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UBC Theses and Dissertations

Theory and applications of compound matrices Thompson, Robert Charles 1956

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THEORY AND APPLICATIONS OF COMPOUND MATRICES by ROBERT CHARLES THOMPSON A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE DEPARTMENT OF MATHEMATICS We accept t h i s t h e s i s as conforming to the standard 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 of the DEPARTMENT OF MATHEMATICS. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1956. ABSTRACT ( If A i s an n-square matrix, the p-th compound of A i s a matrix whose entries are the p-th order minors of A arranged i n a doubly lexicographic order . In th is thesis some of the theory of compound matrices i s given, including a short proof of the Sylvester-Franke theorem. This theory i s used to obtain an extremum property of elementary symmetric functions of the k largest (or smallest) eigenvalues of non-negative Hermitian (n.n.h) matrices. Extensions of theorems due to Weyl and Wielandt are given. The f i r s t of these relates elementary symmetric functions of singular values of any matrix A with the same elementary symmetric functions of the eigenvalues. The second gives an extremum property of arbitrary eigenvalues of n.n.h matrices and enables inequalit ies relat ing the eigenvalues of A, B with the eigenvalues of A + B to be given (A, B, n .n.h. ) • F i n a l l y , a norm inequality for an arbitrary matrix i s given. ACKNOWLEDGEMENTS The author wishes to express h i s thanks to Dr. M.D. Marcus f o r suggesting the t o p i c of t h i s t h e s i s and f o r h i s i n v a l u a b l e a d v i c e d u r i n g i t s p r e p a r a t i o n , and a l s o to Dr. B.N. Moyls f o r h i s h e l p f u l s u g g e s t i o n s . He g l a d l y acknowledges h i s indebtedness t o the N a t i o n a l Research C o u n c i l o f Canada f o r the f i n a n c i a l a s s i s t a n c e t h a t made t h i s study p o s s i b l e . TABLE OF CONTENTS I . I n t r o d u c t i o n . (1) I I . The Grassman A l g e b r a (2) I I I . L i n e a r Transformations on V , n t o V n p P? Induced by L i n e a r T r a n s f o r m a t i o n s on V n t o V n . (11) IV. Two Important R e s u l t s (23) 1. An Extremum P r o p e r t y o f E i g e n v a l u e s (23) 2. A New I n e q u a l i t y f o r P o s i t i v e Numbers (35) V. A p p l i c a t i o n s . (37) 1. E x t e n s i o n of a Theorem Due to H. Weyl (37) 2. E x t e n s i o n of a Recent Theorem of Wielandt (39) V I . A Norm I n e q u a l i t y (49) V I I . H i s t o r i c a l Survey (51) V I I I . B i b l i o g r a p h y (53) 1. IMTRODUOTIOK: The o b j e c t o f t h i s t h e s i s i s t w o f o l d : t o develop-, and apply the Grassman e x t e r i o r a l g e b r a t o e i g e n -value-problems f o r f i n i t e m a t r i c e s . The f i r s t p a r t o f the t h e s i s d e a l s w i t h the c o n s t r u c t i o n of the Grassman a l g e b r a . This i s done by d e f i n i n g a mapping from an n-dimensional v e c t o r space t o another ( n) = — — dimensional v e c t o r IE j n - P space t h a t s a t i s f i e s c e r t a i n m u l t i l i n e a r and a l t e r n a t i n g p r o p e r t i e s . We s h a l l prove some of t h e more important p r o p e r t i e s of t h i s mapping, i n c l u d i n g the w e l l known S y l v e s t e r -Franke Theorem. Some of t h e p r o o f s developed here w i l l be s h o r t e r than those t h a t are u s u a l l y g i v e n . T h i s m a t e r i a l w i l l be f o u n d i n s e c t i o n s I I and I I I . The second p a r t of t h e t h e s i s i s concerned w i t h a p p l i c a t i o n s o f the Grassman a l g e b r a . These a p p l i c a t i o n s w i l l be concerned with t h e l o c a t i o n o f e i g e n v a l u e s o f n x n m a t r i c e s A over t h e complex f i e l d ; t h a t i s , t o l o c a t i n g t h e complex r o o t s o f t h e polynomial e q u a t i o n i n \ det (\I - A) * 0 where I i s t h e n x n u n i t m a t r i x . Using the Grassman a l g e b r a , c e r t a i n maximum and minimum c h a r a c t e r i z a t i o n s o f the eige n v a l u e s of a matrix A w i l l be proved. These maximum and minimum p r o p e r t i e s w i l l be extensions o f e a r l i e r p r o p e r t i e s g i v e n by Fan (3 ). These r e s u l t s w i l l enable us to g i v e an e x t e n s i o n o f a theorem due t o Wielandt (15). An. e x t e n s i o n o f a theorem due to Minkowski (13) w i l l a l s o be g i v e n . For the Minkowski theorem i t w i l l be necessary t o s t a t e , without p r o o f , a r e c e n t r e s u l t due s e p a r a t e l y t o Marcus (11) and Bohnenblust (2). F i n a l l y a new r e s u l t c o n c e r n i n g t h e norm o f a m a t r i x w i l l be g i v e n ; the proof w i l l be made to depend on t h e extremal p r o p e r t i e s o f e i g e n v a l u e s proved e a r l i e r . A l l t h i s m a t e r i a l w i l l be found i n s e c t i o n s IV, V and V I . F i n a l l y , a b r i e f h i s t o r i c a l survey o f t h e m a t e r i a l covered i n the t h e s i s w i l l be g i v e n i n s e c t i o n V I I . Throughout the t h e s i s , the numbers i n b r a c k e t s r e f e r t o the r e f e r e n c e s g i v e n i n the b i b l i o g r a p h y i n s e c t i o n V I I I . I I . THE GRASSMAN ALGEBRA. The c h i e f instrument to be used i s t h e Grassman e x t e r i o r a l g e b r a . Before c o n s t r u c t i n g t h i s , i t i s necessary to i n t r o d u c e some n o t a t i o n . L e t us agree to denote by V r t h e r - d i m e n s i o n a l v e c t o r space c o n s i s t i n g of a l l r - t u p l e s o f complex numbers. Let e^ denote the u n i t r v e c t o r whose e n t r i e s are a l l zero except f o r a one i n the i t h p o s i t i o n . L a t e r on, we s h a l l have o c c a s i o n to c o n s i d e r s i m u l t a n e o u s l y two v e c t o r spaces, V_ and V , m . The - 3 -phrase " u n i t v e c t o r s " w i l l always mean the v e c t o r s e^ p _ j u s t mentioned. For convenience we s h a l l use £ Q p ^ r t 0 mean V r X V rX ... X V r , to p f a c t o r s ; t h a t i s , t h e c a r t e s i a n product o f . .Vr w i t h i t s e l f p t i m e s . I f p < n l e t Q p n - \ \ ± l f i p \ | 1 < i x < ... < i p < n } ; t h a t i s , the elements of Qp n are s e t s o f p i n t e g e r s chosen from 1, ..., n and arranged -in i n c r e a s i n g o r d e r . Note t h a t Qp n has ( n) elements. I n the seq u e l i t w i l l be necessary to c o n s i d e r a l e x i c o g r a p h i c o r d e r i n g of t h e elements o f Qp n . By t h i s i s meant, r o u g h l y , the scheme used t o or d e r t h e wards i n a d i c t i o n a r y . I f ^ ± l t i p ] , J p \ are two elements of Q p n then, i n the l e x i c o g r a p h i c o r d e r i n g , | i l , i p ^ comes b e f o r e j j ^ i •••» Jp ] i f a n d o n l v i f t h e r e i s an i n t e g e r m < p such t h a t i ^ = » *2 = J*2 » f A^m < Jm • Having d i s p o s e d o f t h e n o t a t i o n a l p r e l i m i n a r i e s , we can now g i v e the f i r s t d e f i n i t i o n . t , P ^ D e f i n i t i o n 2.1: L e t j x ^ , ... , Xp j t rr Q C ) V n . A m u l t i l i n e a r f u n c t i o n f ( x ^ , x p ) i s a mapping from P n <E) V-n t o a v e c t o r space W , such t h a t f i s a l i n e a r 1 ~ i n each v a r i a b l e . and * ( * L . x p ) - sgn TT f ( X t t ( 1 ) , x n ( p ) ) - 4 -where n i s a permutation o f 1, ..., p and