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Pairs of matrices with property L. Chow, Jih-ou 1958-01-19

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PAIRS OF MATRICES WITH PROPERTY L BY JIH-OU OHOW A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE DEPARTMENT OF MATHEMATICS We 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 candidates f o r the degree of MASTER OF ARTS. Members of the DEPARTMENT OF MATHEMATICS. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1958. ABSTRACT Let A and B "be n-square complex matrices w i t h eigenvalues X^, • • •, jX^ and^/^yA , • • • ,jx^ r e s p e c t i v e l y . The matrices A and B3are s a i d to have property L i f any l i n e a r combination aA + bB> w i t h a, b complex, has as eigenvalues the numbers a X + b/^, i . = 1,2, • • -,n. A theorem of Dr. M. D. Marcus, which gives a necessary and s u f f i c i e n t c o n d i t i o n such that two matrices A and B have property L i n terms of the tr a c e s of v a r i o u s power-products of A and B, i s proved. This theorem i s used to i n v e s t i g a t e the con d i t i o n s on B f o r the s p e c i a l cases n = 2 , 3 > and 4 , when A i s i n Jordan canonical form. The f i n a l r e s u l t i s a theorem which gives a necessary c o n d i t i o n on B f o r A and B to have property L when A i s i n Jordan ca n o n i c a l form. In presenting 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 of the requirements 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 Columbia, I agree that the 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of 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 granted by the Head of my Department or by h i s r e p r e s e n t a t i v e . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. J i h - o u Chow Department of - Ma.thema.tins The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date A o r i l 16. 1QS8. A CKNOWLEDGEMENT S I am indebted to Dr. B. N. Moyls and Dr. M. D. Marcus of the Department of Mathematics at the U n i v e r s i t y of B r i t i s h Columbia f o r d i r e c t i n g and su p e r v i s i n g my work. I would a l s o l i k e to express my g r a t i t u d e to the N a t i o n a l Research Council of Canada f o r considerable f i n a n c i a l a s s i s t a n c e . T A B L E OF CONTENTS 1. I N T R O D U C T I O N •" •• 1 2. A N E C E S S A R Y AND S U F F I C I E N T C O N D I T I O N FOR P R O P E R T Y L 2 3. A P P L I C A T I O N S 4 4. THE N E C E S S A R Y C O N D I T I O N ON B SUCH THAT . A AND B H A V E P R O P E R T Y L 15 B I B L I O G R A P H Y 22 1. I n t r o d u c t i o n , D e f i n i t i o n . Let A and B: be- n-square complex matrices with, eigenvalues X, • • •,^ n andyx., >>u.i, • • •, jxn r e s p e c t i v e l y . The matrices A and B-are s a i d t o have property L i f any l i n e a r combination aA + bB, w i t h a, b complex, has as eigenvalues the numbers a\+ t>j*.lt i = 1,2, ''',n. i n 1952 T. S. Motzkin and Olga Taussky ( l ] proved the f o l l o w i n g theorem: Let the n-square matrices A and B have property L. Let t be the number of d i f f e r e n t eigenvalues of A and assume that a l l the eigenvalues A i of A are arranged i n sets of equal ones. Let O K be the m u l t i p l i c i t y of the.eigenvalues XL of A and assume th a t there are m^  independent eigenvectors corresponding to e a c h ^ . Let be the corresponding eigen- * -1 * -1 values of B. Let B:= P BP where A = P AP i s i n Jordan c a n o n i c a l form. Then *- B = ' B l l B 21 B 22 B i t B t l B,. • • • t2 where i s an m^-square matrix ( 1 = 1,2, « »«,t ) and I x l - B*| = .n |xl - B