UBC Theses and Dissertations

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UBC Theses and Dissertations

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. National 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 thesis as conforming to the " required standard. THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C o lumbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f M a t h e m a t i c s  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date S e p t e m b e r 1 7 . 1 9 6 9 ( i i ) ! i ABSTRACT Much i s known about the eigenvalues of some spe c i a l types of matrices. For example, the eigenvalues of a hermitian or skew-hermitian matrix l i e on a l i n e while those of a unitary matrix l i e on a c i r c l e ; t h e i r spectra are "l i n e a r " or " c i r c u l a r " . This suggests the question : What matrices have t h i s property ? Or, more generally, what matrices have t h e i r eigenvalues on plane curves of a simple kind ? Is i t possible to recognize such matrices by inspection ? In t h i s thesis, we make a small start on these problems, exploring some matrices whose eigenvalues l i e on one or more ••; l i n e s , or on one or more c i r c l e s . . / ( i i i ) TABLE OF CONTENTS Page SECTION 1 : Introduction .. 1 SECTION 2 : Matrices with Eigenvalues on Lines or C i r c l e s 6 SECTIOI 3 : The Expanded Matrix of A Complex Matrix 11 SECTION 4 : Nonnegative Matrices 17 SECTION 5 : Tridiagonal Matrices 21 SECTION 6 : Compound Matrices and Some Other Theorems 25 SECTION 7 : Extended Polynomial-problem on • Eigenvalues / 29 BIBLIOGRAPHY 38 / (iv) ACKNOWLEDGEMENTS I am greatly indebted to Professor B. N. Moyls for suggesting the topic of thi s thesis, f o r allowing me a generous amount of his time and f o r his many constructive comments during the preparation of t h i s thesis. I also wish to thank Professor Roy Westwick f o r his c r i t i c i s m of the draft form of t h i s work. The f i n a n c i a l support of the University of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. / 1. Introduction Let a21 a22 11 a 12 A = ( a j k ) \ an1 an2 nn J be any square matrix of order n with elements i n the complex complex entries x^ , ... » x n such that AX = \X i s said to be an eigenvector of A corresponding to the eigen-value X j X i s a root of the ch a r a c t e r i s t i c equation of degree n. Counting m u l t i p l i c i t i e s there are n eigenvalues of A. The set of eigenvalues of A i s called the spectrum of A. There exists a l o t of information about the eigenvalues of some special types of matrices. For example, i t i s ea s i l y seen that diagonal(a^=0 i f and triangular(a^=0 i f j > k ) matrices exhibit t h e i r eigenvalues on t h e i r main diagonal : = ^ j j ' «5 = 1»2,...,n. It i s not quite so t r i v i a l that the eigenvalues of a T T r e a l symmetric matrix A (A = A , where A denotes the transpose of A) l i e on the r e a l axis. This was f i r s t proved by A. Cauchy [ 8 ] i n 1829» and many subsequent proofs have been given, by f i e l d C#. A non-zero vector X = ( x ... , x n ) T with det( A I - A ) = 0 2 . other eminent mathematicians, including Jacobi and Sylvester. This theorem was generalized by Hermite [%2] i n 1855 to matrices for which A = A (A denotes the transpose conjugate of A), and resulted i n such matrices being named a f t e r him. A w e l l -known simple elegent one-line proof of t h i s r e s u l t ' i s the following : For a unit eigenvector X corresponding to X, X = XX*X = X* AX = X*AX = ( X*AX ) * = A. In a sim i l a r way the eigenvalues of a r e a l skew-symmetric matrix (A = -A ) l i e on the imaginary axis. This was f i r s t proved by A. Clebsch [ 9 ] , and l a t e r by Weierstrass [19]. That the same i s true f o r a skew-hermitian matrix (A = -A ) was shown by G. Scorza j t 8 J i n 1 9 2 1 . This i s immediate i f one notes that iA i s hermitian. T The eigenvalues of an orthogonal matrix A (AA = I, where I i s the identity matrix of order n) have absolute value 1 and occurc i n rec i p r o c a l pairs. This was proved by F. Brioschi [7] i n 1 8 5 4 , and again by F. Rahusen £17} i n 1 8 9 4 . The eigen-* it-values of a unitary matrix A (AA = A A = I) also have absolute value 1 . The proof was f i r s t given by H. Aramata [3] i n 1 9 2 7 , and a short proof was given by R. Brauer [ 5 ] i n 1 9 2 8 . The result i s obvious i f we observe that f o r a unit eigenvector X corresponding to A,, .', • 1 = X*X = X*A*AX = ( XX*){XX) = XA. 3. As f a r a s t h e e i g e n v a l u e s o f a g e n e r a l m a t r i x a r e c o n c e r n e d n o t h i n g s p e c i f i c c a n b e . s a i d a b o u t t h e i r l o c a t i o n ; t h e y c a n o b v i o u s l y l i e a n y w h e r e i n t h e c o m p l e x p l a n e . A g r e a t many t h e o r e m s h a v e b e e n p r o v e d a b o u t t h e l o c a l i z a t i o n o f e i g e n v a l u e s ; 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 a n d H. M i n e [15] and M. P a r o d i [ 1 6 ] . The t h r e e t y p e s o f m a t r i c e s m e n t i o n e d a b o v e h a v e s o m e t h i n g i n common. T h e i r e i g e n v a l u e s e a c h l i e o n a l i n e o r o n a c i r c l e ; o r , i n o t h e r w o r d s , t h e i r s p e c t r a a r e l i n e a r o r c i r c u l a r . T h i s s u g g e s t s t h e q u e s t i o n : What m a t r i c e s h a v e t h i s p r o p e r t y ? O r , more g e n e r a l l y , what m a t r i c e s h a v e t h e i r e i g e n v a l u e s o n p l a n e c u r v e s o f a s i m p l e s o r t ? I s i t p o s s i b l e t o r e c o g n i z e s u c h m a t r i c e s b y i n s p e c t i o n ? Th e s e seem t o be r a t h e r d i f f i c u l t \ q u e s t i o n s t o a n s w e r . I n t h i s t h e s i s , we make a s m a l l s t a r t , e x p l o r i n g some m a t r i c e s whose e i g e n v a l u e s 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 . ' I n s e c t i o n 2 we i n t r o d u c e a c l a s s o f m a t r i c e s c a l l e d "HORT" A m a t r i x i s s a i d t o be HORT i f i t c a n be o b t a i n e d f r o m a n h e r m i -t i a n m a t r i x b y a s u i t a b l e r o t a t i o n a n d / o r t r a n s l a t i o n . I t s e i g e n v a l u e s l i e o n a l i n e . I n a s i m i l a r way we d i s c u s s "UOT" m a t r i c e s , w h i c h c a n be o b t a i n e d f r o m u n i t a r y m a t r i c e s b y a s u i t a b l e t r a n s l a t i o n . We o b t a i n n e c e s s a r y a n d s u f f i c i e n t c o n d i -t i o n s f o r a m a t r i x t o be HORT and UOT. 4. With each n x n complex matrix A there i s associated i n a natural way a 2n x 2n r e a l matrix calle d the Expansion of A. In section 3 we discuss some simple relations between A and i t s expansion. In section 4 we examine n x n nonnegative indecomposable matrices. 