sgn TT = + 1 i f n i s even. -1 i f TI i s odd. The f i r s t theorem i s concerned w i t h t h e e x i s t e n c e o f such a m u l t i l i n e a r f u n c t i o n . Theorem 2.2. For each p = 1, 2, n, t h e r e e x i s t s a P ^ m u l t i l i n e a r f u n c t i o n d e f i n e d on TT V_ t o V n , such 1 l p ' t h a t t h e s m a l l e s t v e c t o r space c o n t a i n i n g range f jis V/n> V P r o o f . L e t x-^, X p be any v e c t o r s i n , where xj_ = Ullt x l n ) x p = * x p l > *••» ^ n ^ t h e r i g h t hand s i d e s of the i d e n t i t i e s (1) may b e t h o u g h t o f as f o r m i n g a pxn m a t r i x x l l » •••» x l n j ( 2 )  x p l > "••» ^ n / C o n s t r u c t ( n) numbers as f o l l o w s : S e l e c t any p columns from (2) and form t h e determinant o f t h e pxp submatrix so o b t a i n e d . C l e a r l y t h i s may be done i n (p 1) ways. By a r r a n g i n g these (p1) numbers i n l e x i c o g r a p h i c order a c c o r d i n g t o t h e manner of s e l e c t i o n of columns from ( 2 ) , we can c o n s t r u c t an (p1) v e c t o r . D e f i n e t h i s (p) v e c t o r t o be P f l x i , . . . . X p ) . I t i s c l e a r t h a t f c a r r i e s TT (g) V n i n t o v/m • Because o f t h e elementary p r o p e r t i e s o f V - 5 -determinants, i t i s a l s o c l e a r t h a t f s a t i s f i e s the m u l t i l i n e a r and a l t e r n a t i n g p r o p e r t i e s o f D e f i n i t i o n 2.1. Only the l a s t statement of the theorem remains to be proved. L e t us c o n s i d e r the set of u n i t v e c t o r s j e^, ..., e n ^ i n V n . L e t j^l» •••» *"p\ ^ e a n e l e m e r r t °^ Qpn » a n c * c o n s i d e r t h e v e c t o r f ( e i ^ ) •••> e ± § ^ * n ^{ n) ' ^° f i n d the c o o r d i n a t e s o f t h i s v e c t o r we f i r s t have t o f o r m a l l pxp submatrices of t h e pxn m a t r i x i n which t h e j t h row c o n s i s t s o f zeros except f o r a one i n t h e i j column, j « 1, ..., p • The submatrix formed from t h e columns ijL, i p w i l l be a u n i t m a t r i x and t h e r e f o r e have determinant one. Any o t h e r submatrix w i l l c o n t a i n a row o f z e r o s , and so w i l l have determinant z e r o . Thus ^ ( © 1 ^ , ©ip) i s an (°) v e c t o r with a one i n the { ^ l * •••» i p \ p o s i t i o n and zeros elsewhere. Consequently, as [ i ^ , i p j ranges over a l l the elements of Qp n, we o b t a i n the complete set of (p) u n i t v e c t o r s i n v\ nx . T h i s completes t h e p r o o f . P T h i s theorem s u p p l i e s the answer to t h e q u e s t i o n o f the e x i s t e n c e of m u l t i l i n e a r f u n c t i o n s . Henceforth, t h e o n l y m u l t i l i n e a r f u n c t i o n t h a t we s h a l l c o n s i d e r w i l l be the one j u s t d e f i n e d . Having c o n s t r u c t e d the m u l t i l i n e a r f u n c t i o n , i t i s n a t u r a l t o enquire how the image of a g i v e n s e t of p v e c t o r s i n V n i s determined by t h e images o f s e t s of p v e c t o r s chosen from a b a s i s set i n V n . The next - 6 -theorem w i l l p r o v i d e an answer t o t h i s q u e s t i o n . F i r s t , however, we i n d i c a t e a use o f the Qp n n o t a t i o n i n t r o d u c e d e a r l i e r : I f co - j i i , i p ^ * Qpn and x±, are v e c t o r s i n V n , we d e f i n e xco = f (xj_]_ i • • •» x i p , ) • • • •» x n IP' I n p a r t i c u l a r , the proof of Theorem 2.2 shows t h a t i f e w = f ( e i ] L , e i p ) , then the set | e w j co * Qp n j i s the complete s e t of u n i t v e c t o r s i n Vin\ • V Theorem 2.3. L e t y^, •.., y p belong t o V n , and l e t n ( i • 1, ..., p) where x^, . •., x n a l s o belong t o V n . Then co « Q. cco xco pn where. i f co = { i ^ , i p j 6 Qp n , t h e n and 7 P i j ; x ^ = f(x. i.2» •••» x i ; p ) • P r o o f . We have y. = > ^ y.._• x . , so, u s i n g the m u l t i l i n e a r i j = i X J J p r o p e r t i e s o f f , we get f(ylf y p ) = f ( y l a.x., ]Ty2*y ^/pjV J l J l -1 J l » , # * » J p J l J 2 r Now, because o f t h e a l t e r n a t i n g p r o p e r t y of f , i t i s c l e a r t h a t i f two i n d i c e s i n f ( x ^ , ..., x. ) are the same, t h e n J l Jp the cor r e s p o n d i n g value o f f i s ze r o . Again, because o f the a l t e r n a t i n g p r o p e r t y of f , we can r e s t r i c t o u r s e l v e s t o summing over those terms i n which 1 < < j'2 < • • • < j p S as f o l l o w s : *(yi» yp) = ) ( ZZylTT/, x...yp1T(.j )Sgn n)f (XJ ...x- ) l<j 1 < 0 2<...<jp<n TT 1 T T ( J 1 ) P ^ V J i Jp where ) [ means summing over a l l s e t s o f d i s t i n c t p e rmutation TT o f , ..., jp . Now c l e a r l y JZ^i^i* ••• v^v s s n n B d e t so that f(yn, .... yJ - . ) / or - B -as r e q u i r e d . Theorems 2.2 and 2.3 g i v e t h e f o l l o w i n g i n t e r e s t i n g r e s u l t Theorem 2.4« L e t x-^, ..., x n be a b a s i s set f o r V n , Then the set ^ x w j co e Q p n ^ i s a b a s i s s e t f o r n » Proof. Because {x-^, ..., x^ i s a b a s i s set f o r V we know t h a t any u n i t v e c t o r e. ( j = 1, ..., n) can be w r i t t e n as a l i n e a r combination of t h e v e c t o r s x-^, x n . L e t T T, co be elements Of Q p n • By Theorem 2.3 i t f o l l o w s t h a t e^ i s a l i n e a r combination o f v e c t o r s x^, where co runs over a l l the elements of Q p n • T h i s holds f o r every TT Qp n . Since t h e s e t [ e n TT & Qpn^ i s t n e complete s e t o f u n i t v e c t o r s i n v7 n» , i t i s c l e a r t h a t any v e c t o r P w i t h ( p) c o o r d i n a t e s can be w r i t t e n as a l i n e a r combination of t h e v e c t o r s of [ x u | to e Qpn]* Hence t h i s s e t o f (£) v e c t o r s must form a b a s i s f o r V(n\» P Theorem 2.5» L e t x-^, ..., Xp be v e c t o r s i n V n . Then f(x-j_, ..., Xp) - 0 i f and o n l y i f ^x-^, x p j i s a l i n e a r l y dependent s e t . Proof. L e t x± = ix* , ,,.yx± ), i = 1, . . . , p . Then 1 p f(xjL, Xp) = 0 i f and o n l y i f a l l p x p minors o f the matrix \ x p l x p n , are z e r o . T h i s happens i f and o n l y i f rank X < p - 1 , t h a t i s , i f and o n l y i f x^, ..., Xp are l i n e a r l y dependent. Up to t h i s p o i n t we have used f.(xj_, x p ) t o denote the m u l t i l i n e a r f u n c t i o n d e f i n e d i n Theorem 2.2: To conform with u s u a l p r a c t i c e an a l t e r n a t i v e n o t a t i o n w i l l now be i n t r o d u c e d . We w r i t e f ( x ^ , .... x p ) as X ^ A ... ^Xp t h a t i s X^ y\ . . . A X p 8 8 f ( X^ y . . . , Xp ) • H e n c e f o r t h these two n o t a t i o n s w i l l be used i n t e r c h a n g e a b l y . I n t h i s n o t a t i o n , i f co » { i-j_, i p j € Q p n , then i X',£ = X^ A ... A X^p Theorem 2.6. I£ x^ A ... * x p , y^ A ••. A y p 6 V. , p then (xj_ A ... i X p , A ... A y p ) - det ( ( x ^ , y^ ) Remarks: ( i ) ( , ) r e f e r s t o the u s u a l u n i t a r y i n n e r product• ( i i ) The Theorem r e f e r s o n l y to those v e c t o r s i n V ^ n j t h a t are of t h e form x^ A ... A x p f o r s u i t a b l e x l >' • • • > x p i n ^n • P r o o f . The p r o o f c o n s i s t s of t h e f o l l o w i n g c a l c u l a t i o n . L e t - 10 -x i ~ ( x i i > •••> x i n ^ y-i ( y ^ i f •••» y-in^ i f d - 1, p Then ( x i , y j ) - x i r y j P . 3 0 det ((x^y-j)) _ n _ n _ n ^j xpr y l r 7 2 r ' 4 * X P r V n n n ZZ yir 3 x i r i & x i r 7 2 r * **** 5 X i r y p r X P r y P r ' * * *' rCi Xpr 7pr ^Or^ x l r 2 » •••» 2-^ x l r y p r n _ X p r 2 ' * * *' § V V Z r i » r 2 » , , # *rp y l r y 2r? yprT x l r -I ( ^ y i T T ( r 1 ) - - - y p n ( r _ ) ^ ff) x l r x x l r p V * ! x pr p - 11 -l < r - j < . . .