Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Maximum and minimum problems in functions of quadratic forms Westwick, Roy 1957

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1957_A8 W3 M28.pdf [ 1.41MB ]
Metadata
JSON: 831-1.0080646.json
JSON-LD: 831-1.0080646-ld.json
RDF/XML (Pretty): 831-1.0080646-rdf.xml
RDF/JSON: 831-1.0080646-rdf.json
Turtle: 831-1.0080646-turtle.txt
N-Triples: 831-1.0080646-rdf-ntriples.txt
Original Record: 831-1.0080646-source.json
Full Text
831-1.0080646-fulltext.txt
Citation
831-1.0080646.ris

Full Text

MAXIMUM AND MINIMUM PROBLEMS IN FUNCTIONS OF QUADRATIC FORMS  by  ROY WESTWICK  A Thesis Submitted In P a r t i a l Fulfilment Of The Requirements For The Degree Of MASTER OF ARTS In The Department of Mathematics  We accept t h i s thesis as conforming to the standard required f o r candidates f o r the degree of MASTER OF ARTS.  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1957  ABSTRACT  Let  be  A  the s e c o n d  a^,  ajc  and  this thesis  E2  [ ( A x ^ , X-J^) , •  let  2  x n  hermitian  symmetric  C2(A)  be  (^-k:' k^-J x  matrix,  function  X  a n c  * *  n e  m  i  2  ~  1  a )  l f  k  2  1  n-space  and  product.  4,  m  u  (\)  a l l sets x^  These  /\  )]  and  of  m  are c a l c u l a t e d .  of  x^^  k  The  orthonormal  d e s i g n a t e s the  r e s u l t s depend on t h e  ( l( l, E  {(Ax^,  A .  2  are taken over  in unitary  i )  2  a  compound m a t r i x o f det  n  ^2^ l>  let  of the l e t t e r s  the second  (C (A) k  Grassman e x t e r i o r E (a  n  t h e maximum and minimum "of  maxima and minima vectors  an  elementary  In  / . 1 ± i \ ^ i  be  a ))  a  inequality  2  k  w  h  i  c  h  i  s  h  e  r  e  k established ^2 x-,,  (  x  l ' "*•» x  f o r a r b i t r a r y r e a l numbers, x  1,  on  t h e minimum o f  k)  where the minimum i s t a k e n o v e r a l l v a l u e s o f k k q q_ such that ^ = ^3 and ]>_^ — zZ. i x  k  a l l sets ..., k .  a  s  i=l for  and  of Here  q  distinct OC^ —  ...  i=l integers =t a.y •  i=l s.,  s  1  i=l taken  from  ACKNOWLEDGMENTS  I wish to express my sincere thanks to Dr. M.D. Marcus f o r the help he gave me when working on t h i s thesis and to both Dr. Marcus and Dr. Mbyls f o r t h e i r suggestions and t h e i r time taken up i n reading through the t h e s i s .  I am also indebted to the National Research  Council of Canada f o r the f i n a n c i a l assistance that made t h i s study possible.  TABLE OF CONTENTS  PAGE  I II  Introduction and Definitions . . . . Statement of General Problem and H i s t o r i c a l Survey  III  1  Results Bibliography  4 5 27  MAXIMUM AND MINIMUM PROBLEMS IN FUNCTIONS OF QUADRATIC FORMS  I  INTRODUCTION  The main results contained i n this thesis are the solutions to the problems of determining; min and max det (Axi,  Xj)  ,  1  - i,  j  ^  k;  where A i s an n by n non-singular Hermitian matrix and the vectors X j are constrained to run over orthonormal (o.n.) sets. To this end two lemmas and a corollary on elementary symmetric functions are established. Definition 1.1  Let 1 ^» p  be positive integers. The  elementary symmetric function of degree p on the k letters a^,• • » , a is the coefficient of t "P i n k  k  TT (t  +  i=l  H)  fc  2 and written as Ep(a^, • • • , a ) . k  I t w i l l a l s o be convenient to define* EpA)  = E (a , • • p  aj.!, a  x  D e f i n i t i o n 1.2  Let x-^, • •  x  p  , • •  i + 1  W  a ). k  any vectors i n V  n  where  p ^ n and, l  x  = ( ll» * ' •* x  ^  = (*plt  '  '  '*  x  l  n  )  ^n)  •  Consider the matrix ,  /  x  (1.1)  x  / I  l l Xpl  ' * * *ln * ' * p n / t. x  Construct the ( ) possible p by p determinants from t h i s matrix by p  selecting p columns a t a time.  