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, and i t s index of imprimi-t i v i t y i s a l most 2 (Theorem 4.2). Its eigenvalues l i e on a c i r c l e i f and only i f i t s index of imprimitivity i s n (Theorem 4 . 3 ) . In section 5 we introduce two further classes of matrices called "almost" hermitian (Definition 5.2) and "almost" HORT (Definition 5 . 3 ) , and obtain necessary'and s u f f i c i e n t conditions for a matrix to be "almost" HORT. In section 6 we discuss the relationship between the eigen-values of the r compound matrix C r(A) of A and those of A. F i n a l l y , i n section 7, we looked at the extended polynomial -problem on eigenvalues ; that i s , the determination of the roots X of det( A r A 0 + 7i. r~ 1A l +...+ A r) = 0, where the A^ are n x n complex matrices. In the case where the degree r i s 1 or 2, we give some conditions f o r the roots to l i e on a l i n e or on a c i r c l e . I n t h i s t h e s i s we s h a l l u s e t h e f o l l o w i n g n o t a t i o n : C# t h e c o m p l e x number f i e l d . H h e r m i t I a n m a t r i x . S s k e w - h e r m i t i a n m a t r i x . I i U u n i t a r y m a t r i x . ; (p><l) a p o i n t i n t h e c o m p l e x p l a n e . E x p A e x p a n d e d m a t r i x o f A. Q, — - t o t a l i t y o f s t r i c t l y i n c r e a s i n g s e q u e n c e s o f k i n t e g e r s c h o s e n f r o m 1,2,...,n. t h C r ( A ) r compound m a t r i x o f A. A j x j y ] s u b m a t r i x o f A u s i n g r o w s numbered x and c o l u m n s numbered y. H e r e x and y a r e s e q u e n c e s o f i n t e g e r s . d e t ( A ) d e t e r m i n a n t o f A. / 2. M a t r i c e s w i t h e i g e n v a l u e s o n l i n e s o r c i r c l e s . D e f i n i t i o n 2.1 A m a t r i x A i s m - l i n e a r ( m - c i r c u l a r ) i f a l l i t s e i g e n v a l u e s l i e on m l i n e s ( c i r c l e s ) i n t h e c o m p l e x p l a n e . T h i s d e f i n i t i o n i m p l i e s t h a t i f A i s m - l i n e a r , i t i s a l s o p - l i n e a r f o r a l l p^m. One c o u l d a r g u e t h a t A s h o u l d be m - l i n e a r i f i t s e i g e n v a l u e s l i e o n m l i n e s b u t n o t more t h a n m l i n e s . I n t h a t c a s e , h o w e v e r , we s h o u l d s o m e t i m e s be f a c e d w i t h some p e s k y c o m b i n a t o r i a l c o n s i d e r a t i o n s , w h i c h m e r i t d i s c u s s i o n o n l y when we a r e i n t e r e s t e d i n t h e l e a s t m f o r w h i c h A i s m - l i n e a r . I t a p p e a r s r e a s o n a b l e t o u s t o d e f i n e m - l i n e a r a s i n D e f i n i t i o n 2.1. T h o s e m a t r i c e s w i t h j u s t one d i s t i n c t e i g e n v a l u e ( t h e s p e c -t r u m i s a p o i n t ) a r e r e t h e r s p e c i a l . We c o u l d c a l l them O - l i n e a r and O - c i r c u l a r , b u t we s h a l l be c o n t e n t t o i n c l u d e them among t h e 1 - l i n e a r and 1 - c i r c u l a r m a t r i c e s . T h e s e m a t r i c e s a r e n o t t h e o n l y o n e s t h a t a r e b o t h 1 - l i n e a r a nd 1 - c i r c u l a r . S u c h m a t r i c e s a r e t h o s e 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 two p o i n t s . E v e r y 2x2 - m a t r i x i s 1 - l i n e a r a nd 1 - c i r c u l a r . S u p p o s e t h a t t h e m a t r i x A i s 1 - l i n e a r . I f t h e l i n e L o f e i g e n v a l u e s o f A p a s s e s t h r o u g h a = r + s i w i t h a n g l e o f i n c l i n a -t i o n 0, t h e n p a r a m e t r i c e q u a t i o n s f o r 1 a r e : where t i s a r e a l p a r a m e t e r . L e t a = r + s i , q/p = t a n 0 i f pj^O, and 0 = n/2 i f p=0. Now t h e m a t r i x B = (A - al)e~^ h a s r e a l r o o t s . C o n v e r s e l y , i f t h e r e e x i s t a c o m p l e x number a and a r e a l number 0 s u c h t h a t B h a s r e a l r o o t s , t h e n A i s 1 - l i n e a r . Hence x = r + p f \ y = s + q t j Theorem 2.1 A i s 1-linear i f and only i f there exist a complex number a and a r e a l number 0, 0^0 <"r such that (A - al)e"" i^ has r e a l roots. One might be tempted to say that the problem of recognizing 1-linear matrices r e a l l y amounts to recognizing matrices with r e a l eigenvalues. But i t i s not thi s simple. The c r i t e r i o n of Theorem 2.1 may not have too much value i n recognizing 1-linear matrices from those with r e a l roots. However, matrices related to hermitian matrices can be recognized. D e f i n i t i o n 2.2 Let A = ( a ^ ) be an n-square matrix with elements i n C#. We say that A i s HORT "hermitian on rotation and/or translation" i f (A-aI)~ i^ i s hermitian f o r some complex number a and-real number 0. If A i s HORT and H=(A-aI)e~ i^ i s hermitian, the eigenvalues of A are A.