<Tp<n ^ 1 x l r r y n r p i * i y l r T r p r T h i s c a l c u l a t i o n completes t h e p r o o f . C o r o l l a r y 2.7. Let |x-p ..., x n j be an o r t ho normal set of v e c t o r s i n V n . Then the s e t j X w J co t Q p n ^ i s an orthonormal s e t i n V,n. . V P r o o f . Consider ( X ^ A . . . A X J ^ , X J ^ A . . . A X J ^ ) « det | l x ^ ^ X j ^ ) g Case ( 1 ) . Suppose there i s an i n t e g e r m which l i e s i n j i ^ , i p | but not i n Jj^ , •••» J p j • T n e n t h e c o r r e s p o n -d i n g row of the m a t r i x ( ( x ^ , x..- }) w i l l c o n s i s t e n t i r e l y of zeros because of t h e o r t h o g o n a l i t y p r o p e r t i e s of the x f s . Hence (x,« A . . . A X J , x. A . . . A X J ) » 0 . x l ^p J i Jp Case (2). I f • j ^ , .... i p • j ' p , then the m a t r i x ( (XA , xn- )) i s a p x p i d e n t i t y m a t r i x and so has determinant s J t 1. Hence (x^ A ... *x,« , x,* A . . . A X - « ) = 1 • x l -"-p -"-l "T> I I I . LINEAR TRANSFORMATIONS ON V/m TO V/m INDUCED BY V V LINEAR TRANSFORMATIONS ON V n TO V n . I n t h i s s e c t i o n , l i n e a r t r a n s f o r m a t i o n s on V#n\ t o V P ; V(gj t h a t correspond to l i n e a r t r a n s f o r m a t i o n s on V n t o V n are d e f i n e d , and some of t h e i r p r o p e r t i e s are determined. - 12 -Definition 3.1. Let A be a linear transformation on V n to V n • Define the linear transformation Gp(A) on — ^(n) by defining its effect on the basis set [ e a) | oo e Qpn j to be Cp(A)e i l A ••• A. e^ = (Ae^J^v •«»• A (Aeip) Since a linear trarisformation of a vector space is uniquely determined by its effect on a set of basis vectors. Definition 3.1 defines one and only one linear transformat ion on V/n\ to — IpJ -— Theorem V <P°> • Then 3»2. Let y 2 A . . . A y p f V ( p 1 ) ' C p ( A ) y 1 A . . , A y p = (Ay 1)A . . . /v(Ayp) Proof. Let A be defined by n Ae-i = a i k e k > 1 = l j and let y i " * y i l ' , 8 * » y i n J » 1 " 1 » •••» p Then, by Theorem 2.3 A e J l A -\ 7 . . . A Ae •? = ^ > p l<k]<.. .<kp<n a. -: i_ . . . d . , JpKp *1 and - 13 -y 1 A ... A y p n l<Jl<...<Jp<n y i J ] L - yu-yP J l — y P J p 6 • A • • • A © J J l J] so t h a t , by t h e d e f i n i t i o n o f C p ( A ) as a l i n e a r t r a n s -f o r m a t i o n C p ( A ) y 1 A . . . 4 y T 1 I > l<j 1<...<j p<n 7 U ! — 7 U , 7PJ1 7 PJ, (Ae ) A . A ( A e . ) j l Jp l<j 1<...<j p<n 7 U , — y 7 P J n - y p l 1<KX<.. .<K_<n a . . • . . a J v. j i k p J p f c l J P K P 1 K T Consequently the j i ^ , i p | c o o r d i n a t e o f C p ( A ) y 1 A ... A y p i s HZ l < J l < . . . < j p $ i 7^h "' 7 U , 7 P«L V a . a. J l x l J l x p a . . . • • a . . To complete the proof, i t i s necessary t o show t h a t t h i s i s e x a c t l y the j i ^ , ..., i p | c o o r d i n a t e o f ( A y ^ A . . . A (Ay p) . We proceed t o do t h i s . From - 14 -we obtain A y ± = y ± 1 Ae_. 1 j = l X 3 3 a 5 { § y « a ^ ) e k so t h a t t h e Ji - j ^ i p | c o o r d i n a t e o f (Ay-^ i s j u s t j=l J l l T' y P J a i i 0=1 P J 3 P I T> a Jl>*»«»dr) 1 1 p p P J l Z l < J i<. . .<3p^ 7 l ^ l y U y P J l yP0' a. . V i this completes the proof. - 15 -T h i s theorem shows t h a t t h e r e i s nothing e s s e n t i a l i n u s i n g t h e b a s i s v e c t o r s e^. • «., e n t o d e f i n e Cp(A); i n f a c t , i f x-^, x n i s another b a s i s f o r V n , we c o u l d g i v e an e q u i v a l e n t d e f i n i t i o n o f Cp(A) as Cp(A) A • •• A x. » (Ax* ) A. • •• A (Axi ) x l x p 1^ p Before proceeding t o t h e next theorem, i t i s n e c e s s a r y t o g i v e a word o f e x p l a n a t i o n . Suppose A • ( a ^ j ) i s any n x n m a t r i x . From A , we can c o n s t r u c t a new m a t r i x B as f o l l o w s . Given any set o f p rows o f A , we f i x our a t t e n t i o n on t h e p x n submatrix of A d e f i n e d by the g i v e n p rows, and use t h i s p x n submatrix t o c o n s t r u c t a v e c t o r w i t h (p) c o o r d i n a t e s , as i n Theorem 2.2. That i s , we w r i t e t h e p x p minors o f t h i s p x n submatrix i n a row v e c t o r form, i n l e x i c o g r a p h i c o r d e r a c c o r d i n g to the manner o f s e l e c t i o n of columns o f A . Thus, f o r a g i v e n s e t o f p rows o f A we o b t a i n a v e c t o r w i t h ( n ) c o o r d i n a t e s . I f t h i s c o n s t r u c t i o n i s c a r r i e d out f o r each p o s s i b l e s e l e c t i o n o f p rows o f A , we w i l l o b t a i n a s e t o f ( n) row v e c t o r s , P each w i t h (£) c o o r d i n a t e s . I f t h e s e row v e c t o r s are used as the e n t r i e s o f a column v e c t o r and ordered i n l e x i c o g r a p h i c f a s h i o n according t o the manner o f s e l e c t i o n of rows o f A , we o b t a i n the m a t r i x B . B i s s a i d to c o n s i s t of the p x p minors o f A , arranged i n doubly l e x i c o g r a p h i c o r d e r . Note t h a t B i s an (p1) x (£) m a t r i x . - 16 -Theorem 3 .3 » I f x-^, • • •, x i s a b a s i s f o r V n , t h e n  the r e p r e s e n t a t i o n o f C P ( A ) r e l a t i v e t o the b a s i s  xo> ^ e Qpn^ — — m a t r i x whose e n t r i e s are t h e p x p minors of the r e p r e s e n t a t i o n of A r e l a t i v e t o x^, ••• arranged i n doubly l e x i c o g r a p h i c order. P r o o f . The method of proof i s a now f a m i l i a r type of c a l c u l a t i o n . L e t A X J ^ = aJ£ X j , i = 1, n t h e n G n(A)x4 A . . . A x ± • ( ) 'a.™ X . J ) A . . . A (T~'a 1 i -2 ^ a. . •»• a . • X J A . J l , . . . , j p V P J L l<Jl<nT^p<n V " ^ l ^ l n ( j p ) i p 6 J l A • . z 1<0"l<« • • <JpS n a V l J1 XT • a V P X J A . . . A J l T h i s completes the p r o o f . Because o f Theorem 3:.2, f o r any v e c t o r X ^ A . . . A and any two l i n e a r t r a n s f o r m a t i o n s A , B we have - 17 -G_(AB)x n- A . . . A X i . •• (ABxi. ) A . . . A ( A B X I ) p 1 p 1 P - C D ( A ) ( B X i l ) / v . . . A ( B X I ) p 1. P • C p(A) C p ( B ) x i A . . . A x i Since t h i s c a l c u l a t i o n i s v a l i d f o r a l l b a s i s v e c t o r s x^, A . . . A Xi , i t f o l l o w s t h a t Cp ( A B ) = C p ( A ) C p(B) . C l e a r l y t h i s a l s o h o l d s f o r the matrix r e p r e s e n t a t i o n s o f A , B r e l a t i v e t o a g i v e n b a s i s . Consequently we have proved the f o l l o w i n g theorem. Theorem 3.4. I f A , B are l i n e a r t r a n s f o r m a t i o n s on V n to V n , or the m a t r i x r e p r e s e n t a t i o n s o f l i n e a r t r a n s -formations r e l a t i v e to a g i v e n b a s i s , t h e n Cp ( A B ) = Cp ( A ) C p ( B ) C o r o l l a r y . I f A"^ - e x i s t s , then Gp(A) e x i s t s , and C p(A) " 1 = C P ( A - 1 ) . P r o o f . From AA" 1 - A" 1 A « I i t f o l l o w s t h a t I ( n - C p(A) CptA" 1) = Cp(A-l) C p ( A ) P J so t h a t C - ( A - l ) = C„(A) " 1 The next theorem g i v e s a p r e l i m i n a r y r e s u l t t h a t w i l l be needed i n order t o determine the ei g e n v a l u e s o f G p(A) . - 18 -Theorem 3.5* Let A possess n l i n e a r l y independant  e i g e n v e c t o r s x^, ..., x n b e l o n g i n g to the e i g e n v a l u e s X^, ..., X n r e s p e c t i v e l y . Then the s e t of e i g e n v e c t o r s o f Cp(A) i s the s e t (z w | to i Q p n j and the set of eig e n v a l u e s i s the set P r o o f . We have Ax± = X i