Arrange these numbers i n lexicographic  (1.1).  order according to the manner of s e l e c t i o n of the columns from Define t h i s (j*) vector to be x i A • • - A X D e f i n i t i o n 1.5  p  ^(j*) •  J  For p ^ n define  v{t i» 1  h>3  • • .  \  I  i * i i <  • •-<i *  n  P  that i s Qp has as elements sets of p d i s t i n c t integers taken from n  1, 2, • • *, n i n Increasing order.  v  By x w e s h a l l mean w  X  <j  s x  il  / N  *  D e f i n i t i o n 1.4  * *  A  x  i  ; p  ^ l * *  * *  ' p) i  e ( ; i  pn •  Let A be a l i n e a r transformation on V  n  to V . n  Define the l i n e a r transformation C_(A) on V/ \ to V/ \ by defining i t s n  . e f f e c t on the basis set \ e  w  n  \ ;  p  p  c o £ Qp V to be_, n  v; p  5 Gp(A) e j ^ A • • • -A ©ip = Aej^A« • • A  where e  8  is_ the n vector with  Ae^  i n the i * h entry.  For a detailed discussion of the theory and applications of Compound Matrices see  £6 J •  II  THE GENERAL PROBLEM;  In the r e s t of the thesis by  we s h a l l mean (AXJ_, XJ_) .  The general problem i s to examine the algebraic structure of extremal sets of o.n. vectors f o r (2.1) (2.2)  E (A , • • ' , A ) y A  21 ^e^kn  1  k  O  ( k(A)*u,, C  and to compute the extreme values.  For A semi-definite both problems have been completely solved f o r both max and min. £j J »  J  The max and min f o r E i ( A i , * * * > ^k) Hermitian  *  8  known f o r A simply  ^2 J . For r = k i n (2.1) we are dealing with the product of k  quadratic forms.  The invariance under A of L(x^, • • • , x^), the sub-  space spanned by an extremal set x j , has been obtained f o r A nonsingular. Hermitian and the extreme values have been calculated  .  For k a n i n (2.2) we are dealing with det (Ax^, X j ) j 1 - i , j - n. Max min and min max r e s u l t s have been established f o r A definite  [1].  Ill  Let A be non-singular Hermitian on V  Theorem 5«1 Let  RESULTS  • • • , Xjj be_ an o.n. set of vectors i n V . n  orthogonal projection of  n  to V , n  Let P be_ the  to L ( x ] , * * * , x ) . Let A-|, • • • , ^ j be k  c  the eigenvalues of PAP i n L and u j , » • • , u^ the corresponding eigenvectors.  Then. k  det ( A x ^ x ) = r  TT  k  A  i  =  TT  (A^,  Prooft Since x^, X j are i n L (AXi, xj) - (APx  Pxj) = (PAPxi, X j )  it  Hence det (Axi, X j ) = det (PAPXi, X j ) . Now k =  Z  <V  u  j )  u  j  •  x^) .  6  (PAP  v  p) = ( E ( v  j>  u  X  *J J.» U  U  PP U  k  Hence det (PAPx^, x^) = det  (xrx*)  = det (XX*P )  3=1 since  - {<v =  *,)-  f  ]  f V-f I  Theorem J.2  (  Let f(x-), • • • , x ) = det (Ax^, X j ) where A i s k  n by_ n,non-singular Hermitjan neither non^negative nor non-positive and where x^, • • • , x^ i s an o.n. set of vectors. I f x^, . « . , x^ i s an extremal set then L ( x i , • • • , Xy) Prooft Then  i s invariant under A.  ( k < n).  Define U j and P as i n theorem 1 f o r the subspace L(x^, . . . x ) . k  det (Ax^, xj) = det (PAPx^ X j ) = detfPAPui, U j ) and L(x , . . . , X j j . ) = L(u x  lf  .. . ,u ) . k  Assume L(x^, . . • , x ) i s not invariant, and say O = (Auj^, v) / 0 where k  v e L^Cuj, . . . , u ) , U v || = 1. k  Set Ul U  U  Ji  'l  =  - tpV * t * , ,2 ?  