(A) = ( H j e ^ + a, j=1,2,...,n and they l i e on J J the l i n e y = tan 0-(x-r) + s i f 0 ^  ir/2 and on the l i n e x = r i f 0 = ir/2. Note that a skew-hermitian matrix S i s HORT since (S - 0-I)e~ i 7 r > / 2 = - i S i s hermitian. Theorem 2.2 A i s HORT i f and only i f there are complex numbers v and w with i wI = 1 such that a j j = T " a j j + ( V - W ) ' and a k j = w a j k i o v j ,k=1 ,2,... ,n. 8. Proof. I f A i s HORT, there exist, by d e f i n i t i o n , a complex number a and a r e a l number 0 such that H = (A- a I ) e _ i ^ i s hermitian. That i s . (A - aDe""^ = (A* - a l ) e ^ or A* = e" 2 i^(A - al) '+ a l , thus a, . = e " 2 i ^ a i i + (a - ae""2i^) a_j = e ~ 2 l ^ a 3 k f o r J^k» 0,k=1,2,...,n. Put v = a and w = e"~2i^. The converse i s immediate by reversing the order of the ab ove i; argument. In practice the recognition of an HORT matrix i s even simpler than the c r i t e r i o n of Theorem 2.2. I f 0 ^ 0 , a(=v) can be assumed r e a l . I f 0 = 0 , a can be assumed to be pure imaginary, 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 either (1) or (2) holds. / • • (1) &^ = w a j j + v(l-w), aj.j = wa J k, j,k=1 ,2,... ,n for some r e a l v and complex w such that | w I = 1. (2) a^j = a^. - 2bi, = a ^ , j^k, j ,k=1 ,2,.., ,n f o r some r e a l number b. HORT matrices are t-l i n e a r , but of course they are by no means the only 1-linear matrices, just as hermitian matrices are not the only ones with r e a l eigenvalues. In'fact the example P a \ A = | shows that a matrix with r e a l elements and r e a l lo 2) eigenvalues can be just about as "unhermitian" as one can conceive. Suppose that the matrix A i s 1-circular. The equation of the c i r c l e of eigenvalues must be of the form (x - r ) 2 + (y - s ) 2 = b 2, where b i s a non-negative r e a l number. Analogous to the case of HORT matrices, we make the following d e f i n i t i o n . D e f i n i t i o n 2.3 Let A = ( a..^ ) be an n-square matrix with elements i n C#. A i s UOT "unitary on translation" i f (A - al)b~^ i s unitary f o r some complex number a and r e a l number b^O. / 1 I f A i s UOT and U = (A - al)b"' i s unitary, the eigenvalues of A are \j(A) = Xj(U)b + a, j=1,2,...,n and they l i e on the c i r c l e (x - r ) 2 + (y - s ) 2 = b 2 i f a = r+si. to Theorem 2.4 A i s UOT i f and only i f there exist a comple number v and r e a l number w^ O such that n _ 11 y a_.„a..„ - va^ . - v a ^ + | v|^ - = 0 m=1 n _ _ V ajm akm ~ v a k j " v a j k = °' 3» k=1»2,...,n. m=1 Proof. I f A i s UOT there are, by d e f i n i t i o n , a complex number a and a r e a l number bj£0 such that U = (A - al )b"~^ i s unitary. That i s UU* = (A - a l ) b ~ 1 ( A * - a D b " 1 = I or AA - aA - aA + |a| I = b I, n thus _ I a 0 m a j m - **U " a a j j + | a | 2 = b 2 m=1 n _ _ _ y ~ ~ Z j n 3 - ^ " a a i c j ~ a a j k = ° ' d»k=1 ,2,.. .',n. m=1 Put v = a and w = b. ; The converse i s immediate by reversing the order of the above argument. ^ 11 3. The e x p a n d e d m a t r i x o f a c o m p l e x m a t r i x I n l o o k i n g a t 1 - l i n e a r , o r more g e n e r a l l y m - l i n e a r , m a t r i c e s t h e r e a r e r e a l l y 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 : c o m p l e x m a t r i c e s a n d r e a l m a t r i c e s . We s h a l l n o t c o n f i n e o u r s e l v e s t o r e a l m a t r i c e s , b u t e x p l o r e b o t h p r o b l e m s . We n o t e , h o w e v e r , t h a t c o r r e s p o n d i n g t o e a c h c o m p l e x m a t r i x A t h e r e i s a r e l a t e d r e a l m a t r i x c a l l e d t h e "expanded m a t r i x o f A", denoted by E x p A, and defined for A = ( a j k } = ( V * c d k i ), b by °11 b 1 2 °12 b 1 n c 1 n / ~ C 1 1 b 1 1 " c 1 2 b 1 2 . ~ c t n b 1 n ;" b 2 i c 2 1 b 2 2 c 2 2 ... b 2 n c 2 n • E x p A - C 2 1 • b 2 1 • ~ c 2 2 b 2 2 . • '* ° ~ c 2 n • • b 2 n • u °n1 • • c n 2 • • nn • °nn b n 1 -°n2 b n 2 nn b^^ nn b j k ' c j k r e a 1 ' I n 1 960, E. G o t t [10] p r o v e d / t h a t d e t ( E x p A ) = | d e t ( A ) | 2 . A o n e - l i n e p r o o f o f t h i s 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 ] i n 1961 b a s e d o n a n i n t e r e s t i n g t h e o r e m o f S. N. A f r i a t [ 1 ] Re d e t ( A ) Im d e t ( A ) d e t ( E x p A ) = d e t -Im d e t ( A ) Re d e t ( A ) j = | d e t ( A ) | 2 . 12. B r e n n e r a l s o showed t h a t t h e c o l l e c t i o n o f t h e e i g e n v a l u e s o f E x p A c o n s i s t s o f t h e e i g e n v a l u e s o f A and t h e i r c o n j u g a t e s . T h i s i s p a r t i c u l a r l y i n t e r e s t i n g f o r m a t r i c e s A w i t h r e a l r o o t s . The r o o t s o f E x p A a r e j u s t t h o s e o f A c o u n t e d t w i c e , c e r t a i n l y r e a l and 1 - l i n e a r . We g i v e some s i m p l e r e s u l t s r e l a t i n g A and E x p A i n Theorem 5.1 L e t A = ( a..^ ) he a n n - s q u a r e m a t r i x w i t h c o m p l e x e l e m e n t s . E x p A i s n o r m a l i f and o n l y i f A i s n o r m a l . E x p A i s s y m m e t r i c i f A i s h e r m i t i a n , E x p A i s s k e w - s y m m e t r i c i f A i s s k e w - h e r m i t i a n and E x p A i s u n i t a r y i f A i s u n i t a r y . P r o o f . E i r s t we show t h a t E x p A i s n o r m a l i f and o n l y i f A i s , n o r m a l . S u p p o s e a ^ = + c ^ i , a n d l e t S i n c e a * n * m=1 m=1 / i f and o n l y i f . n n 7~ a,im akm = Y~ a,im akm j ,k=1,2,... ,n. m=1 m=1 ( E x p A ) ( E x p A ) * - ( ( f^^Xm ) j k ) m=1 = ( ( 2_ A j m A k m ) j k ) = ^ A ) ^ A> m=1 13. i f a n d o n l y i f * n _ _ n _ * M = ( ( TZ a j m a - n ) j k ) = ( ( 21 a j m a k m ) j k ) = A A ' m=1 m=1 Thus A i s n o r m a l i f and o n l y i f E x p A i s n o r m a l . I f A i s h e r m i t i a n , i t s e i g e n v a l u e s a r e r e a l , a n d , s i n c e t h e e i g e n v a l u e s o f E x p A a r e t h o s e o f A and t h e i r c o n j u g a t e s , E x p A h a s r e a l r o o t s . However A b e i n g h e r m i t i a n a l s o i m p l i e s E x p A n o r m a l . Now a n o r m a l m a t r i x w i t h r e a l e i g e n v a l u e s must be h e r m i t i a n . Hence E x p A i s h e r m i t i a n , a n d i n d e e d , s i n c e i t s e l e m e n t s a r e r e a l , E x p A i s s y m m e t r i c . S i m i l a r l y , one c a n p r o v e t h e l a s t two s t a t e m e n t s , i f one n o t e s t h a t t h e c o n j u g a t e o f a p u r e i m a g i n a r y i s a g a i n p u r e i m a g i n a r y and | a + b i | = 1 = | a - b i | . Theorem 5.2 L e t A = ( a ^ j . ) be a n n - s q u a r e m a t r i x w i t h c o m p l e x e l e m e n t s . Assume t h a t A i s HORT; i . e . , (A - a l ) e " " ^ i s h e r m i t i a n f o r some c o m p l e x number a = r + s i a n d r e a l number 0 w i t h 0 = 0<T . E x p A i s 1 - l i n e a r i f ( 1 ) 0=TT/2, o r ( 2 ) s=0 and 0=0, o r ( 3 ) s p e c t r u m o f A i s a p o i n t . E x p A i s 2 - l i n e a r o t