x^, so C p t A j x ^ A ... A X j [ p - ( A x ^ ) A ... A ( A x i p ) = X-? ...X-: A . . . A X±_ 1 p 1 P Since t h e set |Xj^ ^ . . . A x i p | i s a s e t o f (p) l i n e a r l y independant e i g e n v e c t o r s , Cp(A) can have no other e i g e n -v a l u e s t h a n those s t a t e d . Theorem 3.6. Let A be any l i n e a r t r a n s f o r m a t i o n on V n  t o V n , w i t h e i g e n v a l u e s X^, • • •, X n • Then t he s e t of  ei g e n v a l u e s of C_(A) i s the s e t P r o o f . Because of the importance of t h i s theorem, two p r o o f s w i l l be g i v e n . F i r s t P r o o f . Case ( 1 ) . A has d i s t i n c t c h a r a c t e r i s t i c r o o t s . I n t h i s case A possesses a s e t o f n l i n e a r l y independent e i g e n -v e c t o r s , and so the r e s u l t f o l l o w s f r o m the p r e v i o u s theorem. - 19 -Case ( 2 ) . Given A , t h e r e e x i s t s a m a t r i x B whose elements d i f f e r a r b i t r a r i l y l i t t l e from t h e elements o f A and which has d i s t i n c t e i g e n v a l u e s . The theorem i s then t r u e f o r B , by Case ( 1 ) . Since t h e compound o p e r a t o r Cp i s continuous, and t h e ei g e n v a l u e s of a m a t r i x are continuous f u n c t i o n s of t h e e n t r i e s , i f we choose a sequence of m a t r i c e s B approaching A , and f o r which the theorem i s t r u e , i t f o l l o w s t h a t t h e theorem i s t r u e f o r A • Second P r o o f . T h i s proof depends on t h e f o l l o w i n g lemma. Lemma. I f T i s a t r i a n g u l a r matrix, t h e n Cp(T) i s  t r i a n g u l a r . P r o o f . L e t us suppose t h a t t h e lower t r i a n g l e of T i s zero, and attempt t o show t h a t t h e lower t r i a n g l e of Cp(T) i s a l s o z e r o . To do t h i s , c o n s i d e r an entry i n the ( i l ... i p , J ' I ... jp) p o s i t i o n of Cp(T) . I f t h i s e n t r y i s t o be below the main d i a g o n a l we must have, f o r some k < p , i l = J*l> A 2 5 J2> •••» *k = Jk» i k + l > J'k+1 • T h i s i m p l i e s i p > i p - 1 >••• > i k + l > J'k+1 so t h a t the ( i i ... i p , J i ... j p ) e n t r y o f C p ( T ) i s the determinant of t h e f o l l o w i n g minor of A , e x h i b i t e d i n a b l o c k form: 1 1 / S j ik+l ^+2 - 20 -J k Jk+1 Jk+2 P « 0 ¥ k O o 0 o 1 \0 1 1 o / Here the numbers down the s i d e and a c r o s s the t o p i n d i c a t e the rows and columns of A used i n c o n s t r u c t i n g t h i s minor o f A . Because of the form of t h i s matrix, i t s determinant i s z e r o . T h i s completes the p r o o f o f t h e lemma. To prove t h e theorem we use the f a c t t h a t g i v e n A , t h e r e always e x i s t s U such t h a t U"^ - AU = T , a t r i a n g u l a r m a t r i x . Hence C p ( U ) - 1 C p(A) C p(U) = G p ( T ) where Cp(T) i s a t r i a n g u l a r m a t r i x whose diagonal e n t r i e s are p r e c i s e l y the numbers g i v e n i n t h e statement of t h e theorem. The second proof of t h e theorem i s now complete. Theorem 3 * 6 - e n a b l e s a simple p r o o f to be g i v e n of t h e w e l l known S y l v e s t e r - F r a n k e Theorem. Theorem 3.7. |c p(A)| = j A | P " P r o o f . JCp(A)j i s t h e product of the eigenvalues of Gp(A) . In t h i s product each eigenvalue of A o c c u r s e x a c t l y (pZ±) t i m e s . Q.E.D. By theorem 2 .4 we know t h a t i f j x i , x n ^ i s a b a s i s s e t f o r V n , t h e n | x w | oo e Q p n j i s a b a s i s set f o r ^ ( n ) • Using Theorem 3 . 7 , we can complete theorem 2 .4 as f o l l o w s . C o r o l l a r y . L e t j , ..., x n j be a set o f v e c t o r s i n V n . Then j x w j co t Q p n j i s a b a s i s set f o r Y^nj i f and o n l y i f |XJL, ..., x n ^ i s a b a s i s s e t f o r V n . P r o o f . Let A be t h e m a t r i x whose i t h row i s the v e c t o r x i • Then G p(A) i s the (p) x (p) mat r i x whose s u c c e s s i v e rows c o n s i s t o f the v e c t o r s x* A . . . A X^ , arranged i n , 1 , p ( ^ 4 ) l e x i c o g r a p h i c o r d e r . Since jC p{A)J = |AJ p-1' i t f o l l o w s t h a t | C p ( A ) | ^ 0 i f and o n l y i f |AJ / 0 . The next, and l a s t , theorem i n t h i s s e c t i o n g i v e s a resume 1 of p r o p e r t i e s o f Cp(A) i n terms of assumed p r o p e r t i e s of A . Theorem 3 . 8 . The f o l l o w i n g r e l a t i o n s a re v a l i d . - 22 -(2) (1) C p(A) _ 1 = C p U - 1 ) C p(A*) = C p ( A ) * (3) I f A i s normal. C p(A) i s normal (4) I f A i s H e r m i t i a n . C p(A) i s Hemitian (5) I f A i s p o s i t i v e d e f i n i t e (or non-negative) H e r m i t i a n t h e n G p(A) i s p o s i t i v e d e f i n i t e (or non-negative) H e r m i t i a n . P r o o f . (1) T h i s has a l r e a d y been proved (2) We have, u s i n g Theorems 2.7 and 3.2 (Op(K¥)x1 A ... A Xp, Y± A ... A y p ) = ( ( A * x i ) ... A (A Xp), y-L A ... A Jp) * U y p ) I n p a r t i c u l a r i t f o l l o w s t h a t e<o) = ^eTT» C p U ) e O j ) f o r any rr, co e Q__ . Hence C p ( A ) = C p ( A * ) . (3) I f AA* = A * A , t h e n C p ( A > C p ( A ^ = C p < A * ) C P ( A ) - 23 -so that C p(A) G p(A) = C p(A) C p(A) (4) This follows immediately from (2). (5) Using Theorem 3.6, i t follows that i f the eigenvalues of A are positive (or non-negative), then the eigenvalues of C p(A) are positive (or non-negative). IV, Two Important Results. In this section two important theorems w i l l be given. Both theorems w i l l be needed later , but only the f i r s t w i l l be proved. Because of their dissimilar nature, this section i s divided into two subsections, 1, An Extremum Property of Eigenvalues. The chief result i s Theorem 4«2. Definition 4.1. Let 1 < p < k be positive integers. The  elementary symmetric function of degree p on the k let ters  a l> •••> a k i s the coefficient of t^'P i n k TT (t + a^  ) i=l r and i s written as ... » a k ) o r f as w i l l later be seen to be convenient. - 24 -Theorem 4*2. L e t 1 < p < k < n and l e t A be an n-square p o s i t i v e d e f i n i t e Hermitian t r a n s f o r m a t i o n w i t h e i g e n v a l u e s 0 < < . < an . Then max Z s (CpCAjx^, x j = E p ( a n , a n _ k + 1 ) « < Qpk min ^ \ (GptAjx^, x w ) «= E (ax, ak) <° * Qpk where x w «= Xj. A ... A i f to = | i ^ , ..., i p f Q p k , and the max and min are t a k e n over a l l s e t s o f k o r t ho normal v e c t o r s j x j , • •., x k j i n V n . P r o o f , The proof i s i n s e v e r a l s t e p s . For n o t a t i o n a l s i m p l i c i t y we s h a l l l e t - 5 Z g ( x 1 } x k ) = 2 • (C (Ajx^, x w ) . co « Q p k ( i ) F i r s t i t i s c l e a r t h a t a set o f maximizing (minimizing) orthonormal v e c t o r s f o r g e x i s t s . T h i s i s e a s i l y seen u s i n g a standard c o n t i n u i t y argument. ( i i ) I f k • n , then ) I ( C p U J x u , x w ) - t r a c e C p(A) = E p f c ^ , an) c o e Q p n s i n c e the Xco (co e Qpn) are an orthonormal b a s i s i n V/ n. • Vpi T h i s completes the p r o o f f o r t h e s p e c i a l case k = n . - 25 -( i i i ) L e t [y±, y ^ be a maximizing (minimizing) se t f o r g ( x i , ..., x^) • Consider t h e l i n e a r subspace of V n o f dimension k L = ^ ( y i , y k ) spanned by y^, • •., y-^ • L e t P be the p e r p e n d i c u l a r p r o j e c t i o n onto t h i s space. Consider PA : L •* L . Because of t h e p r o p e r t i e s o f p e r p e n d i c u l a r p r o j e c t i o n s , P = P^ = P and Px • x f o r any x 5 L . I f x, y * L , the n (PAx, y) - (Ax, Py) - (Ax, y) - (x, Ay) = (Px, Ay) = (x, PAy) so t h a t PA i s a He r m i t i a n t r a n s f o r m a t i o n on L to L . Let u^ , % be orthonormal e i g e n v e c t o r s o f PA ( i n L) Then: HI ( C p ( A ) y ( 0 , y j = 5^ Cp ( A ) y j Ll^ • • A 7 i p ' 7 i l ^ * ' A 7 ± ^ = £ > t ( ( A y v y i t ) S j t = 1 | . . . > p ) = E d e t ( ( P A y i s ' y H ) } • H < c p ( P A , y " > yco) t r a c e C p(PA) - ^ ( C p t P A j u ^ , u w ) = Edet((PA)ui , u± )) *a t - ZZd«t((Attis; u ± t ) ) - C ( C (A)uto, uj - 26 -( i v ) We c l a i m t h a t L i s an i n v a r i a n t space o f A ; t h a t i s A L < 1 , The proof i s by c o n t r a d i c t i o n . Since UJL, u k i s a b a s i s f o r L , l e t us assume t h a t A L . Then there e x i s t s v be l o n g i n g to the orthogonal complement of L such t h a t f - (Au x, v) J 0 Let u x - t f v 1 A • t 2 i ? t z u' = u ( j = 2, k) where t i s a r e a l number. I t i s easy t o v e r i f y t h a t |u£, .... u^ j i s an orthonormal s e t . Since g ( u ^ , % ) i s an extremum i t f o l l o w s t h a t g(u£, ..