'l = i » i -» • • • » U  2  k  •  Consider fCu'^ . . . u*j) (Au» , yx\)  (Au^, u» )  (Au» , u'i)  (Au' , u» )  k  1  k  k  (Au{ , u^) (Au'p u ) 2  1 t"| ol* +  (Au ,  \x\)  (Au ,  M\)  2  k  k  . . (Au'  lf  Note that (AUJ, u ' x ) Note also that Hence  t = 0  2  2  (Au , u ) k  =0 = (AUJ, \xi) = (PAPuj, ux) = 0 for j > 1  (Au'  lf  u^) ^t=0 = -2 |  k  (Au , u )  Since the U i are an extremal set we must have  Wt=G  u)  k  8  (If)  •2 lC-1 t=0  / w v  2  0  -  (AU2, U2)  0  (Au , u ) k  k  A l l other terms are zero since d i f f e r e n t i a t i n g the i  t  h  row  ( i > 1) and setting t = 0 gives r i s e to a determinant with a l l zeros i n the i * *  1  column.  and since  /  Hence  0  TT( 1» j ) = 0 Au  u  Thus either max det (Ax^, X j ) = 0 or min det (Ax^, X j ) = 0 which implies e i t h e r det (Ax^, X j ) 6 0 or det (Ax^, X j ) ^ 0 f o r a l l choices of x^, . . . , x . k  C l e a r l y det (Ax^, X j ) takes on a l l possible products of  k of the eigenvalues of A.  This implies that the eigenvalues are a l l of  the same sign since the product of any k of them must be of the same sign.  But t h i s implies that A i s d e f i n i t e contrary to hypothesis.  Therefore L i s invariant under A.  Therefore we have k min (max) det (Ax-, x.) = lain (max) ~] j A , J i=l r  where ( r i , . . . , r ) 6 0,^ and k  . , . ,*  n  are the eigenvalues of  A. Lemma 5.5  For a^, , . . , a  fc  a r b i t r a r y and real  9 ProofI  Let %  =  %  , 1 . 1 , 2  • • • ,a ) k  = %(ai,  fk(a ) - E k  . . . ,a  " (  2  2  k - 1  ) ( ^ )  , 1 . 1 , 2  )  =2 -  2  E  E  l  •  2  Assume the lemma i s true f o r k - 1 numbers a-^, . . . , a _ ^ k  A  f  *<*k) •» daV [ k E f  + E  a  - *1  -  A 2  - ^  [  a  k  • *f2  2  J  ( k * f) a  E  and 2 i~" ^  ;^2" k( k) = da f  a  Hence f ( a ) i s a maximum a t - • f k ( k ) a  k  k  a  everywhere.  0  =  0»  t  h  a  t  i  s  when a  k  = —±  k  At t h i s point 2  P  A  k - 2 2  d  6 0 For k = 2 we have f  Lemma 5.4 k (1)  (2)  2  = a  (F¥  \2 -an2 „ o 2 * ( l * 2 j = " 2* 6 a  &  2  Let ci^. k  a  .  * s -R  ^  <*  8=1  ^  s  a  \ .  £ -E  S=l  A \2  . 1  ,(il,  E^  k — 1  0  0  If  . . . , ij)€Q  1^ j£ k - 1  j k  .  10 then Min E ( X ! , . . . , Xfc) » Ezi^i  • * • »*k) •  2  Proof:  The value i s taken on by setting x^ =  i n which case (2)  holds because, d-^ + , . . + OL^ i s the sum of the largest j OL's while Xj^ • . . . * i j  *>  x  e t  n  e  s  Either min E 2 ( * i ,  u  m  a  a r b i t r a r y set of yd's.  n  . . . , x ) i s taken on when a l l of the k  i n e q u a l i t i e s ( 2 ) are s t r i c t , i n which case  (5)  r  k  J L [ E (X,  l  • . . . • x  l t  = 0L  -  T  R)j  =0  t ^ k - I,  6  H  L  k  or i t i s attained when f o r some t , 1 (4)  *  x )  + . .  l  .  +<*  t  »  that i s , when one of the i n e q u a l i t i e s becomes an equality.  Case 1. E  I f (5) i s true f o r each i , then  l ( * i ) "/< =  R « 7  G  . . . , x ) = x k  ±  + B j ^ ) ,  LL = R - x i , k  k u = kR '  .  21  x  i »  1=1  k^. = (k - 1)R ,  x  /  -  ^  i  -  I .  •  R  E ( x , • • • » k^ x  2  1  =  (2)  (I)  = ( )^ l ( ^ l k  • • • ^-k)j  a* E 2 ( 0l-\ . . . °^k)  2  b y l e m m a  5*5  11  That i s , any relative min i n the interior of the region defined by (1) and (2) i s 2s E (<*i, . . . d ) (5)  .  k  2  Assume that for each U>, 2 - Co ^ k - 1, the lemma i s true  for a l l sets a^ ^ . . . ^ a . w  Case 2.  (4) applies.  