h e r w i s e . / P r o o f . I f 0=n/2jthe e i g e n v a l u e s o f A l i e o n x = r and so do t h o s e o f E x p A. I f s=0 and 0=0, t h e e i g e n v a l u e s o f A and E x p A a r e a l l r e a l a n d l i e o n y=0. I f t h e s p e c t r u m o f A i s a p o i n t , E x p A i s o b v i o u s l y 1 - l i n e a r . O t h e r w i s e , t h e e i g e n v a l u e s o f A l i e o n t h e l i n e w i t h e q u a t i o n y = tan0*(x - r ) + s. 14 T h ose o f E x p A l i e on y = tan0«(x - r ) + s and y = -tan0*(x - r ) - s. T h i s p r o v e s o u r t h e o r e m . Theorem 5.5 L e t A = ( ) be a n n - s q u a r e m a t r i x w i t h c o m p l e x e n t r i e s . Assume t h a t A i s UOT; i . e . , t h e r e a r e c o m p l e x number a = r + s i and r e a l number b^O s u c h t h a t (A - a l ) b i s u n i t a r y . E x p A i s 1 - c i r c u l a r i f s = 0 , o r i f t h e s p e c t r u m o f A i s a p o i n t . E x p A i s 2 - c i r c u l a r o t h e r w i s e . P r o o f . By t h e r e m a r k s shown i n S e c t i o n 2, t h e e i g e n v a l u e s o f A l i e o n t h e c i r c l e ( x - r ) 2 + ( y - s ) 2 = b 2 . Now, i f s = 0 , t h e a b o v e e q u a t i o n may be w r i t t e n ( x - r ) 2 + y 2 = b 2 w h i c h i s s y m m e t r i c w i t h r e s p e c t t o t h e r e a l a x i s . T h e n t h e e i g e n v a l u e s o f E x p A a l l l i e o n t h i s c i r c l e f o r t h e y a r e c o n j u g a t e i n p a i r s . I f s^O, t h e e i g e n v a l u e s o f E x p A w i l l l i e on two c i r c l e s : / U - r ) 2 + ( y - s ) 2 = b 2 and ( x - r ) 2 + ( y + s ) 2 = b 2 . t5. More g e n e r a l l y , one c a n s e e t h a t i f A i s 1 - l i n e a r t h e n E x p A i s e i t h e r 1 - l i n e a r o r 2 - l i n e a r . C o n v e r s e l y , i f E x p A i s 1 - l i n e a r , t h e n A i s 1 - l i n e a r ; i f E x p A i s 2 - l i n e a r , t h e n A c a n he e i t h e r 1 - l i n e a r o r 2 - l i n e a r . I t i s o b v i o u s , h o w e v e r , t h a t i f E x p A i s ©(-linear, c( =1 , 2 , 3 , • •. , t h e n E x p ( E x p A) = E x p ( A ) i s a l s o °< - l i n e a r . No new l i n e s o f e i g e n v a l u e s a r e a d d e d . T h i s i s n o t s u r p r i s i n g s i n c e E x p (A) i s A " b l o w n u p " w i t h e a c h r o o t d u p l i c a t e d . A s i m i l a r d i s c u s s i o n a p p l i e s t o < * - c i r c u l a r m a t r i c e s . A more i n t e r e s t i n g q u e s t i o n i s t o a s k f o r t h e number o f l i n e s on w h i c h t h e e i g e n v a l u e s o f B m l i e , where (1) BQ = B i s 1 - l i n e a r , ( 2 ) B k = E x p ( a k B k _ 1 + b k I ) ; a k , b k c o m p l e x , k=1,2,...,m. I n g e n e r a l , e a c h t i m e we make a n E x p a n s i o n , we d o u b l e t h e number o f l i n e s o f e i g e n v a l u e s , h e n c e t h e maximum l i n e a r i t y i s 2 m . B u t o f c o u r s e B c a n be 2 ^ - l i n e a r f o r e a c h q, m 0 =^ q ^ m, f o r s u i t a b l e c h o i c e s o f B, a^. and b k . C o n c e i v a b l y , B c o u l d a l s o be <*Z-linear, where << i s n o t a power o f 2. J u s t when Bffi i s c< - l i n e a r f o r a n y i n t e g e r °C = 2 m seems t o be a c o m p l e x 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 p o s s i b i l i t i e s we g i v e t h e l i n e a r i t y o f B^ and B 2 when B Q i s h e r m i t i a n , skew h e r m i t i a n a n d u n i t a r y . ( T a b l e s 1 a n d 2 ) . 16. L e t a.j=p,|+q.j i ^ O , b ^ = r ^ + s ^ i ^ O , a 2 = P 2 + < l 2 i ^ a n d b 2 = r 2 + s 2 i ^ 0 . T a b l e 1. B Q = H ( S ) C o n d i t i o n s o n P 1 , q 1 , r 1 , s 1 E x p B 1 C o n d i t i o n s o n P 2 » ^ 2 ' r 2 * s 2 E x p B 2 j q 1 = 0 & s ^ O ( p 1 = 0 & 8^=0) 1 - l i n e a r ( y = 0 ) q 2 = 0 & s 2 = 0 p 2 = 0 & s 2 ^ 0 o t h e r w i s e 1 - l i n e a r ( y = 0 ) 1 - l i n e a r ( p e r p . t o y= 0 ) 2 - l i n e a r P 1 = 0 & r ^ O ( q 1 = 0 & r 1 = 0 ) 1 - l i n e a r ( x = 0 ) P 2 = 0 & s 2 = 0 q 2 = 0 & s 2 ? ^ 0 o t h e r w i s e 1 - l i n e a r ( y = 0 ) t - l i n e a r ( p e r p . t o y= 0 ) 2 - l i n e a r P 1 = 0 & s ^ O ( q 1 = 0 & s 1 = 0 ) 1 - l i n e a r ( p e r p . t o y= 0 ) q 2 = 0 ' P 2 = O 1- l i n e a r ( p e r p . t o y= 0 ) 2 - l i n e a r ( p a r a l . t o y= 0 ) q 1 = 0 & r 1 = 0 ( p 1 = 0 & T^Q) 2 - l i n e a r ( p a r a l . t o y= 0 ) q 2 = 0 & s 2 = 0 q 2 = 0 & s 2 ? ^ 0 P 2 = 0 2 - l i n e a r ( p a r a l . t o y= 0 ) 4 - l i n e a r ( p a r a l . t o y= 0 ) 2 - l i n e a r ( p e r p . t o y= 0 ) o t h e r w i s e 2 - l i n e a r q 2 = 0 & s 2 = 0 q 2 = 0 & s 2 ^ 0 P 2 = 0 & s 2 = 0 P 2 = 0 & S 2 J ^ O 2 - l i n e a r 4 - l i n e a r 2 - l i n e a r 4 - l i n e a r T a b l e 2 . B = I 0 C o n d i t i o n s o n P 1 , q 1 , r 1 , s 1 J E x p B 1 C o n d i t i o n s o n P 2 » ( 3 . 2 ' r 2 ' ^ 2 E x p B 2 s ^ O 1 - c i r c u l a r s 2 = 0 o t h e r w i s e 1- c i r c u l a r 2 - c i r c u l a r o t h e r w i s e 2 - c i r c u l a r s 2 = 0 o t h e r w i s e 2 - c i r c u l a r 4 - c i r c u l a r m a t r i x P s u c h t h a t P A P T = ( I w h e r e B a n d C a r e 17. 4. N o n n e g a t i v e m a t r i c e s . D e f i n i t i o n 4.1 A r e a l n - s q u a r e m a t r i x A = ( a ^ k ) i s c a l l e d n o n n e g a t i v e , i f a ^ k = 0 f o r j ,k = 1 , 2 , . . . , n . We w r i t e A = 0. D e f i n i t i o n 4.2 A n o n n e g a t i v e n - s q u a r e m a t r i x A = ( a ^ ) ( n > 1 ) i s s a i d t o b e d e c o m p o s a b l e i f t h e r e e x i s t s p e r m u t a t i o n B 0 C D s q u a r e m a t r i c e s . O t h e r w i s e A i s i n d e c o m p o s a b l e . T h e f u n d a m e n t a l t h e o r e m o n i n d e c o m p o s a b l e n o n n e g a t i v e m a t r i c e s i s t h e P e r r o n - P r o b e n i u s t h e o r e m [ J 5 ] , w h i c h we s t a t e a s f o l l o w s : T h e o r e m 4•1 L e t A b e a n n - s q u a r e n o n n e g a t i v e i n d e c o m p o s a b l e m a t r i x . T h e n : ( 1 ) A h a s a r e a l p o s i t i v e e i g e n v a l u e r ( t h e m a x i m a l e i g e n v a l u e o f A ) w h i c h i s a s i m p l e r o o t o f t h e c h a r a c t e r i s t i c e q u a t i o n o f A. I f 7V.