«, u£) must be zero at t = 0 . Now f o r t = 0 d / Ax u-j - t ? v Un - t f v ~ l C p l A J A U j A . . . A U j , . — A U j A . . M U j ) d t ^T7-2 2 1 + tr ! ?l f (Gp(A)v ^ U i 2 A . . . ^ u i p , u i 2 A . . . A U ^ ) f ( G p ( A ) u 1 A . u ^ A ... A U j [ p , v A u i 2 A . . . A U i p ) (Av, u j ) ( A u i 2 , u ^ ) o ( A u i p . u i p ) - 2 7 -( A u 1 } v) o (Aui (Au - 2 l ? l 2 P (Au, a=2 Here we have used the f a c t t h a t , i f s, t > 2 and s # t , U u i s , u ± t ) = ( P A u i g , u ^ ) = 0 s i n c e the u f s are orthonormal e i g e n v e c t o r s of PA . P Furthermore, n (Au, , u, ) 0 (because A i s a=2 x a ^ p o s i t i v e d e f i n i t e ) , and f ^ 0 , so i t c l e a r l y f o l l o w s t h a t —rr g(u', u') i s not zero at t » 0 . T h i s i s a dt 1 k c o n t r a d i c t i o n u n l e s s Au^ t L . Hence L(y-p y^) i s an i n v a r i a n t space of A • ( v ) . I t i s now easy to complete the p r o o f . I f j y ^ * 7^ i s an e x t r e m i z i n g set of orthonormal v e c t o r s , t h e n L ( y ^ , y^) i s an i n v a r i a n t space o f A ; l e t B = A | L , the r e s t r i c t i o n of A to L • Then B i s a p o s i t i v e d e f i n i t e H e r m i t i a n t r a n s f o r m a t i o n on a k d i m e n s i o n a l space. The eigenvalues o f B are some k o f the e i g e n v a l u e s of A . Hence by ( i i ) then - 28 -xjUJyo), y w ) - ( B ) y w , 'CO' CO CO - E p ( a ± i , . . . . a l f c ) f o r c e r t a i n e i g e n v a l u e s , cu of A . Thus 1 k Ep( al» •••» a k } ^ S ( x x , .... x k ) < E p ( a n , o n _ k + 1 ) f o r any orthonormal v e c t o r s x^, x k . Since e i t h e r e q u a l i t y can be o b t a i n e d by choosing x^, ..., x k to be s u i t a b l e orthonormal e i g e n v e c t o r s of A , t h e p r o o f i s complete, Two i n t e r e s t i n g c o r o l l a r i e s can be g i v e n . C o r o l l a r y 1 . min E p ( ( A x ^ xj_), .... (Ax^, x k ) ) = E p ^ a l » •••» a k ^ where t h e minimum i s taken over a l l s e t s of k orthonormal  v e c t o r s i n V n . P r o o f . Because ((Ax. , Xj ) • , - ) i s a p o s i t i v e V H s , t = 1 , p d e f i n i t e H e r m i t i a n matrix, i t f o l l o w s from t h e Hadamard determinant i n e q u a l i t y t h a t ( C p ( A ) x w , x j = det ( ( A x i s , « i t ) S f t . 1 > # . . > p ) < " ( A * i a . x i ) ; s=l s s so t h a t E p ( ( A x x , x ^ , (Ax k, x k ) ) > E p ( a 1 , . . . . a k ) E q u a l i t y can be a t t a i n e d by use o f s u i t a b l e orthonormal e i g e n v e c t o r s o f A , so t h a t t h e proof i s complete. G o r o l l a r y 2. max E p {(Axlt x±), (Ax k, x ^ ) k where the maximum i s taken over a l l s e t s of k orthonormal  v e c t o r s i n Y p . P r o o f . Theorem 4.2 , i n the case t h a t p = 1, s t a t e s max £3 (AXi, X i ) = E 1 ( a n , a n _ k + 1 ) because C^(A) = A . Since f o r any p o s i t i v e members a l » •••» a k ( 9 , p.49) V * i , - k ) < <5> ( E l ( a i ' "1L!!E1) p k 1 i t f o l l o w s t h a t E p ( ( A x 1 , x^Y, (Axjj., x k ) ) < (£) ( ( A x ! ^ ) - * - ... + ( A x k , x k ) j P k * «p' ( 1 ) To complete the proof we r e q u i r e t h e f o l l o w i n g lemma. 30 -Lemma 4.3» Let B be any t r a n s f ormation on V n t o V n . Then t h e r e e x i s t n orthonormal v e c t o r s x^, ..., x n such  t h a t tr> \ t r a c e B . _ [Bx±f x±) « — - i • 1, n . Proof. The set j (Bx, x) | (x, x) = 1 ; x £ V n j i s known t o be a c l o s e d convex set o f complex numbers c o n t a i n i n g the e i g e n v a l u e s of B , so t h e r e e x i s t s one v e c t o r x-[ w i t h (BXT , ) = . The p r o o f i s by i n d u c t i o n . Assume h t h e r e e x i s t k orthonormal v e c t o r s x^, •»., x k w i t h ( B x i , Xj.) « , i = 1, .... k . Let U be a u n i t a r y n n-square m a t r i x i n which the f i r s t k columns are t h e v e c t o r s x 1 ? x k . Then U*BU « / B l 1 B l 2 \ B 2 1 B22 t r B where B ^ i s a k x k m a t r i x w i t h diagonal e n t r i e s — — , and B 22 i s an (n-k) x (n-k) m a t r i x . We have k + t r B ? o • t r B n so t h a t t r B 2 2 a t r B n-k n Now t h e r e e x i s t s an (n-k) x (n-k) u n i t a r y m a t r i x V such t h a t the ( 1 , 1 ) entry o f V ^ B 2 2 V i s t r ^ . L e t n-k w - Ik <±> v • Then (UW)*B(UW) has ^ L l a s i t s f i r s t k + 1 n d i a g o n a l e n t r i e s , so t h a t the f i r s t k + 1 columns o f UW t r B are orthonormal v e c t o r s t h a t s a t i s f y (Bx, x) = • Hence n the i n d u c t i o n step i s complete and the Lemma i s proved. To complete the p r o o f of C o r o l l a r y 2, l e t X]_, x k be orthonormal e i g e n v e c t o r s of A c o r r e s p o n d i n g to a n , a n _ k + i r e s p e c t i v e l y . L e t B be the r e s t r i c t i o n of A t o L ( X " L , Xfc) . Then t h e r e e x i s t orthonormal v e c t o r s y^, y k such t h a t ( A 7 i , y ± ) - ( B 7 i , y ± ) = a n +' ''' + «n-k+l k Hence "p((*Jl, n), . . . . Uy k , yk)> = (J) f » ° " - i ^ ) P By use of a standard c o n t i n u i t y argument we may e s t a b l i s h Theorem 4.3. Theorem 4«2 and i t s two C o r o l l a r i e s remain  t r u e i f A i s a non-negative H e r m i t i a n t r a n s f o r m a t i o n . Theorems 4.2 and 4«3 and t h e C o r o l l a r i e s a r e due t o Marcus and McGregor [10 ). I t i s i n t e r e s t i n g to note t h a t p r o o f s o f t h e C o r o l l a r i e s were g i v e n b e f o r e Theorem 4.2 i t s e l f was proved. D e f i n i t i o n 4.4» ( i ) A f u n c t i o n of one r e a l v a r i a b l e f (x) d e f i n e d on [ a , b j i s s a i d to be convex i f f o r any xlt x 2 e [ a , b j - 32 -f (x l + x 2 ) < f ( x x ) • f ( x 2 ) 2 2 ( i i ) A f u n c t i o n f i s concave i f — f i s convex  Lemma 4»5» I f f i s continuous and convex and i f n r i " 1 , o*i - 0 » then *( Zyai x i } ^  23 a i f ( x i ^ 1=1 1=1 P r o o f . T h i s i s w e l l known i n t h e t h e o r y of convex f u n c t i o n s . I f H i s any H e r m i t i a n m a t r i x with e i g e n v a l u e s h^ £ ... Z h n c o r r e s p o n d i n g t o orthonormal e i g e n v e c t o r s u i , u n , then any v e c t o r x has a unique r e p r e s e n t a t i o n i n the form h x «• £ ^ (x, U ^ J l l j and h Hx • 2^ (x, u i ) h i \i± T h i s remark motivates D e f i n i t i o n 4.6. For any f u n c t i o n f d e f i n e d on the spectrum  o f the Hermitian m a t r i x H , d e f i n e f (H)x « £ ^ (x, u ± ) f ( h ± ) u ± . Theorem 4.7. I f f i s a r e a l continuous f u n c t i o n of one  v a r i a b l e d e f i n e d on the spectrum of the H e r m i t i a n m a t r i x H, - 33 -t h e n f (H) i s H e r m i t i a n . I f f i s convex (concave) t h e n  f o r any u n i t v e c t o r x : f( ( H x , x ) ) < ( > ) ( f ( H ) x , x ) Proo f . The Hermitian nature of f ( H ) e a s i l y f o l l o w s from D e f i n i t i o n 4.6. I f f i s convex, then u s i n g Lemma 4»4 f ( ( H x , x ) ) = f( "Ll u ± ) | 2 h ± ) i = l < g \l*> ^ i ) l 2 * ( * i > = ( f ( H ) x , x) . Theorem 4» 8. I f f i s d e f i n e d on t h e spectrum o f t h e Herm i t i a n m a t r i x H and f ( x y ) = f ( x ) f ( y ) then f ( C p ( H ) ) = C p ( f ( H ) ) P r o o f . By d e f i n i t i o n C p ( f ( H ) ) x 1 A ... A X p = f (H)xi A . . . A f(H)Xp = C ( x l > V f ( h a > a - - - - ' I T { x P ' % ) f ( h a K = Cdet((xs, ^ i s ) 3 ! 