Define E of a single number to be zero. Then 2  E  (xi» . . . , x ) = E ( x  2  k  2  iif  . . . x^) + E ( x ^ 2  • (x^ + . . . • X i ) ( x t  (6)  i t n  i  • . . . • x ) ik  = E (x , . . . x ) • E ( x 2  + (<V . . . +<* )(* t  il  it  2  . .  tn  +i>  i t + i  . . . x^) .  , ...,x ) ifc  , 1* t*k -1 ,  since (4) and (1) imply that (7)  x  * ...*x  i t + 1  i k  =*  t  * . . . . * <*  n  .  k  Now, E (<X , . . . , 0 C )  E (x , . . . , x ) 2  il  it  2  t  1  because by (4) and (2)  t  H  •  E  8=1  t 8=1  £ /? * II * X  B  8=1  . . • ,/Oj)e(il» . . . , i )  B  t  8=1  ^  '  J  1 * j ^ t - 1  we can apply the induction step (5) for co = t and a i = OC^ . For t = 1 we have by definition E (xi) = 0 = E ( # i ) , E (xi) =>E (Ot ) . 2  2  2  2  1  Also E2(xi , . . . , Xi ) ^ E ( C t t+1  k  2  t + l f  . . . , 0C ) k  12 because by (2) x  • . . . + x  i x  + x  i t  t  +  1  + . . . • x  t  +  j  ^ Od * . . . + 0L + . . . + a X  1  t + j  and therefore by (4)  together with (7) we can apply the induction step (5) f o r ui = k - t and a  l  =  rt  t l» +  • • • » k - t * **k . a  Hence by (6) • • • k) - 2(^l»  E2(*l>  E  x  • • • »<*k) 2(<*t+1» • ' • »°^k) +  E  = E ( < ^ i , . .. . » ^ k ) • 2  To complete the induction we need to show that f o r U) = 2 the lemma i s true. x  x  In t h i s case l * <*  1 •  2 ^ ** 1 »  x +x =<^x  + 0 i  2  1  Therefore x^ £ <*  , x 3J ^  2  a n <  2  2  *  2 w  e  m  U  8  min E (xj. x ) = OC^ CX2  E  Since ~  2 ( l 2) = x  [ l(^l x  x  l -  2  ^"1 »  t  h  x  x  x  r  t  show that  ..  2  l2 " l (  x  t  l  + ( X  2 " l) x  + C  ^ 2 ** l ) ] = -1 < 0 i n the entire i n t e r v a l ,  e  "d-^  x  11581 a  r  e  attained a t x^ = O f and x\ = 2  and  i n both cases we get Lemma 5.4 i s an extension of a r e s u l t contained i n  to the  15 second elementary symmetric function.  I f the variables were r e s t r i c t e d  to positive values then the r e s u l t i s contained i n the r e s u l t  referred  to above. 5.5  Corollary  I f CL 4£ - ( Ot + . . . • <* ) 2  i  = 2, . . . k ; £ x i = 0 and i=l  H  then  2  £ x s=l  £ -  i s  Min E ( x x , . . . , Xk) « = - (& +  S e t * !  . . . +^ ).  2  ^ <* s=j+l  , (i  s  [<*2> • • • » <*k>  2  Proof«  . 0t  k  Then  k  ^  # ^. 2  l f  -  *  ±n  a  ±j  . . . ij)€Qj  ( *2 • • • <  +  . . -<*  k  .  5 £  8=1  <*s = - ( *  * . . .  2  <*k) * <*2  +  +  . . . * <*1  k  = -E  8=j+l  *s  Therefore we have  £  x  8=1  i  8  -  £  8=1  and since  Z  8=1  <*s - -  (*2 * • • •  +  k)  a  * <*2  +  • • •  +  <*k -  G  we have  Z <*e •  k  k  E  8=1  H  -  8=1  Apply lemma 5,4 and we have the r e s u l t .  Theorem 5.6 Let  M=max(|i: s=l  Let A be Hsrmitian with eigenvalues  A| ; | £ *.().. a  s=n-k l +  j  Then  • » »~  A n  »  k  14  max  E ( 2  A l  ,  . . . , A)  =  k  ^  *  where max  i s taken over a l l sets of k o.n. vectors xj_, . . . , x  Proofi  By Lemma  k  .  (5.1)  E (Ai, . . . , A ) 2  k  ^(*1>  ±  • » k) j  —  A  2  .  By [2] n  k  Z  *S  s=n-k l +  -  £  k  A  s=l  S  ~  ZL  "As  5-1  ,  which implies E (A 2  . . . , A )-  l t  (^) ( | )  k  •  2  The value i s attained since there e x i s t s k o.n. vectors yj_, . . . , y such that k (A , yi  y i  )  £ t  =  *  ,  8  1 - i - k  ,  k and k o.n. vectors z^, . . . z  k  such that  k  Theorem 5,7 *1  >  • • •^ ^ n •  L e t  Let A be n x n Hermitian with x  l»  • • • » k + l ^2. x  minimizes E (A^, . . . + ^ ) . 2  x  l»  A k  • • • k l» x  • • n  Either of  • .. .