(A). i s a n y e i g e n v a l u e o f A, t h e n |/V:(A ) I = r . J J ( 2 ) I f A h a s h e i g e n v a l u e s o f m o d u l u s r : / V j = r , T ^ * • • • » ^ - ^ t t h e n t h e y a r e t h e h d i s t i n c t r o o t s o f ?\. - r = 0; h i s c a l l e d t h e i n d e x o f i m p r i m i t i v i t y o f A. ( 3 ) I f X-j , '/^2» • * *' a r e a ^ "kke e i g e n v a l u e s o f A a n d 6 - e l 2 l T / n , t h e n 7^9, "X^O,..., 7^6 a r e A 1 , T ^ , . . . , X n 18. i n some o r d e r . (4) , I f h > 1 , t h e n t h e r e e x i s t s a p e r m u t a t i o n m a t r i x P s u c h t h a t w h e r e t h e z e r o b l o c k s d o w n t h e m a i n d i a g o n a l a r e s q u a r e . S u p p o s e t h a t t h e n o n n e g a t i v e i n d e c o m p o s a b l e m a t r i x A i s 1 - l i n e a r . L e t r b e t h e m a x i m a l ( p o s i t i v e r e a l ) e i g e n v a l u e o f A a n d s u p p o s e t h a t t h e r e w e r e a c o m p l e x e i g e n v a l u e , s a y 9 ^ r . S i n c e t h e c o e f f i c i e n t s o f d e t ( A . I - A) = 0 a r e r e a l , •JL i s a l s o a r o o t o f d e t ( A I - A) = 0 a n d | x l = r . T h e n — 2 2 2 7\_and 7\. H e o n o r i n t h e c i r c l e x + y = r a n d a r e s y m m e t r i c a l l y p l a c e d w i t h r e s p e c t t o t h e x - a x i s . T h u s r , A a n d 9^  c a n o n l y b e o n a l i n e i f A.= A . H e n c e we h a v e : T h e o r e m 4.2 I f a n n - s q u a r e n o n n e g a t i v e i n d e c o m p o s a b l e m a t r i x A i s 1 - l i n e a r , t h e n a l l i t s e i g e n v a l u e s a r e r e a l . F u r t h e r m o r e , PAP ,T I t i s t r i v i a l t h a t i f a n d 1 - l i n e a r , t h e n E x p A r o o t s a r e a l l r e a l . A i s n o n n e g a t i v e i n d e c o m p o s a b l e i s a l s o 1 - l i n e a r ; i n f a c t , i t s 19. Part(3) of Theorem 4.1 indicates that A i s a l most n/h-circular. If A i s 1-circular, then a l l the eigenvalues of A s a t i s f y |A.(A)| = r, j = 1,2,...,n, where r i s the positive r e a l maximal eigenvalue of A. Thus a l l A (A) are roots of x n - rn = 0, and h = n. Conversely, i f h = n, the roots of jJ1 - r n = 0 coincide with the eigenvalues of A, and A has 1-circular eigenvalues. Hence Theorem 4.3 An n-square nonnegative indecomposable matrix A i s 1-circular i f and only i f n = h. An example of the matrices appearing i n Theorem 4.3 i s the permutation matrix / 0 1 0 ... 0 p = 0 1 0 0 ... 0 1 A 1 0 ... 0 0 The eigenvalue's of P are the n /roots of unity : A.(P) = e i ( 2 * / n ) J , j = 1,2,...,n, for P i s the companion matrix of the polynomial A n - 1 . P i s a special case of the more general matrix a b 1 0 ... 0 0 a b 2 ... 0 B = 0 ... 0 a b . n—l b^ ... 0 0 n 20. I Here, d e t ( A l - B) = ( A - a) + (-1 ) || b, . The eigen-k=t K values of B are : A.CB) = r e i ( 2 7 r / n ^ + a , j = 1,2,...,n, n even J or A,(B) = r e i ( l r / n + 2 7 rJ/ n> + a , j=T,2,...,n, n odd, J n / n where r = / I I b^ w k =1 The eigenvalues of Exp B are just those of B counted twice; hence Exp B i s again 1-circular. B i s inter e s t i n g i n that i t i s 1-circular even i f the non-zero entries are non-real. Its eigenvalues are XJ(B) = y-prr e i ( , 1 - 1 , 2 . . . . , » . n j r i -where c = || b, = |c|e * . l e t a = p+qi, p,q are r e a l , then k=t * Exp B ' i s 1-circular i f q = 0 and Exp B i s 2-circular other-wise . - • • 21 5. T r i d i a g o n a l m a t r i c e s . D e f i n i t i o n 5.1 A m a t r i x A = ( a ^ ^ ) i s c a l l e d a t r i d i a g o n a l o r J a c o b i m a t r i x i f a..^ = 0 w h e n e v e r j j - k J = 2, T h u s (5.1) L n = b 1 c 1 0 ,0 d 2 b 2 ° 2 0 0 d 3 b 3 c 3 0 0 0 d „ b j c n-1 n-1 n-1 0 d b n n i s a g e n e r a l n - s q u a r e c o m p l e x J a c o b i m a t r i x . S i n c e a n y m a t r i x A i s s i m i l a r t o a J a c o b i m a t r i x [ 1 4 ] , we a r e i n t e r e s t e d i n c o n d i t i o n s u n d e r w h i c h L i s m - l i n e a r ; p a r t i c u l a r l y w h e n i s 1 - l i n e a r . ' \ T h e f o l l o w i n g t h e o r e m w a s p r o v e d b y F . M. A r s c o t t i n 1961 [ 4 ] : T h e o r e m 5.1 ' I f a l l t h e e n t r i e s ' o f t h e J a c o b i m a t r i x L a r e n r e a l a n d c..d.. + 1 > 0, j = 1 , 2 , . . . , n - 1 , t h e n a l l i t s e i g e n v a l u e s a r e r e a l a n d s i m p l e . We s h a l l g e n e r a l i z e T h e o r e m 5.1 a s f o l l o w s : 2 2 . T h e o r e m 5.2 L e t . ML b e t h e m a t r i x L ^ w i t h r e a l b, , 1 = k = n , ————————• j2 II n. a n d c o m p l e x c ^ , d ^ . + 1 s u c h t h a t c J ^ J + - J ^ ^ » 1 = 3 = n-1 .. T h e n t h e e i g e n v a l u e s o f a r e r e a l a n d , s i m p l e . P r o o f . T h e c o e f f i c i e n t o f A k i n d e t ( X l - M ) i s t h e sum o f t h e ( n - k ) - s q u a r e p r i n c i p a l m i n o r s . V / h e r e v e r c . o c c u r s a s a f a c t o r i n s u c h a m i n o r s o d o e s » a n ( * v i c e v e r s a . T h u s t h e s e e l e m e n t s a l w a y s o c c u r t o g e t h e r a s a p r o d u c t c . d . .. I t J J f o l l o w s t h a t t h e c h a r a c t e r i s t i c e q u a t i o n d e t C A l - M ) = 0 h a s t h e same c o e f f i c i e n t s a n d r o o t s a s w h e n a l l c ^ a n d a r e r e a l a n d c d . 0. B y T h e o r e m 5.1 t h e e i g e n v a l u e s o f 3 3+1 a r e r e a l a n d s i m p l e . B y T h e o r e m 5.1 a n d T h e o r e m 5 . 2 , g i v e n a J a c o b i m a t r i x L ^ , i f b^ , b 2 » * . . , b n a r e r e a l , c^ , > • >, c n_-| a n ^ &2> d ^ , . . . , d n a r e c o m p l e x n u m b e r s s u c h t h a t C j d ^ + ^ > 0, 3 = 1 , 2 , . . . , n - 1 , t h e n i t s e i g e n v a l u e s a r e a l l r e a l . A n a l o g o u s t o t h e c a s e o f 1 - l i n e a r m a t r i c e s d i s c u s s e d i n S e c t i o n 2, we make t h e f o l l o w i n g d e f i n i t i o n s : D e f i n i t i o n 5.2 L e t L b e a J a c o b i m a t r i x w i t h c o m p l e x e n t r i e s . * — n • YJe c a l l L " a l m o s t " h e r m i t i a n i f b> , b 0 , . . . , b , a r e r e a l , n 1 2 n -;n_.| a n d d 2 , d ^ , . . . , d n a r e c o m p l e x n u m b e r s s u c h t h a t c j ( i j + 1 > 0, 3 = 1 , 2 , — ,n-1 . 