3 l > . . . > p ) ^ V • • f ( h i p ) U i l A • • • A % • ^ f ( x l A ' • • A X P ) » U J F ( H J ua> f (C p(H) )x1 A . . . AXp Here we have used Theorem 2.3. - 34 -Theorem 4»9 Assume the hypotheses of Theorem 4»3. Let f (x) = x s . Then (a) I f s > 1 max 7~7V f ( ( C ( A ) X a ) , x w ) ) = E ( f ( a n ) , f ( a n . k + 1 ) (b) I f 0 < s < 1 min f ( ( C D ( A ) x 0 0 , x C 0 ) ) = E D ( f ( a 1 ) , f ( a k ) ) c o f Q p k P P r o o f , (a) For s 2 1 , f ( x ) = x s , i s a convex f u n c t i o n and f (xy) •» f (x) f (y) . Hence, by Theorems 4»7 and 4.S f ( C p ( A ) ) = C p ( f ( A ) ) f ( ( C p C A j x ^ x J ) < ( f ( C p U n x u , x w ) . T h e r e f o r e < E p ( f ( h n ) , , f ( h n „ k + i ) ) . E q u a l i t y i s a t t a i n e d i f Xj_, ... e i g e n v e c t o r s c o r r e s p o n d i n g t o a (a) • (b) i s proved s i m i l a r l y . are orthonormal » a n - k + l T h i s proves C o r o l l a r y . Consider any orthonormal s e t ^x (1) I f 0 < s < 1 , E p ( ( A x p X l ) s , (Ax k, x k ) s ) - 35 -I E p ( a j , as) (2) I f s > 1 , Ep ({Axlt X j ) , (Ax k, x k ) ) < ( k) ^<*n + ... + a n - k + l \P k ' Proof, The p r o o f s are a l o n g t h e l i n e s of t h e p r o o f s of the C o r o l l a r i e s of Theorem 4.2. I n (1) above, e q u a l i t y i s a t t a i n e d . 2. A New I n e q u a l i t y F o r P o s i t i v e Numbers. R e c e n t l y Marcus (11) and Bohnenblust ( i ) have proved the f o l l o w i n g theorem: Theorem 4.10. Let 1 < r < k and l e t a^, b^, i = 1, k be p o s i t i v e numbers t h e n l / r l / r E r ^ a l + ^ l » •••> ak + 1 : >k) ~ E r ^ al» *•*» a k ^ + [blt b k ) wit h e q u a l i t y i f and o n l y i f t h e s e t s (a^) and (b^) a r e  p r o p o r t i o n a l . The p r o o f w i l l not be g i v e n . We w i l l be content t o g i v e the f o l l o w i n g consequence: - 3 6 -C o r o l l a r y . L e t A, B be no n-» negative n-square Hermitian  m a t r i c e s . For any n~square m a t r i x M l e t P r ( M ) denote the  c o e f f i c i e n t o f t r i n d e t ( t l - M) , r = 1, ..., n . Then | F R ( A + B ) | 1 / R > |P r(A)| 1 / r • |P P(B)| l A . P r o o f . Let ; a nd , . •., X n denote t h e e i g e n v a l u e s o f A , B , and A+B r e s p e c t i v e l y . Then P r(A)| « E r i * i , «n) P r(B)| = V-^i* •••>Ai) |Pr(A+B)| = E r ( \ 1 , X n) . I f X j , x n are orthonormal e i g e n v e c t o r s of A+B , then ^{k+B)^7 = ti^r ( ( A + B j x i . X i ) , ( ( A + B ) x n , x n ) ) l / r = E r ( ( A X ^ ^ M B X ^ X - L ) ' , ( A x n , x n ) + ( B x n , x n ) ) ^ l / r i - E / ((Ax!, x x ) , (Ax n, x n ) ) 0} l / r +: E r ((Bx!, x1)J (Bx n, x n ) ) > ^ « n > + E r / r J * n ) 8 3 l P r U ) I ^ + | P r { B ) I V r u s i n g Theorem 4«2., C o r o l l a r y 1. - 37 -T h i s i s a g e n e r a l i z a t i o n of a c l a s s i c a l i n e q u a l i t y of Minkowski (11): (det(A+B) ) n > (det A ) n + (det B ) n . V. A p p l i c a t i o n s . In t h i s s e c t i o n we s h a l l g i v e e x t e n s i o n s of two p r e v i o u s l y known theorems. We s t a t e the f o l l o w i n g Lemma. Lemma 5.7. ( i ) I f o r > 0 , 8 > 0 , f f + 8 - l , a > 0 , b > 0 , then a a b 5 < cr a + 5 b ( i i ) I f s > 1 , a s + b s < (a + b ) s I f 0 < s < 1 , a s + b s > (a + b ) s 1. E x t e n s i o n of a Theorem of H. Weyl. The o b j e c t of t h i s s e c t i o n i s to extend a r e s u l t due to H. Weyl (16) . Lemma 5.2. L e t A, B be any two n-square non-negative H e r m i t i a n m a t r i c e s , w i t h e i g e n v a l u e s < ... < a n , A < • • • S j * n r e s p e c t i v e l y . I f ff + 8 = 1 , u ) 0 , S > 0 , and i f t h e ei g e n v a l u e s o f o*A + 8B are < ... < *>n > t h e n  f o r 1 < r < k < n E p U l f " > k > > B?(a l f . . . ,a k -) E^ i filt...$Jtk) P r o o f . Let x^, x k be orthonormal e i g e n v e c t o r s o f o*A + $B corresponding t o X ^ , \ k • Then - 3d -E ^ 1 * ^ , . . . ^ ) = E V r ( ((ffA + 8 B ) X l , x x ) , ...) I ff E J A ( (AX-L , X X ) , . . . ) + 5 Ej/r ( ( B x i j Xly9 . . . ) > ff E j / r ( a l f a k ) + 8 ^r(plf Jt) > E ^ r (a l f a k ) 4 / r(A» - . A ) u s i n g Theorem l+,2 C o r o l l a r y 1, Theorem 4«10 and Lemma 5.1» We can now g i v e the e x t e n s i o n of W e y l r s r e s u l t . Theorem 5.3» L e t A be an a r b i t r a r y n-square m a t r i x w i t h  e i g e n v a l u e s \± such t h a t J X ^ J 2* • • • — j ^ n J * **et ff > 0 , 8 > 0 , ff + 8 = 1 , and l e t ff A*A + s AA* have ei g e n v a l u e s a 1 > ... > a£ . Then f o r 1 < r < k < n and s £ 1 Bp ( a * 8 , ags) 2 E ^ I X J 2 3 , ( X k | 2 s ) . P r o o f . By S c h u r T s Lemma choose orthonormal v e c t o r s x-p x k such t h a t (AXJL, x^) = X^ ( i = 1, k) and ( A x i , x^) = 0 ( i > j ) . Then, i f f ( x ) = x s , E r ( a 2 s j a 2 s ) > 7^  ( C r [ f f f f A ^ A + SAA* ) J X ( 0 , x j - YZ? [(Cr(ffA*A + SAA*)x w, X a ) ) ] [det ( f f ( A * A x i g , x i t ) + 8(AA*x i g, x ^ ) ) ] " ZZf [ (det( ^ i s > X i t } | ^ / d e t ( A A % x i s » X i t }| 8 ] - 39 -aYlf [ (Cr(AlA)xw,x(0)(T (Cr(AA*):xa),xa))8] = H f [(Gr(A)x(0,Cr(A)xa))ff (Cr(A*)xw,Cr(A*)xJS] _>]Tf f |(Cr(A)Xa),xj|20r ((^ (A^J^xJl 25 ] -E* < W\2 ••• K | 2 ) = E r ( \ X 1 \ 2 s , |X k| 2 3 ) Here, i n s u c c e s s i o n , we have used Theorem 4.9, Theorem 4*7, Lemma 5.2 ( i n the s p e c i a l case r = k = n) and the f a c t t h a t (u, u) > J (u, v ) | i f v i s a u n i t v e c t o r . T h i s completes t h e p r o o f . Remarks. T h i s theorem, f o r the case k = n and any s > 0 was g i v e n by H. Weyl i n 1949 (16) . For n o n - s i n g u l a r m a t r i c e s , the case k = n was a l s o extended by H. Weyl to n e g a t i v e s • 2. E x t e n s i o n o f a r e c e n t Theorem of Wielandt. To s i m p l i f y the statements of t h e theorems i n t h i s s u b s e c t i o n , we i n t r o d u c e some n o t a t i o n s . Throughout t h i s s u b s e c t i o n we s h a l l assume t h a t S i s a giv e n s e t of p n a t u r a l numbers < n and t h a t i < j < ... < m are i t s elements. By V^, V j , V m w i l l be meant subspaces of V n w i t h t h e p r o p e r t i e s % h a t - 40 -v.'c v 1 c . . . c v m dimension V f f = CT (cr S) I n g e n e r a l , a s u b s c r i p t to a symbol d e n o t i n g a subspace of V n w i l l i n d i c a t e t h e dimension of t h e subspace. I f r < p we d e f i n e S r p as s r p = | 0 0 = { i l * •••'» A r ] |' 1 S A i < ••• < i r S n» 9 X 1 13 e s We can now s t a t e Theorem 5.4. Theorem 5.4» Let S be a set of p n a t u r a l numbers < n and f o r 1 < r < p < n l e t Spp be as g i v e n above. Let A be a non-negative H e r m i t i a n operator on V n such t h a t & i t • • • 1 ct n and J)tl < •.. < ^ n are two r e p r e s e n t a t i o n s  of i t s