n|. b.,b v. where the summation i s over a l l i<k w i t h ( i , k ) outside of every B ^ ( i = 1,2, • • • , t ). i n the above theorem A i s r e s t r i c t e d i n that i t i s s i m i l a r to a diagonal m a t r i x , i n the present paper we consider the s t r u c t u r e of B f o r more general types of A. We f i r s t - 2 - present a r e s u l t of Professor M. Marcus which gives a necessary and s u f f i c i e n t c o n d i t i o n that A and B have property L i n terms of the tr a c e of various power-products of A and B. We then determine the r e s u l t i n g s p e c i a l c o n d i t i o n s on B f o r the cases n = 2,3 and 4. we conclude w i t h a general necessary c o n d i t i o n that £ and B have property L. L x y x y ^ A XB 1 " - A r B r ) = A'9), l*p*fc,l*k.-n, 2. A Necessary and S u f f i c i e n t Condition f o r Property L»  Theorem 1. A and B have property L i f and only i f "~ p' . x ^ . .. +x r= P «•=• y 1 +...+y r= k-p x l i 0 ' x 2 » * * , ' x r > 0 y r - o , jr_V"'>yp where X,, \, • •; and yx, ,jxi, • • • are eigenvalues of A and B : r e s p e c t i v e l y . Proof. I f A and B have property L, then n trace( aA + bB ) = and t r a c e (aA + bB) 1 2 = £ j a\+ b y A j k ( k = l , 2 , * " , n ). Expanding both side s of the second i d e n t i t y , we have K-I x y x y a k t r a c e Ak+ H a p b k ~ p t r a c e ( 2__i A 1 B 1 " « A r B r ) + b k t r a c e B k P=' x ^ . • .+xr= p y l + * , + y r = k ~ p x-^0, .Xgj • • ,xr>0 k . ( y r ^ o , y r _ 1 , - - . . y 1 > 0 lc T -,k , V p-Js-p , L v ..p k-p -j£y k = a + L a ' b * ( ) 21 M C + B • V = l p = l & t.= l ' P K-P The c o e f f i c i e n t s of a b (p = 0,1,•••,k) i n both sides of the i d e n t i t y must be equal, so we have the f o l l o w i n g c o n d i t i o n s : k n n t r a c e A =TJXlJ , t r a c e B k = yx k and - 3 - L x y x y X„+. . . + X__ = P P i / = l x 1 +... +x r = p yx+« • • +yr = k-p x l * 0 , X 2 ' " * * , x r > 0 y r-o, y^!.*** .y^o Conversely, i f trace( YLi A * ^ 1 - * ^  r A X r B y r ) = (^)(Lx?J5"P). l ^ P ^ l ^ n , then trace( aA + bB ) k = trace( aKAK + L a ^ ' P LA%^. • .A*Vr + b k B k ) = a k t r a c e A k + Z l a p b k s " p t r a c e ( A r* y 1 J 1 B •A B ) + ITt r a c e B 7", / k k y7' p - , k - p / k x _ p k - p , ^ k . k v = fc1( a X , + 2_,a^b f ( p ) + ) = £ ( a\+ b u L ) k . .*. trace( aA + bB ) = J L ( a\+ by**,)*, 1^-k^n. L= i For b r e v i t y of n o t a t i o n we now l e t aA + bB = C and aX+ hjx^z^c^ ( i = 1, 2, • • •, n ). Suppose C- has eigenvalues ( i = 1,2, •••,n ), we want to c 1 1 prove that the o^s are ex a c t l y the Y ^s. Since there e x i s t s a u n i t a r y matrix U such th a t r s. U _ 1 C u and U _ 1C kU = 0 , k = 1,2, O <L-<-rn. are upper t r i a n g u l a r , then n n L ^ i = t r a c e G k = tr a c e U*"1Ck U = L^ K = S k, 1-k-n. By Newton's formula f o r symmetric f u n c t i o n s , we have p l = SV Po = 5 ( s i S Q ), p^ = r a t i o n a l f u n c t i o n of S-^Sg, •••»S|C, - 4 - where p^ ., l ^ k ^ n , are the elementary f u n c t i o n of the n v a r i a b l e s S,, ^ n,and f ( x ) = xn-p-^xn~'1'+• • • + ( - l ) n X p l^k-^n, so p^, 1-k-n, are a l s o the elementary functions of V, , T^, . .., Y„ . Hence these are a l s o the n roots of the c h a r a c t e r i s t i c equation f ( x ) = 0 of C. Since f ( x ) z 0 i s of degree n and has only n roots, tthe o^'s are e x a c t l y the V s . Therefore = aX^+byu^ ( i = 1 , 2 , n ) are the eigenvalues of G = aA + bB. This completes the proof of the theorem. 