2 3 . N o t e t h a t i f L i s " a l m o s t " h e r m i t i a n a n d d = c . , n j + i J 3 = 1 , 2 , . . . , n - 1 , t h e n L n i s h e r m i t i a n . D e f i n i t i o n 5.3 l e t L n b e a J a c o b i m a t r i x . We c a l l L Q ! " a l m o s t " HORT i f ( L - a D e " 1 ^ i s " a l m o s t " 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 a a n d r e a l n u m b e r 0 s u c h t h a t 0 = 0 < TT T h e o r e m 5.3 L n i s " a l m o s t " HORT i f a n d o n l y i f t h e r e a r e c o m p l e x n u m b e r s v a n d w w i t h |w| = 1 s u c h t h a t ( b . - v ) w J 2 i s r e a l , j = 1 , 2 , . . . , n a n d c j d j + - | w ^ °» 3 = 1 , 2 , . . . , n - 1 . P r o o f . I f L n i s " a l m o s t " HORT, b y d e f i n i t i o n , t h e r e e x i s t a c o m p l e x n u m b e r a a n d a r e a l n u m b e r 0 w i t h 0 = 0 < IT s u c h t h a t B = ( L n - a.I)e~^ i s " a l m o s t " h e r m i t i a n ; i . e . , ( b j - a ) e - i ^ i s r e a l a n d c ^ . d ^ + 1 e ~ 2 i ^ > 0, j=1 , 2 , . . . ,n-1 . * ch P u t v = a a n d w = e ~ . T h e c o n v e r s e i s i m m e d i a t e b y r e v e r s i n g t h e o r d e r o f t h e a b o v e a r g u m e n t . / T h e o r e m 5.4 L e t L ^ b e a J a c o b i m a t r i x t h a t i s " a l m o s t " HORT; i . e . , ( L n - a l ) e _ i ^ i s " a l m o s t " 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 a = r + s i a n d r e a l n u m b e r 0 w i t h 0 = 0 < TT . T h e n E x p A i s 1 - l i n e a r i f ( 1 ) 0 = T / 2 o r ( 2 ) s = 0 a n d 0= 0, a n d E x p A i s 2 - l i n e a r o t h e r w i s e . P r o o f . We o m i t t h e p r o o f h e r e s i n c e i t i s s i m i l a r t o t h a t o f T h e o r e m 3 . 2 . / 25: 6. C o m p o u n d m a t r i c e s a n d some o t h e r t h e o r e m s . D e f i n i t i o n 6.1 L e t A = ( a ^ ) b e a n n - s q u a r e m a t r i x a n d 1 =• r = n . T h e r ^ * 1 c o m p o u n d m a t r i x o f A , C r ( A ) , i s t h e ( r ) X (ri m a t r i x w h o s e e n t r i e s a r e d e t ( A [ x j y ] ) , x €" Q r n a n d y 6 Q a r r a n g e d l e x i c o g r a p h i c a l l y i n x a n d y . F o r e x a m p l e , i f n = 3 a n d r = 2 t h e n / d e t ( A [ l ,2|1 ,2] ) d e t ( A [ l ,2|1,3-] ) d e t ( A [ l ,2 i 2 ,3] ) d e t ( A [ l ,311,2]) d e t ( A [ l ,3|1 ,3J) d e t ( A [ l ,312,3] ) ^ d e t ( A [ 2 , 3 M ,2] ) d e t ( A [ 2 ^ 3 i 1,3] ) d e t ( A [2,3| 2,3] ) C 2 ( A ) = I n p a r t i c u l a r , C ^ ( A ) = A a n d C n ( A ) = d e t ( A ) . T h e o r e m 6 .1 . L e t A = ( p + q i ) U , p + q i j4 0. T h e e i g e n v a l u e s o f C y ( A ) , 1 = r = n , l i e o n a c i r c l e w i t h c e n t e r a t t h e o r i g i n a n d r a d i u s j p + q i j . When r = n , t h e e i g e n v a l u e s c o a l e s c e a t t h e p o i n t ( p + q i ) n . I f J p + q i | = 1, a l l t h e e i g e n v a l u e s o f C ( A ) , r = 1,2,...,n, a r e o n t h e u n i t c i r c l e ( F i g . 6 - 1 ) . | p + q i |=1 F i g . 6-1 |p+qi |<1 2 6 . P r o o f . S i n c e t h e e i g e n v a l u e s o f C r ( A ) a r e t h e p r o d u c t s A s ( A ) A . ( A ) . . .A- ( A ) , w h e r e 1 = j . < j 0 < ... < j = n , a n d s i n c e - t h e A . ( A ) a r e o n t h e c i r c l e w i t h c e n t e r a t t h e o r i g i n J a n d r a d i u s j p + q i j , t h e e i g e n v a l u e s o f C r ( A ) l i e o n t h e c i r c l e w i t h c e n t e r a t t h e o r i g i n a n d r a d i u s | p + q i | r . T h e o r e m 6.2 T h e e i g e n v a l u e s o f C r ( S ) , 1 = r = n , l i e o n t h e r e a l a x i s i f r , . i s e v e n a n d o n t h e i m a g i n a r y a x i s i f r i s o d d , w h i l e t h o s e o f 0 ^ ( 1 1 ) , l i e o n . t h e r e a l a x i s . P r o o f . S i n c e t h e e i g e n v a l u e s o f S a r e p u r e i m a g i n a r i e s , t h e e i g e n v a l u e s o f C r ( S ) a r e r e a l i f r i s e v e n a n d p u r e i m a g i n a r y i f r i s o d d . T h e l a s t s t a t e m e n t f o l l o w s i m m e d i a t e f r o m t h e f a c t t h a t t h e s e t o f r e a l n u m b e r s i s a f i e l d . One n o t e s t h a t t h e s e t { c r ( A ) : r = 1 , 2 , . . . , n j i s 2 - l i n e a r i f A = S a n d 1 - l i n e a r i f A = H. T h e f o l l o w i n g t w o t h e o r e m s t e l l u s t h a t we c a n o b t a i n 1 - c i r c u l a r m a t r i x f r o m 1 - l i n e a r m a t r i x b y c e r t a i n t r a n s f o r m a t i o n , a n d v i c e v e r s a . T h e o r e m 6.5 . A = ( I - S ) ( I + S ) ~ 1 i s a u n i t a r y m a t r i x ; i . e . , we c a n o b t a i n a m a t r i x w h o s e e i g e n -v a l u e s l i e o n a u n i t c i r c l e f r o m a s k e w - h e r m i t i a n m a t r i x b y t h i s t r a n s f o r m a t i o n ( P i g . 6 - 2 ) . 2 7 . a F i g . 6-2 P r o o f . " F i r s t n o t e t h a t I + S i s n o n s i n g u l a r s i n c e t h e r o o t s o f S a r e p u r e i m a g i n a r i e s . A l s o , ( I + S ) ( I - S ) = ( I - S ) ( I + S ) . = ( I - S ) ( ( I - S r 1 ( I + S ) r 1 ( I + S ) = I . T h e o r e m 6.4 I f d e t ( I + U ) £ 0, t h e n A = ( I - U ) ( I + U ) ~ 1 i s s k e w - h e r m i t i a n ; i . e . , we c a n o b t a i n a m a t r i x w h o s e e i g e n -v a l u e s l i e o n - t h e i m a g i n a r y a x i s / f r o m a u n i t a r y m a t r i x b y t h i s t r a n s f o r m a t i o n ( F i g . 6 - 3 ) . A A * = ( I - S ) ( I + S ) " " 1 ((I+sr1 ) * ( I - S ) = ( I - S ) ( I + S ) ~ 1 ( I + S * ) " 1 ( I - S * ) = ( I - S ) ( I + S ) ~ ' ( I - S ) " , ( I + S ) ( s i n c e S = - S ) -> 0 F i g . 6-3 We n e e d o n l y t o s h o w t h a t ( u + u r 1 ) * ( i - u ) * ( I + U ) " 1 ( I - U ) ( i + u ~ 1 r 1 ( i - u " " 1 ) ( u - 1 ( u + i ) r l ( u ' " 1 ( u - D ) ( U + I ) " 1 U U ~ 1 ( U - I ) ( u + i ) - 1 ( u - i ) ( U + I ) ~ 1 ( U - I ) ( U + I ) ( U + I ) " 1 ( U + I ) ~ 1 ( U + I ) ( U - I ) ( U + I ) " 1 - ( I - U ) ( I + U ) ~ 1 -A . / 2 9 . 7. E x t e n d e d p o l y n o m i a l - p r o b l e m o n e i g e n v a l u e s . L e t A Q , A ; j , . . . , A r b e r+1 n - s q u a r e c o m p l e x m a t r i c e s . L e t X = ( x ^ , X2,..., x n ) ^ " b e a n o n z e r o v e c t o r w i t h c o m p l e x e n t r i e s . C o n s i d e r ( 7 . 1 ) ( A r A 0 + X r " 1 A 1 +...