e i g e n v a l u e s . I f s, t are any two r e a l numbers such t h a t 0 < s < 1 , t * 1 , t h e n : ( i ) E r(Bg.; er t S) - min max Ep* ((Ax^, x f f ) s ; cr t S) V x t t ( i i ) E J J C X Q . ; a e S) • max min Ej_ ( ( A X Q - , xa) ; cr t S ) V x ( i i i ) E r ( ; cr 6 S) = min max X ( C r ( A ) x M , x J S ° V X C0€ Spp w ( i v ) Ep (a|; cr e S) « max min / 7 ( ^ ( A j x ^ , x ^ V x c o e S r p where , i f co - j" i x , i r j e Spp. t h e n x^ = X ^ A . . . A x ± j and max (min) i n d i c a t e s the max (min) over a l l p o s s i b l e p x x orthonormal v e c t o r s x^, x j , ..., x m s a t i s f y i n g x £ e- V f f (ff * S) (2) - 4 1 -f o r f i x e d subspaces (1), and min (max) i n d i c a t e s t h e '• " *~ v v ~ ~ " min (max) over a l l p o s s i b l e subspaces ( 1 ) . P r o o f , ( i ) : The r e s u l t i s e q u i v a l e n t to t h e p r o p o s i t i o n s (a) and (b) below. (a) : There i s a s p e c i a l sequence Y ^ C . . . C Y M of subspaces o f V n such t h a t f o r every orthonormal s e t x i > x j > •••> xm w i t h XQ. YQ. (ff t S ) , we have E r ( ( A x 0 , x a ) S , ff k S) < E r ( B ^ ; ff t S) . (3) For l e t y-p 7 n be orthonormal e i g e n v e c t o r s of A corresponding t o JfJ^, J J n . Define Y 0 t o be the sub-space spanned by y-p y^ . Then (Ax 0, x 0 ) < $ Q (Xg. « Y A , (xg., XQ.) • 1 ) , hence (3) f o l l o w s . (b) : Let V i } V ^ , V m be g i v e n subspaces of V n s a t i s f y i n g (1) . Then t h e r e are orthonormal v e c t o r s x^, ..., x m s a t i s f y i n g (2) such t h a t E r ((Ax f f, xaf ;<r e S ) ? ^ e S ) (4) The p r o o f i s by i n d u c t i o n on n . F i r s t , l e t S c o n t a i n a l l n a t u r a l numbers ff < n . Then choose x^, x n s a t i s f y i n g (2) but otherwise a r b i t r a r y . Then by Theorem 4.2 C o r o l l a r y 1 ^ ( ( A x x , X T J S , (Ax n, x n ) S ) > E r ( ^ J , . . . , 3 £) Hence (4) h o l d s . T h i s e s p e c i a l l y a p p l i e s t o (r=)n=l . In what f o l l o w s we assume th e r e i s a n a t u r a l number < n which i s not i n S . The l a r g e s t such gap number w i l l be denoted by g . We d e f i n e f t o be the l a r g e s t number i n S which i s < g i f such e x i s t s ; i f not, we d e f i n e f = 0 . I n e i t h e r case 0 < f < g < n . I f f > 0 ', V f i s d e f i n e d by h y p o t h e s i s ; i f f = 0 we d e f i n e If = 0 . The simplest case i s t h e one f o r which n j S , t h a t i s , g = n . Then we choose any subspace Vn_-j_ c o n t a i n i n g . Define A to be the unique H e r m i t i a n operator on Vn_-j_ such t h a t (Ax, x) = (Ax, x) , x f- 7 n - 1 (5) The e i g e n v a l u e s J3-j_ < ... < Jh ^ of A are known ( 2. ) t o s a t i s f y 0 < J 5 X < 5i < J$2< j f 2 < . . . < ^ n _ ! < J n - l ^ n <*> By the i n d u c t i o n hypothesis t h e r e are orthonormal v e c t o r s x ( 7 £ ^ n - l s u e h t h a t xcr c" vcr (cr * s ) and Ep ((Ax f f, x a ) S ; a € S) 1 E r ( J f f S ; or t S) I n view o f (5) and (6) t h i s i n e q u a l i t y i m p l i e s ( 4 K T h i s f i n i s h e s the case g = n . Now l e t n k S , t h a t i s , g < n . We choose o r t h o -normal v e c t o r s 7g +i» •••> ^n °^ A c o r r e s p o n d i n g t o ^ ' g + l » •••» 3xi ' ^ ° S e t n e r > w i t h they span a subspace o f dimension < (n-g) + f < n . Hence we can choose some sub space ^ n _i such t h a t v f c V i » y v * 7 n - l » ( v = S + 1 » •••'» n ) ( 7 ) Since g+1, n S we have v f c v g + 1 c . . . c v n so v f S V i ^  V i < < V i n V i c v n - 1 Since t h e dimension o f V y A ^ n - l i s a t l e a s t v _ 1 w e c a n choose subspaces Vg, V n „ 2 such t h a t \ <• V l ; — ; ?n-2 <= V l <8> V ± 0 ... C 7 f C ? g C V g + 1 0 ... 0 T n . 2 0 (9) We d e f i n e as be f o r e t h e o p e r a t o r A oh Vn_2_ . By the i n d u c t i o n h y p o thesis a p p l i e d to A and the subspaces (S) the r e e x i s t orthonormal v e c t o r s x^ (cr k S) such t h a t x 0 * (<r <g) , xa * V f f - 1 (cr > g) (10) E,. ( (Ax, ,**) 8 ; cr t S) > V J j , J * f > J | > S n - l 1 U s i n g (5), (6) and (5) we f i n d t h a t *cr e V f f (cr e s) (11) E r ( ( A x f f , x a ) s ; crt S) 2 E r ( j , J , . . . , j ^ , J | , . . . ; J Now, by (7) we knew t h a t y g +]_> •••> y n are eigen-v e c t o r s o f A" with e i g e n v a l u e s J$g + i > c ^ n • Hence f o r t h e l a r g e s t eigenvalue s o f A we have - 44 -J «£ n-1' - >^n-2 ; • • • » ^ g + l < J g so t h a t T h i s completes t h e p r o o f of (b) and hence of ( i ) . Proof o f ( i i ) : Obvious m o d i f i c a t i o n s to the above proof and the use of Theorem 4.9 C o r o l l a r y (a) t o b e g i n t h e proof o f the analogue to (b) above w i l l y i e l d the r e s u l t . P r o o f o f ( i i i ) : We have t o e s t a b l i s h t h e analogues to (a) and ( b ) . (a) f o l l o w s e a s i l y u s i n g t h e Hodamard determinant i n e q u a l i t y . (b) f o l l o w s without any a l t e r a t i o n s upon n o t i n g t h a t t h e case i n which S c o n t a i n s a l l n a t u r a l numbers < n i s a consequence o f Theorem 4.9. Proof o f ( i v ) ; The o n l y d i f f i c u l t y here l i e s i n demonstrating the analogue t o ( a ) , namely t h a t subspaces 1^ C Y j C ... C I m e x i s t such t h a t = E r ( <r e s) . min ( C r ( A ) x ( 0 , Xto) = a± ... - 45 -where t h e min i s taken over a l l orthonormal v e c t o r s x^ , x^ w i t h Xj_ . f , j s 1, r . To see t h i s note t h a t any x„. A , , , A X < l i e s i n t h e subspace o f i - ^ x r V n spanned by the s e t of v e c t o r s T h i s subspace of V . n i s an i n v a r i a n t space of C r(A) so l r ) l e t B be t h e r e s t r i c t i o n of C r(A) to t h i s subspace. Then th e s e t of e i g e n v a l u e s o f B i s e x a c t l y the set \ a J l a J P I j l ~ *1 J ; J r ^ ^ j and c t j ^ ... a j _ r i s the s m a l l e s t e i g e n v a l u e o f B . Hence f o r any u n i t v e c t o r z i n the space spanned by (11), (BZ, Z) > ... , so min ( C r ( A ) X a ) , x u ) = ... a i r . Theorem 5»4 i s now c o m p l e t e l y proved. We can r e c a s t Theorem $.4 i n a s l i g h t l y d i f f e r e n t form (Theorem 5*5)» Theorem 5«5. Under the assumptions of Theorem 5.4. ( i ) E r ( a ® ; cr « S) - min max E r((Axo-, x f f ) s ; cr <: S) ( i i ) E-,( cr* S) = max min E, ((Axg-jXg-)* ; ff * S) ff V x -1-( i i i ) E-taf?; <r * S) - min max J ( C - t A ) ^ , x . ) 3 V x a) C S r p ( i v ) E r ( J 3 £ ; ff € S) = max min ) ( C p U ^ , x )* V x c o t S p p - 46 -where, i n ( i ) f o r example, f o r f i x e d subspaces v i - i c V i c — c V i (13) dim V f f - 1 = cr - 1 (cr * S) the max i s taken over a l l orthonormal v e c t o r s x^, x j , ' • •., x m  s a t i s f y i n g x cr * V l ( ( T * S ) ( 1 4 ) and t h e min i s taken over a l l p o s s i b l e subspaces (13) o f V . Proof, ( i ) Given S • j i , j , ..., m l e t T = |n-m+l, n - i + l j . Then by Theorem 5«4 s E r ( a 3 ; a t S) = min max E p ( ( A x ^ x ^ ) ; cr t S) V c r + l xcr ^  Vn-cr+l L e t t i n g be t h e orthogonal complement i n V n t o ^) J we have E r ( a ^ ; cr C- S) = min max Ep((kx. a,x a) S; cr * S) Tcr-1 x a - L T c r - l I f we r e l a b e l T f f - i as v ^ - j . , ( i ) w i l l be proved. ( i i ) , ( ( i i i ) , ( i v ) f o l l o w s i m i l a r l y . Theorem 5 . 