1  ( x ^ , X2,..., x ) ^ " b e  ( A  r  A  The  determination  the  extended  +X  0  "  r  1  A  +...+  1  o f \ such  that  polynomial-problem  an eigenvector belonging  the  n-square  complex  matrices.  vector with  complex  Consider  (7.1)  that  eigenvalues.  a nonzero  n  entries.  be  be  r  on  the eigenvalues  A  r  ) X =  0.  equation  ( 7 . 1 )h o l d s  on eigenvalues.  t o the eigenvalue  X  A .  o f the polynomial-problems  i s called  i s said One  t o  cansee  are exactly  roots of  (7.2)  d e t ( A  The  determinant  r»n  .  I f  A  Q  r  A  +X "" A r  Q  + ...+  1  |  i sa polynomial i snonsingular,  i n A  A ) r  =  0.  o f degree  t h e degree  not  exceeding  o f (7.2)i s e x a c t l y  r»n. If are  1 and  the eigenvalues If  is  r =  r =  referred  1• a n d  A  =  o f A  Q  I , then  the roots  o f  d e t ( A I  + A^) =  0  -A^. a n d A^  are,, a r b i t r a r y  t o as a generalized eigenvalue  matrices  problem  the  [t3] .  problem  30. Theorem  7.1  values  of  The A.B  (7.3) X  of  (7.2)  B.j  such  that  +  Q  ( A B  where  roots  +  Q  B^X  i s a nonzero  eigen-  0,  =  vector,  0  I  are the generalized  0 ... 0 ...  0 0  0 0  ...  0  0  0  -I  0  0  -I  0  0  0  0  o.  0 ...  I  0  B  0 0  -A  and  B„  0 0  0  0 0  Q  If  A  Q  values  =  0 0  0  0  .  0  0  0  0  .  0  I,  then  of  B , 2  A  A  the roots  i s nonsingular,  * *.  ~ r-2  ~ r-1  A  A  0 0  =  - r If  I  . .  of  then  -A  (7.2)  •  I -A  2  are the eigenvalues  the.r o o t s  of  (7.2)  are the  where /  0  0  I  0  0 0  0  <  ;  -A  0  0  0 0  0 0  0  0  I  I  B,  0  0  0  .A"  r-1  o  1  " r-2 o '* A  A  1  -A A "  1  -v  ,-1  of  B . 1  eigen-  31 •  Proof. /-XI  I -XI  0  0  0  I  0  0  0  -XI  0  0  0  0  .  -Xi  x  " r-1  ~ r-2  •' '  " r  / °0 X B  +  o  B, 1  \ °  ..  A  A  - I-A O/ A  A  A  To the f i r s t column add 7v_ times the second column, . A  A.B  Q  =  +  I  0  0  0  0  -XI  I  0  0  0  0  -XI  0  0  0  0  0  -XI  I  -Ar-2  ~ 2  -A.  r-1  +A  P(X) =  Since  T ~ A  + ... +  1  det( A B  1  +  Q  0  -AI 1  = (-D - det(-P(X)) r  1  1  -XI/  0  =(-1) ~ det(-P(X))det  ...  A. _i A  "* t ~ ^ o  A  A  A  V  r  )  I  r  l a s t column, we o b t a i n  0  -P(X) where  r - 1 i t i m e s the  -s  9 *  times  0  0  0  0  I  0  0  0  0  I  0  0  0  -XI  = (-l) '" - det( A A n  r  1  r  Q  + >\ " A r  1  1  +...+  A )  the r o o t s o f ( 7 . 2 ) a r e the g e n e r a l i z e d e i g e n v a l u e s o f X B Now i f  A = I, Q  -B  Q  i s an i d e n t i t y m a t r i x o f o r d e r  R  Q  R  + r«n,  .  32. and  the roots If  be  A  o f (7.2)a r e t h e eigenvalues  i snonsingular,  Q  then  A~^  of  .  exists,  a n d ( 7 . 2 ) may  written det (( A . I + X " A A ~ r  r  1  +...+  1  1  A A^ )A )  =  1  r  0  0,  or d e t I( X Since  d e t (A  (7.4) From  ) ^  Q  d e t ( A what  ,B  2  of  + X  0,  r  we h a v e  eigenvalues  I  r  I  r  ~  1  A  A  1  + ...+  1 0  the roots  +A " A A~ r  1  B,,,  r  0.  0  o f (7.2)a r e t h e r o o t s  +...+  1  t  proved  -1 A A~')det('A )=  above,  A A~  1  p  the roots  )  of  = 0 .  o f (7.4)a r et h e  where  0  I  0  0  0  0  0  I  0  0  0  0  0  0  0  0  0  =  -A A " \ r o  -A  1  0  .A" r-1 o  -A^ k r-2 o  1  -A  0  A  A" 2 o  1  A  Q.E.D.  let  PCX)' =  Definition  7.1  (m-circular) fewer  than  We  r  A  +  Q  A  m  r  "  1  A  / 1  The p o l y n o m i a l i ft h e roots  shall  polynomials  X  lines  give when  of  + ...+ A A P(A)  r  _  1  +  i s said  det P(X) = 0  A^. t o be  m-linear  l i e o n m,  butnot  (circles).  