+ A r ) X = 0. T h e d e t e r m i n a t i o n o f \ s u c h t h a t e q u a t i o n ( 7 . 1 ) h o l d s i s c a l l e d t h e e x t e n d e d p o l y n o m i a l - p r o b l e m o n e i g e n v a l u e s . X i s s a i d t o b e a n e i g e n v e c t o r b e l o n g i n g t o t h e e i g e n v a l u e A . One c a n s e e t h a t t h e e i g e n v a l u e s o f t h e p o l y n o m i a l - p r o b l e m s a r e e x a c t l y t h e r o o t s o f ( 7 . 2 ) d e t ( A r A Q + X r " " 1 A | + ...+ A r ) = 0. T h e d e t e r m i n a n t i s a p o l y n o m i a l i n A o f d e g r e e n o t e x c e e d i n g r » n . I f A Q i s n o n s i n g u l a r , t h e d e g r e e o f ( 7 . 2 ) i s e x a c t l y r » n . I f r = 1 a n d A = I , t h e n t h e r o o t s o f d e t ( A I + A ^ ) = 0 a r e t h e e i g e n v a l u e s o f -A^. I f r = 1 • a n d A Q a n d A^ are,, a r b i t r a r y m a t r i c e s t h e p r o b l e m i s r e f e r r e d t o a s a g e n e r a l i z e d e i g e n v a l u e p r o b l e m [ t 3 ] . 30. T h e o r e m 7.1 T h e r o o t s o f (7.2) a r e t h e g e n e r a l i z e d e i g e n -v a l u e s o f A . B Q + B.j s u c h t h a t ( A B Q + B ^ X = 0, (7.3) w h e r e X i s a n o n z e r o v e c t o r , B I 0 0 . . . 0 0 0 - I 0 . . . 0 0 0 0 - I . . . 0 0 0 0 0 0 0 o. 0 . . . 0 - A a n d B„ = 0 I 0 . 0 0 0 0 I . 0 0 0 0 0 . 0 0 0 0 0 . 0 • I - A r ~ A r - 1 ~ A r - 2 * * . - A 2 -A I f A Q = I , t h e n t h e r o o t s o f (7.2) a r e t h e e i g e n v a l u e s o f B 1 . I f A Q i s n o n s i n g u l a r , t h e n t h e . r o o t s o f (7.2) a r e t h e e i g e n -v a l u e s o f B 2 , w h e r e B, / 0 I 0 0 0 0 0 I 0 0 0 0 0 0 0 0 ; 0 0 0 I < -A . A " 1 r - 1 o " A r - 2 A o 1 ' * - A A " 1 - v ,-1 31 • P r o o f . X B + B, o 1 / - X I I 0 0 / °- - X I I 0 0 0 - X I . . 0 \ ° 0 0 . - X i " A r - 1 ~ A r - 2 •' ' " A r 0 0 0 x - A I - A A O / To the f i r s t column add 7v_ t imes the second column, . A t imes A . B Q + = 9 * -s r-1i t imes the l a s t column, we o b t a i n 0 I 0 0 0 0 - X I I 0 0 0 0 - X I . . . 0 0 0 0 0 - X I I -P (X) -A. r-1 - A r-2 ~ A 2 " * A t ~ ^ A o where P(X) = + A T ~ 1 A 1 + . . . + A . A r _ i S ince V det( A B Q + ) = ( - 1 ) r ~ 1 d e t ( - P ( X ) ) d e t I 0 0 - A I 1 0 0 - X I / I 0 0 0 0 0 0 0 0 0 I - X I = ( - D r - 1 d e t ( - P ( X ) ) = ( - l ) n ' " r - 1 d e t ( A r A Q + >\ r" 1A 1 + . . .+ A R ) R the r o o t s of ( 7 . 2 ) are the g e n e r a l i z e d e igenvalues of X B Q + . Now i f A Q = I , - B Q i s an i d e n t i t y m a t r i x of order r«n , 3 2 . a n d t h e r o o t s o f ( 7 . 2 ) a r e t h e e i g e n v a l u e s o f . I f A Q i s n o n s i n g u l a r , t h e n A~^ e x i s t s , a n d ( 7 . 2 ) may b e w r i t t e n o r d e t ( ( A . r I + X r " 1 A 1 A ~ 1 +...+ A r A ^ 1 ) A 0 ) = 0, d e t I ( X r I + X r ~ 1 A 1 A 0 1 + ...+ A r A ~ ' ) d e t ( ' A 0 ) = 0. S i n c e d e t ( A Q ) ^ 0, t h e r o o t s o f ( 7 . 2 ) a r e t h e r o o t s o f ( 7 . 4 ) d e t ( A r I + A r " 1 A t A ~ 1 +...+ A p A ~ 1 ) = 0 . F r o m w h a t we h a v e p r o v e d a b o v e , t h e r o o t s o f ( 7 . 4 ) a r e t h e e i g e n v a l u e s o f B,,, w h e r e -1 ,B 2 = 0 0 0 I 0 0 0 I 0 0 0 -A A " 1 -A .A" 1 - A ^ 0k \ r o r - 1 o r - 2 o 0 0 0 0 0 0 0 -A A " 1 A 2 A o Q.E.D. l e t PCX)' = X r A Q + A r " 1 A 1 / + ...+ A A r _ 1 + A^. D e f i n i t i o n 7.1 T h e p o l y n o m i a l P ( A ) i s s a i d t o b e m - l i n e a r ( m - c i r c u l a r ) i f t h e r o o t s o f d e t P ( X ) = 0 l i e o n m, b u t n o t f e w e r t h a n m l i n e s ( c i r c l e s ) . We s h a l l g i v e a f e w r e s u l t s o n 1 - l i n e a r a n d 1 - c i r c u l a r : p o l y n o m i a l s w h e n r = 1 a n d 2. 3 3 . F o r t h e c a s e r = 1 , we a r e p r i m a r i l y i n t e r e s t e d i n t h e c a s e w h e n t h e r o o t s o f d e t P ( A ) = 0 a r e r e a l . One m i g h t g u e s s t h a t i f A a n d A. a r e b o t h h e r m i t i a n , t h e n A A ^ + A> h a s o 1 0 1 r e a l e i g e n v a l u e s . I t i s n o t q u i t e t h i s s i m p l e . Some r e s t r i c t i o n s a r e n e e d e d o n A A a n d A.. 0 . 1 T h e o r e m 1.2 L e t H Q a n d b e h e r m i t i a n m a t r i c e s o f o r d e r n a n d H Q b e p o s i t i v e o r n e g a t i v e d e f i n i t e , t h e n t h e r o o t s o f d e t ( A H - H.j) = 0 a r e r e a l [ 2 ] . P r o o f . L e t X b e a n o n z e r o e i g e n v e c t o r c o r r e s p o n d i n g t o t h e g e n e r a l i z e d e i g e n v a l u e o f A H Q - ; i . e . , ( A H Q - H ^ ) X = 0, # j o r A H Q X = H^X. P r e m u l t i p l y i n g b y X , we h a v e ; A ( X*H X ) = X A H X = X*H.X. S i n c e X*H.X i s r e a l , a n d o o 1 1 X H Q X i s r e a l a n d n o t z e r o , A i s r e a l . T h e r e f o r e t h e r o o t s o f d e t ( A H Q - H 1 ) = 0 a r e r e a l . T h e o r e m 7 . 3 I f H a n d H, a r e h e r m i t i a n m a t r i c e s s u c h t h a t O I / H H 1 = a n d d e t ( H Q ) ^ 0, t h e n t h e r o o t s o f d e t ( A H Q - E^)=0 a r e r e a l . P r o o f . We s h a l l f i r s t p r o v e t h a t H QH^ i s h e r m i t i a n i f a n d o n l y i f H a n d H, c o m m u t e . On t h e ; o n e . h a n d , i f H H. = H H, , o 1 o 1 o 1 t h e n H*H* = H ^ H Q = H QH^ . On t h e o t h e r h a n d , i f H Q H 1 = H j H , I I j 3 4 . t h e n ( H ^ ^ H X ^ ^ H ^ H ^ . ! S e c o n d l y , we n o t e t h a t H,H~ 1 i s h e r m i t i a n , f o r E~\ i s t o o | —1 —T h e r m i t i a n a n d H H, = r L H i m p l i e s I L H = H H< . O l 1 0 ^ 1 0 0 1 i Nox<r, t h e e q u a t i o n d e t ( A H Q - ) = 0 may b e w r i t t e n d e t ( X I - R ^ l T 1 ) d e t ( H Q ) = 0. T h e r o o t s o f d e t ( A l - H " 1 ) = 0 a r e r e a l a n d s o a r e t h o s e o f d e t ( A H Q - ) = 0 . T h e o r e m 7.4 I f t h e g e n e r a l i z e d e i g e n v a l u e s o f A B Q + a r e r e a l , ( X . + a ) B Q - bB^ f o r a , b c o m p l e x . n u m b e r s i s 1 - l i n e a r p r o v i d e d b^O a n d B q n o n s i n g u l a r . P r o o f . ' T h e r e s u l t i s i m m e d i a t e i f one. n o t e s t h a t t h e r o o t s o f d e t ( ( A - f - a ) l - b B ^ " 1 ) = 0 a r e o f t h e f o r m a + A..b, w h e r e A . , j=1,2,...,n, a r e t h e g e n e r a l i z e d e i g e n v a l u e s o f J 7\B + B , . o 1 We now l o o k a t t h e c a s e r = 2. T h e o r e m 7.5 I f A Q = H Q , A 1 = H 1 , A 2 = - I a n d H Q i s p o s i t i v e d e f i n i t e , t h e n t h e r o o t s o f d e t ( A 2 A Q + A A 1 + A 2 ) = 0 a r e r e a l . 