•  0  (1) ^n  +  ' ' •  +  A  s e t  °JL  vectors which  Let L be_ the subspace spanned by  Then L i s Invariant under A i f t  +  (a)  0  eigenvalues  n-k+l  >  0  k  15  is true or: (b) both of > o  A, + . . . + A  *  (2)  *n  +  '- • * Vk*l  <  0  and one of (i) (ii)  A  >  x  - (A  - ( A  N  + . . . *A _ N  • . . . •A _  n  N  K  +  K  +  )> *  2  )  2  ,  > - ( a  x  2  + A  N  -  K  +  5  . n' ' (5)  (iii) - ( A + 2  -  (*  + 2  . . . +>  4  . . . + A  1  2 - i^ k- 2  + A +  1  N  _  K  +  I  N  • . . . A )> X +  n  x  >  +...•A ) ,  *  n  ,  (iv) - (* + . . . • A 2  • 7V )> A  k - 1  n  L  > « ( ^ + . . . + 7^), 2  are satisfied, Notet  The theorem states that for any Hermitian matrix with  distinct eigenvalues and whose largest eigenvalue,.  i s not equal to  any of the k numbers (i) - (* (ii) - ( A  2  + ... * X )  2  • . . . •  k  A  ^  ^  • . . . *  7VJ  2 * i^ k -1  ( i i i ) - ( - A ^ ^ • . . . + TV ) n  L i s invariant under A. Proof i  Assume L i s not invariant under A and that Ax^+i £ L.  Therefore there exists a unit vector v e L" such that (Axjc+i, v) = ^ £ 0. 1  Let x'i = X i for i = 1, . . . , k ,  16* x  k  n  - tpv  *'k+l • A»i = (Ax'  lf  C l e a r l y x ' i , . . . , x' +i k  x'i) . i s an. o.n. set of vectors.  x^, . . . , x +x i s a minimizing k  Since  set f o r the function £2(^1> . . • , Ak+l)  we have  . ..  [& ^<I  VI>U •  A,  A  0  or  Since  we have  %(AVi)  = 0  ,  and we a r r i v e a t the r e s u l t that (4)  Min E (A!, . . . A  (5)  B (A , . . . , A )  2  k + 1  )  =  E^Ai, . . . , A ) k  and 1  1  = 0  k  Case ( a ) . I f one of the expressions  (1) i s true then by £ 2 ]  one of k  max  Yl A^  <  0  min  2_. A^  >  0  1  k i s true which implies that  ^  £  0 f o r any o.n. set xj_, . . . , x .  But t h i s contradicts (5) which i n turn implies that L i s invariant  k  17  under A .  Case (b). Both of (2)  <«)  hold and one of (5)  I T • *. * C  *s  A  8=1  and  ^  [2]  8=1  H 3  A  Y  ^  = -  S  A  Y L  8=1 k  k-j  .jrj  A ^  s=j+l j: s=l  A  *  r  -  ( ) 5  8  n  S  A,  s=l  8  by  ,  Y  8=j+l  IT  (7)  holds.  A  ,  8  s-ntk+jn  a  £  A  by [2]  8  s=n-k+j+l  8  The plan of the proof i s to determine a lower bound f o r min under the r e s t r i c t i o n !  A ) K  (5), (6)  and  (7)}and to  that t h i s i s greater than an attainable value of ^ ( A ] . ,  Case 1.  (2)  Cty,, = K^*  .  Then  J  which with (7)  k  and ( J i ) holds. n  Set  . . . , A +x)  (4),  which w i l l be a contradiction of  £ s=n-k+j+l  k = Zl s=j+l s  A  8  gives  to.  (8) S=l  8=j+l  •  filso  (9) and since  <  &  2  =  < \ _  - (^n-k+2 • • • \ ) ha' +  (10)  ,  <*i  +  w e  k  +  1  and by (2),  6  C* < - (tf + . . . + OLy,) 2  2  show  <  <*s  18 since - (<x  + . . . + . tf ) « - ( A  2  k  • . . , ^  n - k 4 2  n  )  .  Because (5), (8), (9) and (10) are v a l i d we can apply c o r o l l a r y J.5 to get E ^ A i , . . . , A ) =s E 2 [ r t , . . . , c * , k  2  = 2[ n-k+2» • • • » n » -( n-k+2 A  E  A  = 2 ( n-k+2» . . . »  A  .  +<* ) k  A  A  n  .  • • • * n)]  +  A ) - ( -k+2 +  A  E  -(Otg*.  k  A  n  )  n  •  2  By choosing appropriate eigenvectors f o r the x± an attainable value of E (A , . . . A 2  1  k + 1  E  ) is  2( l» n-k+l A  A  A  - l n-k+l * ( A  A  A  l  +  A  n)  n - k + l ) ( n-k+2 A  * 2( n-k+2» = l(^n-k+2 A  • • •  +  Ai-k+l^l  +  +  A  and by (5i)  ^n)  +  n) *  n-k+2  + • • • *  A  * 2( n^k+2» • • » * A  E  • • •  *n)  A  E  +  A  A  n)  n)  and (2) t h i s l a s t expression i s * • • •  <  - ( n-k+2 A  • . .  +  ^ n ) * 2( n-k+2» • • • »  +  X  E  +  A  n)( l A  + X  n-k+2  n-k+2> • • • » n ) " C^n-k+2  "  A  A  +  +  • • •  • • •  +  A  +  A  A  n)  n)  n>  2  - B2U1. . • • A ) k  which contradicts (4), Gases 2 and 4 are special cases of 5 but i n order not to get involved i n any ingenious devices we have s p l i t i t into three cases. Case 2. Set <X  2  (2) and ( J i i ) apply. a  -(  A  j « 0, 1, . . . , k - 3.  + . . . +.. ) = *^2 A  a  n  n  d  X  •  A n  .  k + 5  +...•*„)  Then by ( J i i ) since  n  _  k  +  2  <  .(*Aj_  = "A _  k-j  n  +A _ + n  k  ^ 5 = _ + j we have Ot^ < CV A  n  t h i s and because the j ' s are ordered, A  A  and < *  k  2>  3  3  From  ,  ]  (11)  ±  <X  19 2,  i =  ±  -(7^ +>  By ( J i i ) A > 2  + . . . *  n - k + 5  ^2 *1 = "(^2 • • •' * \c) <  +  (12)  a  Oi <  -(^  2  w  > ) n  we have  GC  n XI ^s s=n-k+j+l  *  a  s=j+l k  2 - j ^ k - 1 while  -  (15)  By (5),  (11),  f  o  r  Therefore by (6)  and (7)  we can apply corollary (J.5)  to get  we have  YL<x  8=3+1  B  (12)  *1.  =  s=2  + -  8=1  and since  2  •  k  k By d e f i n i t i o n of  cX  =  ^ e  e  • • .•••<* )  + 2  1.  . . . k -  and (1?)  E2(A , . . . Aj,) ^  Es[dV , . . . e * , -(<* + . . . + ^ ) ^ ]  X  = E [-(^i  2  n  2  = E  ^ j , ^n-k+J,  2  2  . . . + ^ ) t Vk+J  +  +  k  k  » • • -.^m  (*1 n-k+5 2 +  +?l  • » n) A  K  ' n/ An attainable value of E ( A i , . . . , A +^) i s 2  k  2( 1. *2» \-k+2 » • • • •• ^n)  E  A  * < *2  = *2 V w * E (^ ,^ 2  1  - n-k+2( *1 A  n  .  k + 5  +  A  n-k )( V  , . . . ,  n-k^ « ' +  *n>  Ai)  "*2 ' W j  '. -  +  +  A  +2  +  ^n)  • "A (A + Vk*5 *• • ' *n> +  2  ^n-k5»» •  + E (^i, and by ( J i i ) t h i s expression i s +  2  <  *n-k+5  +  • (  X  +  . • .* * n ) ( * l  Vk+5  l  +  •» ^ n )  • .  +  *n)(*2)  + E ( ' X , "A^+J, . . . » 2  = E (^ 2  1  l t  A ) N  A^+J ...... AN)  " (^1n-k+5 +?l  +  »••••» ^ n )  2  *2  \-k*5  +  • .*  *a>  \  20  * E2(A  ... , A)  LF  K  which contradicts (4). ( 2 ) and ( 5 i i i )  Case 5 . Set  * c*  =  s  A  s = 2, . . . , i  s  - - ( A  i + 1  ^k-s  apply.  r  +  .  .  ^  •A _, n  c  +  i  4  2  +.  .  s = 0 , 1,  . . . , k-  c* =  • . . . + <X  = n-s A  A ) n  i- 2  .  k  k- 1  For  ^j  53,  > i ,  8  =  +1  s=j +1 n  "A-  + . , . •  N  ^"n-k+j+l  EI 'S  s  a  n  d  therefore by (7)  s=n-k j+l +  8=1 For i ^  (* -  • • •  +  +  k  ( A  X  = -  • . +  (15)  .  <*  S=j+1  8  ja  i+2  • \  k  i , . <jT *  -  e  s=j*l  )  • . . . •  • ' \ .  , . • + ^j)  k  n  «  2  .  .  +  - n A  A„)  +A  +  i+2  • • • *n-k+i+2 +  ••... +  = - " j ^ A ^ and therefore by (6)  .  8=1  S=j+1  8  We may combine (14) and (15) to get  (16)  £A.  8=i  8  ^ 2 ^ s=jn  s  In order to show that ^ j + i — l+2~ ^ i + i ~  1* J > k - 1  .  we need only check that  since f o r a l l other i n d i c i e s the cOs are ordered i n  the same way as the A's. By ( 5 i i i ) and the d e f i n i t i o n of  we have  21 • \  -^k+i+i >  A  >  • A  .  An  or ~ A-k+i+l  -C£  >  i  +  1  >-  ^  i  +  i  or > and since  > "A^kn+l  = ^ ± > ^i+1  »  ^ i + 2 = ^n-k+i+2 ^ ^n-k+i 1  we have  +  so that  (17)  « y  Also since (is) Because  (5),  j  ^ ^  n  = -(^2 a  <-  2  (16),  • • • * ^k) *  +  (17)  ^3^k-i .  2  an<  (*  • • . ^)  +  2  ^2  +  ^ 2  =  w  have  e  .  k  and (18) are v a l i d we can apply c o r o l l a r y  5*5"to  get  %(n»« • •»k) ~ ^L" ^*- • •» ^k* A  0  = £ 2 ^ 2 , . . .,  -(  . .+  • •» n»  A  - <*k)D +  "( <*2 - • +  A-kn+2 *• •  +  ^n)»  A  A  = E ( A » ' . ., Ai» n-kn+2» * • •» ^n) A  2  - (A  .  *  + A n i e n s - • •* +  ±  An attainable value of E2(Ai, . . . , A n )  B  K 2 ( A I , . . .Xi+i »A»-k+i*l» A+1  + (X  =  A i  +  F  1  T  >  K  •A  +  i n  (A  1  + E (Ai,. . 2  +  -  k  n  • • • » ^n)  •E (?v. ,. . 2  1  n)(^i .  +  . A  ^ 1 * - • •* A + l  +. . •+ A  >i»>n-kn 2»» • •  1  +  n  Ai-kn+1^  • 7l  I  4  +  +  +  Ai-kn+2 *•  Afc-k+i*2 *•  •AHc+i+2 *• * •  A»A*-kn+2»* ' •»  and by ( J i i i ) t h i s expression i s  n)  1  k  •  A  A)  •  +  "AJ  *n)  • •* An)  + 7 L  n)  22  <  +. . .+ A  * A -k+i*2 . • ^ +  -n) i+l  ? y  A  =  2('^lf  + E  +. .  A  • E C^i,. .  n  + (A  +  n  +  .+ A ) (  + ^ 4 ^  t  +  i  .  +.  x  A  +. . + A  l  i n  V * * * * ! " * +. . .  • • A i , A ^ + j + g , . . ., 7t ) n  V w ^ 2 +• • •* ^ n )  2  A i , An.k+i+2** • •» ^-n)  2  E ( 1 » * • •» A )  ~  A  2  k  which contradicts (4). Case 4.  (2) and ( J i v ) apply.  oL = A  Set  B  s = 2, . . . , k - 1  g  *k = " ( l A  • • • * k-l)  +  •  A  By (6*) and because  k k-J 2^ ^ s = - X_» S=j+1 6=j+l  -  J -*k = £ Xs 8=1  we have  (19)  £ Ay 8=1  =  -  £ S=j+1  B  To show that Since  • A j ^ x ) we have  (20)  ^k  >  a  n  d  <*-  a  n  b  B  y  (5),  o^ <  2 * J*k - 1  %  + 2  2  a  n  d  (21)  w  e  c  a  k  n  = - ( A  x  +. .  .+A  w  e  P P l y c o r o l l a r y 5.5 to get  -  k  , -(A  k - 1  1  )  2  An attainable value of E (A],,, . . A 2  <*k)  2>  k  +. . .  .  ^ E 2 [ c x . . ., <x , -(<* \  = E [A ,. . .,A 2  a  > -(  k  .  k  E^Ai,. . . A ) 2  ^  therefore  d  - ( C * • . . . + <* )  2  (19)f.(20)  (5'iv) we have  v  Also because Oi^ = "Ag and ^ i = * ( ^  (21)  .  7  s i n c e  °^]g_— ^ k - 1  O i ^  s  we need only check i t i n case j = k - 1.  ^j+l— = ^ k-1  1 *j >k-1  tf  x  +. . , + A ^  +E (A ,. 2  k + 1  1  ) isE (A 2  . l t  . .+ C * ) J  2  .  A  K  ), ^ ] k  -  1  )  .  . . ., A , k  A- ) n  2 5  * ( A» • E  •  2  = A +  (A . . .+ A  n  +  ( * l  E 2  + E ( 2  = -(A ~  E  ^k-l)  %  .  A  8  .  k  A  .  )  A^)  +. . .+ A  2( lt  k  . .^A-i  A,. x  + A ) + A (  k - 1  )  k - 1  + B (A .  2  2  l#  . ..Ak.i)  A) k  which oontradicts (k) and the proof i s complete.  Theorem 5.8 7^i~  . . . - A.  Let A be_ Hermitian with eigenvalues  Then  n  Min E ( 2  where x^, , . . , %  A l  , , . . ^  = E ( n , 2  range over a l l  x  . . .  7v ) n  sets of n o.n, vectors.  Proof1  E2( , . . . . Al  =  l^i^ig^n  An) ( ii» i i ) ( i p » i o ) Ax  x  Ax  ( i  » i ) V  A x  l*il<i «n  x  +  8  2  (G (A)x 2  x  il  A.*i  0  s = 1, 2 t « 1, 2  , X i . ^ Xi )  l ^ i ^ n = trace 02(A) =  E 2  (  • • • »  A  n)  and the value i s taken on. Theorem 5,9 Ai^  . . .  - A„.  Let A be Hermitian with eigenvalues Then f o r 1 *  k £ n ,  /l^i )  24 Min E ( A 2  . . .  l f  A )  = Min E ^ A ^ ,  k  . . . , A  I F C  )  ,  (il,...,i ) £(5^ k  where x^, . . , , x  Proofl  run over a l l sets of k o.n. vectors.>  k  There e x i s t sequences £j =  »•••>£*,...  i = 1,.  }  . ., n  such that f o r each j  (i)  ej), . . . , U „ • e")  ( *  s a t i s f y the hypotheses of theorem 5.7 and also Lim  £ ^ = 0 f o r each i .  This i s true because v i o l a t i o n of the hypotheses implies a f i n i t e number of linear e q u a l i t i e s to be v a l i d which can be made into s t r i c t i n e q u a l i t i e s by a r b i t r a r y small changes i n the A ^ ' s . For  each j l e t A ^ be the Hermitian matrix with eigenvalues (1), e  l e t i ( A ^ = ( A ^ x^, x^), and l e t x'^, . . . , x ' be a set of k o.n. £  k  vectors which minimizes E the  2  subspace L ( x ' x , . . .  ^(Agj)x» « » • » (^epk J  • By theorem 5*7  x' ) i s invariant under A and i f we minimize k  €  A. r e s t r i c t e d to t h i s subspace then by theorem 5*8 t h i s minimum i s equal j . . . » £x" ) where Ol^ are the eigenvalues of A r e s t r i c t e d  to E ( 2  k  £  to L ( x ' x , . . . , x ' ) . Since L ( x ' x , k  . . . » x' ) i s invariant under A . k  e  J  we must have that °^x» . . . » °^Jc  i  values V j »  . .  c  s a  y  s  a  *i*£y  cii  o i c e of some k of the eigen-  • • » i ^j • A  k  be eigenvectors we can a t t a i n a l l the values EjjK ( A  I F C  +  £j )J k  and therefore  B y ch  °osing  t  h  e  *i *°  + j ), . . . , fc  25 MinE [(A 2  6 j  )  ( €j) ] A  1  k  Min  *2[(*  e} )  +  (A  1  h  -€J )]  +  k  k  •>ik) Qkn c  Taking the l i m i t as j —>  Lemma 3*10 f(x  gives the desired r e s u l t .  Let  . . . x ) = XI (°r( ) «» " Qrk A  x  x  J  X  k  €  gC^l  . . . z ) = E r f C A z ^ z ) , . . . , (Az , z ) ] k  x  k  k  where A is_ n x n Hermitian and both (*]_, . . . , x ) and (z^, . . . , z ) k  range over o.n. sets of vectors. Rng ( f ) C Proof*  k  Then  Rng (g)  .  Let P and u^ be defined as i n Theorem J . l . f o r L ( x i , . . . , x ) , k  f(x  x  (0 ( ) >  . ..x )=  A  k  < J €  =  X W  r  X  J  ^rk  ZI weQj.  (C (PAP)x , x ) r  w  w  k  = t r tC (PAP) ] r  = YZ a) e Q  (Cr(P ?)  O  A  r k  det \ (PAP u i , u i ) £ +  l*ii<...<i *k r  ZI s^l( l*il«...*i *k  p A P  U  V  u  i >  P  •  Z-I s ^ l < is' ^s> liil^...<i *k A u  r  • g( l» • • • » % ) u  Theorem 5.11 A  - . . . - A . n  •  Let A be Hermitian with eigenvalues  Then  6  2.6 Min  l^ij<i £k  (MM*!!  * *i » i i  Ax i  x  2  2  )  2  Min  where x^, . . . , x Prooft  E (  »  A j  •  •  • »  range over a l l sets of k o.n. vectors.  k  5.10  By lemma  (C (A)Xi /\ Xj. , Xi 2  H  A  )  > Min E [ ( A , z ) , . . . , (Az , z ) ] 2  and by Theorem 3.9 =  x  k  Min  (i ...  E ( Aj,, . . . ,  Ai )  2  i )e k  t  Q k n  £ ^ i s the eigenvector corresponding to  H (0 (A) S i , A 5 , l<i <i *k 2  1  k  t h i s expression i s  l f  If  Zl  i2  ^  then  * i ,  ifi )  A  2  (Ar ,  s^)  ix  (Af^, t ) i 2  l^ij<i *k 2  ( n » A  2  =  ^ii  ^ i  r  i  x  )  (  A r  i » *i ) 2  2  = 2 ( *ii» • • • » E  2  *l*i2Hence  Min (i ,...,i )feQ, 1  k  E ( A j 2  k n  . . . , ^i )  i s  f  k  taken on.  27  BIBLIOGRAPHY  A.R. Amir-Moez, Extreme Properties of Eigenvalues of a Hermltlan Transformation and Singular Values of Sum and Product of Linear Transformations, Thesis f o r Doctor of Philosophy i n Mathematics, University of C a l i f o r n i a a t Los Angeles, June 1955, pp. 24-25. Ky Fan, On a Theorem of Weyl Concerning the Eigenvalues of Linear Transformations I, Proc. Nat. Acad. S c i . U.S.A., v o l 55  (1949), pp. 652-655. M. Marcus and J.L. McGregor, Extremal Properties of Hermltlan Matrices, to appear i n Can. J . Math, M. Marcus and B". Moyls, Maximum and Minimum Values f o r the Elementary Symmetric Functions of Hermitian Forms, submitted to J . London Math. Soc. A. Ostrowsky, Sur Quelques Applications des Fonetions Convexes et Concaves au sens de I. Schur, J . Math. Pures Appl., vol 50-51 (1951-52), page 265, Thm IV A. R.G.  Thompson, Theory and Applications of Compound Matrices, Masters thesis i n Mathematics, University of B r i t i s h Columbia, A p r i l I956.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080646/manifest

Comment

Related Items