a few results r =  1  and  2.  on  1-linear  and  1-circular:  33. For  t h e case  case  when  that  i f A  real  eigenvalues.  are  the roots and  o  needed  on  of  A.  1.2  and  H  .  Let  d e t ( A H  - H.j) = 0  Proof.  Let  generalized or  AH  A(  X*H X o  X H X  X  7.3  and  Q  o r negative  of  1  I f  H  =  1  are  then  simple.  Some  guess  + A>  has  1  0  restrictions  matrices then  o f order  the roots  n  of  A H  eigenvector -  Q  ;  o  X  = X*H.X.  ) = 0  H  A  i . e . , ( A H  by  Since  1  corresponding  X  #  ,  we  X*H.X  - H^)X =  Therefore  0,  j  have  ;  i s real,  1  i s real.  Q  to the  and  the roots  are real.  and  H,  I  det(H )^ Q  0,  are hermitian  matrices  such  that  / then  the roots  of  d e t ( A H  Q  -  E^)=0  real.  Proof. i f  and  AA^  definite,  Premultiplying  O H  this  then  be h e r m i t i a n  and not zero,  - H  hermitian,  One m i g h t  A..  be a nonzero  ) = X A H  Q  are real.  i nt h e  are real [ 2 ] .  = H^X.  d e t ( A H  Theorem  H  0  interested  1  eigenvalue  i s real  Q  of  Q  X  det P(A) =  and  A  be p o s i t i v e  Q  are primarily  I t i snot quite  0 Theorem  we  are both  1  A  1,  r =  H  o  We and  shall H,  prove  commute.  1  H*H* = H ^ H  f i r s t  Q  = H H^ . Q  that  H H^ Q  i s hermitian  On t h e ; o n e . h a n d ,  On t h e o t h e r  i fa n d  i f H H. o 1  hand,  i f H H Q  only  = H H, , o 1  1  = HjH ,  I I  j ( H ^ ^ H X ^ ^ H ^ H ^ .  then  Secondly,  hermitian  we n o t e  H H, = r L H  and  Nox<r,  that  1  O l  t h eequation  Q  are  real  Theorem  are  and so a r ethose  7.4  real,  numbers  of  = H  ) = 0  may  1 0  -  The roots  o f  d e t ( A H  I ft h egeneralized  Q  1-linear  'The r e s u l t  b^O  i simmediate  A . , j=1,2,...,n,  )  d e t ( A l  -  ) = 0 .  o f  i  written  -  A B  H" ) = 1  +  Q  f o r a, b complex .  provided  1  be  o f  eigenvalues  Q  d e t ( ( A-f- a ) l - b B ^ "  where  Q  f o r E~\ i s o|  —T H< . 0 1  —1  I L H  ^  ( X.+ a ) B - bB^ i s  Proof.  i sh e r m i t i a n ,  implies  = 0.  1  1  t o  d e t ( A H  det( X I - R^lT )det(H )  !  H,H~  0  and  B  nonsingular.  q  i f one. n o t e s  = 0  that  a r eo f t h eform  a r et h egeneralized  the roots a + A..b,  eigenvalues  o f  J + B, .  7\B  1  o  We n o w l o o k  Theorem  7.5  positive  a t t h e case  I f  real.  = H ,  Q  definite, det( A  are  A  then 2  A  Q  +  r = 2.  A  Q  1  = H , 1  A  theroots o f A A  1  34.  + A  2  ) =  0  2  = - I  and  H  Q  i s  0  35, Proof.  One  notes  that  / 0 -B  = ( 0  is  1  I  V  0  d e t ( B  det( A  A  Q  7.2,  By Theorem  - B^ ) = 0  A  i s nonsingular  2  2  o  are  Thus  are real  det( A A ( - A ) ~ satisfies  the roots  are real.  2  I f  and  B.  - '  1  H  o f d e t (-B,A o the roots  + B,)  t  =  0  of  7.1.  Theorem  and  +M (-A )~  1  2  the conditions  by  I  = j  V  + A.A.J + A ) • = 0  Q  Remark  A  definite  "  hermitian.  or  i s positive  1  -  1  2  i n Theorem  7.5,  0  I) =  then  a l l i t s  A  -H  roots  real.  Theorem  7.6  det(H )  j4 0  Q  det( A  A  I f and  +  Q  A  AA  ^  + A  2  H  A  Q  0  H  1  = H ,  Q  = H  1  = H-j »  then  H  0  0  ) =  1  are  and  =  2  such  Q  the eigenvalues  that  of  real.  o Proof.  The  7 det( A ^ I + Theorem  / - I B  o  1  Q  the roots of  A B  Q  Q  + A  A^  0,  since  =  of the last  + B^,  f 0  det(B ):£0 Q  B.  + A ) 2  = 0  may  det(H ) Q  equation  are the  be 0.  written By  generalized  where  o\  =  real.  A  f  I)det(H )  and  hermitian, are  ' 1 AB^PT -  7.1,  eigenvalues  d e t ( A  equation  =  I .  .  a n d B B^= - B j B , b y T h e o r e m  Since  B  7.3,  the  . B. a r e  roots  36. Turning results  :  Theorem  7»7  roots  of  complex  to  1-circular  I f  U  detCXU^  +  The  Since  det(U )  ^  o  equation  are  l i e on  7.8  eigenvalues  1  I f  A  = U  d e t ( X  complex  Proof.  the  A  ,  A  B  n  unitary,  by  )  the unit  then  circle  =  1  + A  Q  may  the  i n the  and  0  A^  so  and  +  be  written  =0.  i s unitary,  circle  A ) 2  A  =  the roots are those  2  =  U , 1  0  +  AA  1  0  \  =  and  Theorems  +  A ) 2  of  then  l i eon  B.  the  ./' 0 (  = 0  u  and  are  .  l" i  "V  7.1  =  7.7  the  the  unit  on  the unit  I  0  eigenvalues  2  A  of the  plane.  \°:  det ( A  following  matrices,  ) = 0  )det(U  / - I Since  +  Q  b^U~  the unit 0.  of  i n the  and  =  1  d e t ( A U  + I^U"  0  on  .+ U )  0  Theorem  are  have  are unitary  ) = 0  equation  det( A l  circle  U|  we  plane.  Proof.  d e t ( A U  and  Q  polynomials  circle.  of  above  37. Theorem  7.9  I f the generalized  on- t h e u n i t a  =  r+si  Proof.  circle,  i s a  ( A  complex number  Since  the roots  the form  a •+ b A . ,  generalized  eigenvalues  of  with is  center  + a ) B  at  1-circular.  of  ( r , s ) and  -  i s  and  det(( A A . ,  where o f .A  Q  eigenvalues  B  +  radius  b  A  of  + B.j  b 0  1-circular,  i s a nonzero  where  real  + a ) I - bB^B" ) = 0 j=1,2,...,n, ,  they  b; i . e . ,  are  number.  are  the  a r e on a (A+a)B  are  circle  Q  - bB^  BIBLIOGRAPHY  Afriat, Oxford,  S. N . , Composite matrices, Quart. S e r . ( 2 ) , 5, 1 9 5 4 , pp.81-98.  Aitken,  A.  C ,  A r a m a t a , H., Tohoku Math.  Determinants  and  J.  matrices. 4th  Uber einen Satz f u r u n i t a r e J . Vol.28, 1927, pp.281.  Math.,  ed.,1946.  Matrizen,  A r s c o o t , F . M., Latent roots of tri-diagonal matrices, Proc. Edinburgh Math. Soc. Vol.12, 1961, Edinburgh Math. Notes, No. 44, p p . 5 - 7 . B r a u e r , R., Tohoku Math.  Uber einen Satz f u r unitare J . Vol.50, 1928, pp.72.  B r e n n e r , J . L., Expanded m a t r i c e s from complex elements, SIAM Review Vol.3, 1961, pp.165-166. B r i o s c h i , F., pp.253-256. Cauchy, Vol.9,  J . Math,  A., Anciens pp.174-195-  C l e b s c h , A., pp.232-245.  G o o t , E., A theorem on Vol.2, No.4, October,  Hermite,' C. pp.181-183.  R.  appl.  Exercises.  J. reine.  H a d e l e r , K. P., A r c h . R a t . Mech.  pures  Matrizen,  matrices with No.2, A p r i l ,  Vol.19,  1854,  1829-1830-Coll.  angew. Math.,  Vol.62,  determinants, SIAM 1960, pp.288-291.  Works I I ,  1863,  Review,  Eigenwerete von Operatorpolynormen, A n a l . 20, 1965, pp.72-80. Acad.  S c i . ;  Paris,  Vol.41 ,  1855,  Hojdar, J . , Eigenwertabschatzungen f u r e l n Polynomialproblem, A p l i k a c e Math. S v a z e k 14, .1969, pp.120-133. H o u s e h o l d e r , A . S., The t h e o r y o f m a t r i c e s i n n u m e r i c a l analysis. Blaisdell, New Y o r k , 1964. M a r c u s , M. & M i n e , H., A survey o f m a t r i x t h e o r y and matrix inequalities. A l l y n a n d B a c o n , I n c . , B o s t o n , '. 1964.  39.  P a r o d i , M., La l o c a l i s a t i o n des valeurs caracteristiques des matrices. Gauthier-villars, Editeur-imprimeur-libraire, Paris, 1959. Rahusen,  A.  pp.392-394. Scorza,  6.,  pp.133-179.  Weierstrass,  E.,  W i s k u n d i g e Opgaven,  Corpi  S.  B.  numerici  Vol.5',  e Algebre,  preuss.  Akad.  /  Viss.,  1893,  Messina,  1879.  1921,  

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