35, P r o o f . One n o t e s t h a t / 0 I -B = ( I i s p o s i t i v e d e f i n i t e a n d B. = j 0 1 0 V " V 1 - H ' i s h e r m i t i a n . B y T h e o r e m 7 .2 , t h e r o o t s o f d e t ( - B , A + B , ) = 0 o t o r d e t ( B Q A - B^ ) = 0 a r e r e a l . T h u s t h e r o o t s o f d e t ( A A Q + A.A.J + A 2 ) • = 0 a r e r e a l b y T h e o r e m 7 . 1 . R e m a r k I f A 2 i s n o n s i n g u l a r a n d d e t ( A 2 A o ( - A 2 ) ~ 1 + M 1 ( - A 2 ) ~ 1 - I ) = 0 s a t i s f i e s t h e c o n d i t i o n s i n T h e o r e m 7 .5 , t h e n a l l i t s r o o t s a r e r e a l . T h e o r e m 7.6 I f A Q = H Q , A 1 = H 1 a n d A 2 = - H Q s u c h t h a t d e t ( H Q ) j4 0 a n d H 0 H ^ = H-j H 0» t h e n t h e e i g e n v a l u e s o f d e t ( A A Q + A A 1 + A 2 ) = 0 a r e r e a l . o P r o o f . T h e e q u a t i o n d e t ( A A Q + A A^ + A 2 ) = 0 may b e w r i t t e n 7 ' 1 f d e t ( A ^ I + A B ^ P T 1 - I ) d e t ( H Q ) = 0, s i n c e d e t ( H Q ) 0. B y T h e o r e m 7 . 1 , t h e r o o t s o f t h e l a s t e q u a t i o n a r e t h e g e n e r a l i z e d e i g e n v a l u e s o f A B Q + B^, w h e r e / - I o\ f 0 I B o = a n d B. = . . S i n c e B . B. a r e h e r m i t i a n , d e t ( B Q ) : £ 0 a n d B B^= -BjB , b y T h e o r e m 7 .3 , t h e r o o t s a r e r e a l . 36. T u r n i n g t o 1 - c i r c u l a r p o l y n o m i a l s we h a v e t h e f o l l o w i n g r e s u l t s : T h e o r e m 7 » 7 I f U Q a n d U | a r e u n i t a r y m a t r i c e s , t h e n t h e r o o t s o f d e t C X U ^ + ) = 0 l i e o n t h e u n i t c i r c l e i n t h e c o m p l e x p l a n e . P r o o f . T h e e q u a t i o n d e t ( A U Q + ) = 0 may b e w r i t t e n d e t ( A l + I ^ U " 1 ) d e t ( U ) = 0 . S i n c e d e t ( U o ) ^ 0 a n d b ^ U ~ i s u n i t a r y , t h e r o o t s o f t h e a b o v e e q u a t i o n a r e o n t h e u n i t c i r c l e a n d s o a r e t h o s e o f d e t ( A U 0 .+ U 1 ) = 0. T h e o r e m 7.8 I f A = U , A 1 = 0 a n d A 2 = U 1 , t h e n t h e e i g e n v a l u e s o f d e t ( X A Q + A A^ + A 2 ) = 0 l i e o n t h e u n i t c i r c l e i n t h e c o m p l e x p l a n e . P r o o f . / - I 0 \ ./' 0 . I S i n c e Bn = a n d B. = ( \ ° : " V l " u i 0 a r e u n i t a r y , b y T h e o r e m s 7 . 1 a n d 7.7 t h e e i g e n v a l u e s o f 2 d e t ( A A + A A 1 + A 2 ) = 0 a r e o n t h e u n i t c i r c l e . 3 7 . T h e o r e m 7.9 I f t h e g e n e r a l i z e d e i g e n v a l u e s o f A b 0 + B.j a r e on- t h e u n i t c i r c l e , ( A + a ) B Q - i s 1 - c i r c u l a r , w h e r e a = r + s i i s a c o m p l e x n u m b e r a n d b i s a n o n z e r o r e a l n u m b e r . P r o o f . S i n c e t h e r o o t s o f d e t ( ( A + a ) I - b B ^ B " ) = 0 a r e o f t h e f o r m a •+ b A . , w h e r e A . , j = 1 , 2 , . . . , n , a r e t h e g e n e r a l i z e d e i g e n v a l u e s o f . A B + , t h e y a r e o n a c i r c l e w i t h c e n t e r a t ( r , s ) a n d r a d i u s b ; i . e . , ( A + a ) B Q - bB^ i s 1 - c i r c u l a r . B I B L I O G R A P H Y A f r i a t , S. N., C o m p o s i t e m a t r i c e s , Q u a r t . J . M a t h . , O x f o r d , S e r . ( 2 ) , 5, 1 9 5 4 , p p . 8 1 - 9 8 . A i t k e n , A. C , D e t e r m i n a n t s a n d m a t r i c e s . 4 t h e d . , 1 9 4 6 . A r a m a t a , H., U b e r e i n e n S a t z f u r u n i t a r e M a t r i z e n , T o h o k u M a t h . J . V o l . 2 8 , 1 9 2 7 , p p . 2 8 1 . A r s c o o t , F . M., L a t e n t r o o t s o f t r i - d i a g o n a l m a t r i c e s , P r o c . E d i n b u r g h M a t h . S o c . V o l . 1 2 , 1 9 6 1 , E d i n b u r g h M a t h . N o t e s , N o . 4 4 , p p . 5 - 7 . B r a u e r , R., U b e r e i n e n S a t z f u r u n i t a r e M a t r i z e n , T o h o k u M a t h . J . V o l . 5 0 , 1 9 2 8 , p p . 7 2 . B r e n n e r , J . L . , E x p a n d e d m a t r i c e s f r o m m a t r i c e s w i t h c o m p l e x e l e m e n t s , S I A M R e v i e w V o l . 3 , N o . 2 , A p r i l , 1961, p p . 1 6 5 - 1 6 6 . B r i o s c h i , F . , J . M a t h , p u r e s a p p l . V o l . 1 9 , 1 8 5 4 , p p . 2 5 3 - 2 5 6 . C a u c h y , A., A n c i e n s E x e r c i s e s . 1 8 2 9 - 1 8 3 0 - C o l l . W o r k s I I , V o l . 9 , p p . 1 7 4 - 1 9 5 -C l e b s c h , A., J . r e i n e . a n g e w . M a t h . , V o l . 6 2 , 1 8 6 3 , p p . 2 3 2 - 2 4 5 . G o o t , E . , A t h e o r e m o n d e t e r m i n a n t s , S I A M R e v i e w , V o l . 2 , N o . 4 , O c t o b e r , 1 9 6 0 , p p . 2 8 8 - 2 9 1 . H a d e l e r , K. P., E i g e n w e r e t e v o n O p e r a t o r p o l y n o r m e n , A r c h . R a t . M e c h . A n a l . 2 0 , 1 9 6 5 , p p . 7 2 - 8 0 . H e r m i t e , ' C. R. A c a d . S c i . ; P a r i s , V o l . 4 1 , 1 8 5 5 , p p . 1 8 1 - 1 8 3 . H o j d a r , J . , E i g e n w e r t a b s c h a t z u n g e n f u r e l n P o l y n o m i a l -p r o b l e m , A p l i k a c e M a t h . S v a z e k 14, . 1 9 6 9 , p p . 1 2 0 - 1 3 3 . H o u s e h o l d e r , A. S., T h e 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 a n a l y s i s . B l a i s d e l l , New Y o r k , 1 9 6 4 . M a r c u s , M. & M i n e , H., A s u r v e y o f m a t r i x t h e o r y a n d m a t r i x i n e q u a l i t i e s . A l l y n a n d B a c o n , I n c . , B o s t o n , '. 1 9 6 4 . 39. P a r o d i , M., L a l o c a l i s a t i o n d e s v a l e u r s c a r a c t e r i s t i q u e s d e s m a t r i c e s . G a u t h i e r - v i l l a r s , E d i t e u r - i m p r i m e u r - l i b r a i r e , P a r i s , 1959. R a h u s e n , A . E . , W i s k u n d i g e O p g a v e n , V o l . 5 ' , 1893, pp.392-394. S c o r z a , 6 . , C o r p i n u m e r i c i e A l g e b r e , M e s s i n a , 1921, pp.133-179. W e i e r s t r a s s , S. B. p r e u s s . A k a d . V i s s . , 1879. / 

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