6 . Let A, B, C b_e non-negative H e r m i t i a n o p e r a t o r s  on V n such t h a t C = A+B . L e t a v , J3 V , y„ (a», J? v* , yj) be t h e r e s p e c t i v e eigenvalue s of A, B, G numbered i n d e c r e a s i n g ( i n c r e a s i n g ) o r d e r . Then, under the assumptions of Theorem 5 .4 ( i ) E j / r ( ; cr * S) > [ E J / ' C O ' ; cr * S) + E ^ / r ( J 3 i , J 5 p ) ] S - 47 -t r ( i i ) Er( y t ; cr * S) < (?) ( a i * * J + * ' ' ^ m* J V « ' •+ fip> P r o o f »(i) By the p r o o f of Theorem 5 . 4 t h e r e are subspaces C V j C ... C V m such t h a t E j A ( V » S ; a e S) =» max s j / r ( (Cx £ y,x 0.) S ; ff * S) 0 ff Keeping t h e subspaces f i x e d we choose orthonormal v e c t o r s zff such t h a t za <• Va (ff fr S) and such t h a t ^ r ( ( A Z ( y , z f f ) ; ff t S) = max E ^ r ( (Ax a, x a ) ; ff <• S) x f f <rVa Then u s i n g Lemma 5 .1 and Theorem 4 .2 C o r o l l a r y 1 , E r / r < ^ f f S ; ff * s ) * ^ ( ( C z ^ z ^ ) 3 ; ff * S) > E 3 / ' r ( ( C z c r , z c r ) ; ff e S) > ( E j / r ( ( A a f f ; Z ( y ) ; ff * S) + E y r ( ( B a f f , a f f ) ; ff * S ) ] 3 ( i i ) As b e f o r e we choose subspaces C ...CV m such t h a t s E r ( y t ; ff £ S) = min y^" 7. ( ( C r ( C ) X ( j 0 , x )* and orthonormal v e c t o r s zff such t h a t - 4* -E 1 ( ( A z f f , z f f ) ; cr « S) = min^ E X ( ( A x ^ x ^ ) ; cr S) Then *cr fYcr E ^ y a ; 0 * S) < ^ ( C R ( C ) z , B J * < ZZ<° zi > " i ) * a=l a a = Ey ( ( Cz-, ZoO* ; o- e s) r / 7 7 ( ( A + B ) z f f , zcr)* V (ZJ ( A z ^ z ^ + ( B z ^ z j y ) ! t r <( p) ( ^ s a - + ^ + - ^ p ) t r P r Remarks: Theorems 5.4 ( i i ) , 5»5 ( i i ) and 5.6 ( i i ) i n the s p e c i a l cases r=s=t=l were g i v e n by Wielandt i n 1955 ( 1 5 " ) . As g e n e r a l i z a t i o n s of t h e s e s p e c i a l cases, the co n t e n t s o f t h i s s u b s e c t i o n are b e l i e v e d t o be new. However, the p r o o f s l e a n h e a v i l y on W e i l a n d t T s o r i g i n a l p r o o f . I n p a s s i n g we may n o t i c e t h a t t h e value of max min E r ( (Ax^ jX^ ); cr f S) i s as yet unknown. - 49 -V I . A Norm I n e q u a l i t y D e f i n i t i o n 6.1. For any n-square matrix A d e f i n e ||A||2 = t r a c e AA Theorem 6.2. For any H e r m i t i a n matrix H l e t X V(H) denot e  the e i g e n v a l u e s o f H i n i n c r e a s i n g o r d e r . Then i f A, B are Proof. For any non-negative H e r m i t i a n m a t r i x H , Theorem 4.2 s t a t e s i f t h e x's are orthonormal v e c t o r s . I t can be shown t h a t t h i s i n e q u a l i t y h olds f o r any H e r m i t i a n matrix H . ( I n f a c t , i n the s p e c i a l case p = 1, the p o s i t i v e d e f i n i t e assumption i n Theorem 4.2 i s not needed). I n what f o l l o w s we s h a l l l e t Xj_, ...,x n be orthonormal e i g e n v e c t o r s o f A+A^ c o r r e s p o n d i n g to ^ ( A + A * ) , \ n(A+A* ), and l e t y x , y n be any n-square m a t r i c e s orthonormal e i g e n v e c t o r s o f i c o r responding to \ (A-A*\ • y i A-A*, Kl >> •••» x n l ~ ) Then : - 50 -2 | |(A +B|1 2 - ||A||2 - ||B||2j 2 t r a c e (AB* + BA* ) ^ G # # . . » * W T ^ T * I . /A-A?,B-B* n ^ ( ( ( ( A + ^ X B + B * ) • (*=£)(£=£ ) ) X j ) Xj)J ^ ( ( A + A * ) < B + B * ) * . , + g ( (is^H&SVj. y j ) § ^ ( A # A * ) ( ( B + B * ) x j > x j ) • ^ ( ^ H ^ J y j . 7 j ) X 5^1 3 3 + (X 2(A+A*) - ^ ( A + A * )) ( ( B + B ^ J x j ^ . ) + ... + (X n(A+A* ) - X n - 1 (A+A* ) ( ( B + B * ) x n , x n ) + a s i m i l a r decomposition f o r t h e second sum n < X x (A+A*) £ J Xj (B+B*) + ( X 2 (A+A*) - X-^A+A*)) X j (B+B*) • • • ©*tc • The o t h e r i n e q u a l i t y i s proved i n a s i m i l a r f a s h i o n . V a r i o u s i n e q u a l i t i e s can be ob t a i n e d as s p e c i a l cases o f Theorem 6 . 2 . I f we l e t B = -A we o b t a i n . - 51 -C o r o l l a r y . For any n-square m a t r i x A 1 A £ [ y A + A * X _ j t l ( A + A * ) + X j l ^ V ^ nAn2 n-j+l i 5=* J J i 1 V I I . H i s t o r i c a l Survey. The m a t e r i a l c o n t a i n e d i n s e c t i o n s I I and I I I concerning compound ma t r i c e s has been known f o r some time. The S y l v e s t e r -Franke theorem was f i r s t proved about 1&50. The p r o o f g i v e n i n t h i s t h e s i s i s b e l i e v e d t o be one of the s h o r t e s t g i v e n . Other r e c e n t p r o o f s have been g i v e n by Tornheim (14) and F l a n d e r s (7). The c o n t e n t s of s e c t i o n s IV and V have i n t e r e s t i n g h i s t o r i e s . Weyl's theorem, mentioned i n s e c t i o n V . l , was g i v e n i n 1949 (16). S h o r t l y t h e r e a f t e r , i n a paper devoted to Weyl's theorem, Fan (3) gave t h e s p e c i a l case p = 1 o f C o r o l l a r i e s 1 and 2 of Theorem 4.2. Other s p e c i a l cases of t h e s e C o r o l l a r i e s were l a t e r g i v e n by Fan i n problem form (5, 6). I t was not u n t i l 1955 t h a t Theorem 4.2 i t s e l f was c o n j e c t u r e d and proved by Marcus and McGregor (10). - 52 -The u s e f u l i n e q u a l i t y Theorem k*10 was f i r s t g i v e n i n 1955, p r o o f s are due ( s e p a r a t e l y ) t o Marcus (11) and Bohnenblust (1). The o r i g i n of t h e c o n t e n t s of s e c t i o n V .2 i s a w e l l known minimax p r i n c i p l e o f Courant (2) (see a l s o (3)). WIelandt's (1955) theorem (15) i s r e a l l y an e x t e n s i o n of t h e e a r l i e r Courant p r i n c i p l e . F u r t h e r extensions have been g i v e n i n t h i s t h e s i s . The c o n t e n t s o f s e c t i o n VI a r e b e l i e v e d t o be new. V I I I . B i b l i o g r a p h y 1 . H.F. Bohnenblust, I n e q u a l i t i e s between symmetric f u n c t i o n s . t o appear. 2. R. Courant, D. H i l b e r t , Methoden der mathematischen  Physik. B e r l i n , 1931. p. 2g. 3. Ky Fan, On a theorem o f Weyl concerning; the e i g e n v a l u e s  of l i n e a r t r a n s f ormat i o n s . I . Proc. Nat. Acad. S c i . U.S.A.' v o l . 35(1949) pp. 652-655. 4. , On a theorem o f Weyl concerning the ei g e n -v a l u e s o f l i n e a r t r a n s f o r m a t i o n s . . I I , Proc. Nat. Acad. S c i . U.S.A., v o l . 36. (1950), pp. 31-34. 5. , Problem 4429, Amer. Math. Monthly, proposed v o l 58 (1951) p. 194, s o l u t i o n v o l . 60(1953) pp. 4#. 6. , Problem 4430, Amer. Math. Monthly, proposed v o l 53 (1951) p. 194, s o l u t i o n v o l . 60(1953 p. 50. 7. H. F l a n d e r s , A note on the S y l v e s t e r - F r a n k e theorem. Amer. Math. Monthly, v o l . 60 (1953) , pp. 543-5. 8. P. Halmos, F i n i t e Dimensional Vector Spaces. Annals o f Mathematics S t u d i e s No. 7, P r i n c e t o n , 1953 p. 157. 9. G. Hardy, J . L i t t l e w o o d , G. Po l y a , I n e q u a l i t i e s . Cambridge, 1934. 10. M. Marcus and J.L. McGregor, Extremal p r o p e r t i e s o f H e r m i t i a n m a t r i c e s , t o appear i n Can. J . Math. 11. M. Marcus, Some i n e q u a l i t i e s f o r H e r m i t i a n m a t r i c e s , t o appear. 12. M. Marcus, Symmetric f u n c t i o n s o f s i n g u l a r v a l u e s , t o appear. 13. H. Minkowski, D i s k o n t i n u i t a t s b e r i c h f u r a r i t h m e t i s c h e  A q u i v a l e n z . Jounr. f u r Math., v o l . 129 (1905) pp. 220-274'. 14. L. Tornheim, The S y l v e s t e r - F r a n k e theorem, Amer. Math. Monthly, v o l . 59, (1952) pp. 3^9-391. 15. H. Wielandt, An extremum p r o p e r t y o f sums of ei g e n v a l ue s, Proc. Amer. Math. S o c , v o l . 6 (1955), pp. 106-110. 16. H. Weyl, I n e q u a l i t i e s between the two k i n d s of e i g e n v a l u e s  of l i n e a r t r a n s f o r m a t i o n s . Proc. Mat. Aca. S c i . U.S.A., v o l . 35 (1949) PP. 403-411. 

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