3• A p p l i c a t i o n . In t h i s s e c t i o n we s h a l l determine e x p l i c i t l y necessary and s u f f i c i e n t c o n d i t i o n s on the c o e f f i c i e n t s of an n-square complex matrix B, when n = 2 , 3 and 4 , such that A and B have property L by u s i n g the Theorem 1. We'assume that A i s i n Jordan cannonical form. has eigenvalues y/.,andjx^ Case 1. when n=2, suooose B= There are three cases to be considered. A, o ( i ) Let A = , \jz\, then A and B have property L o i f and only i f °2.21°21 ~ °* proof. By Theorem 1, A andB have property L i f and only i f the f o l l o w i n g r e l a t i o n s hold: t r a c e (B) = b]_]_ + b,2 = / A i + / u a ' t r a c e (AB) = 0^,+ h22\=/it\+/zK' or (/,-b 1 1) + (^ z-b 2 2) = 0, (/ lrbn)^.+ ^ - b 2 2 ^ a + = °* (2) - 5 - Since 1 1 X, X2 22 Now u nu„ = b- _, b„^ - b b = u U - b b ^1T2 11 22 12 21 r l r 2 12 21 . . b b = o . 12 21 Conversely i f b b = o, then u = b ,/A = b and (2) x^ ^x J_ IX 2 22 holds. ( i i ) Let A= f A, 1 ° A, ; then A and B have property L i f and only i f b =o. 21 Proof. (trace (B) = b., n + b = JU. + JU 11 22 O ^ 2 Urace* (AB) =A(b + b } + b„ = A.(u +u ) 7 1 11 22 21 1 / l 2 . . b 2 1 = o.... This i s c l e a r l y necessary and s u f f i c i e n t , ( i i i ) I f A =Al, then, f o r any B, A and B have property L. Proof. The c h a r a c t e r i s t i c equation of a A + ^ B i s (3) ^|<xA+(aB-<Tl| = o , or ' i J o { A l + ^ B - < T l | = o . I f (3 = o, then the eigenvalues of A are both <* X . I f (3 ^  o , (3) becomes I B -( £ ) I | = o I f B has eigenvalue sy /C^ andyx^ , then each — — — and <5- . = o( X + ^ ^ A . , i= 1,2. - 6 - Case I I . n = 3. We l e t B = (bjj), ( i j = 1,2,3) have eigenvalues There are s i x cases to he considered, but we s h a l l give the proof only f o r the f i r s t case and s t a t e the necessary and s u f f i c i e n t c o n d i t i o n s f o r the r e s t without g i v i n g t h e i r p r o o f s . ( i ) Let A =|o Xi o , where A, ^ A^and A 3 are d i f f e r e n t , then [o o X j A and B have property L i f and only i f ( 4).u,=b„ . kx „= b UL _=b ^l=Dll-> M 2 = D22*>M- 3 33 ; 12b21 • "3l"l3 • ~23~32 Proof I f A and B have property L, then from (1), U,„ on ; b,nb,, i b _ b _ = ( X{ -Xz) : (X^-Xj) : (\-A 3), (.5) t r a c e B = b 1 1 + b £ 2 + b ^ = ^  . tra c e AB = b A, + b X + b A~„ =M X, +u. X + u 11 1^  22 | 33 3 ^ 1 1 ^ 2 L ~3 3 5 t r a c e A 2B = b^X* + b^A** b ^ A ^ ^ A * +^ 2X^ +^ 3A* . trace B 2 = £ £ b . ^ . = £ ^ / i=) k«l 1=1 -3 -3 t r a c e AB 2 = ( b l kb ) A +( L b 2 k b k 2 ) V k=i k=l 3 2 ; + ( L b 3 k \ 3 ) A 3 ^ 1 X 1 + ^ 2 V / 3 X 3 ' For example, the t h i r d equation of (5) i s derived from (1) when k=3, p=2; i n t h i s case (1) becomes. (i)lX/-;= t r a c e (A 2 B +ABA + BA 2) = tr a c e A^B + trac e ABA + t r a c e BA =3trace A 2B, . sin c e , f o r any matrices M,N, trac e MN = t r a c e E M . - 7 - Prom the f i r s t three equations of (5) we have [LK. - b ) + ( M. - b ) + ( M - b ) = o. v / l 11 ^ 2 22 ^ 3 33 V ^ l 11 \\ ( M. - b ) + A (JA, \ r 1 11 2- r ) + A ( u - b )+A„(u - b ) = o . 7 2- J 2 22 3 / 3 33 - b ) + A , ( / A - b ) = o 2 22 5 J 3 33 Since 1 1 1 Aj Aj. A3 Ai A 2 A 3 = ( \ - \ ) ( ( A 3 - X J ) jt o (6) M. =b , X A = b , JJ< = b ^ 1 11 ^ 2 22 ^ 3 33 Prom the l a s t two equations of (5) and ( 6 ) , we have •(b12 b21 + W ^ l + <b21b12 + b 2 3 b 3 2 U 2 + ( V l 3 + = o , b, r,b„„ + b„ b + b b = o -12 21 31 13 23 32 or , b b A + b b X + b b A. = o j 23 32 1 13 31 2 12 21 3 ^b O I 7 b + b b + b b = o . 23 32 13 3 1 12 21 b b : b b : b b = ( X - X ) : (A -X ): {X -\ ) 23 32 13 31 12 21 2 3 3 1 1 2 Hence a necessary c o n d i t i o n i s (4) = b J 1 "11 ,/X2 \ = b 22 = b 33 ' b _ b : b b ; b b = (A - X ) i (A - A ) : (A - X ). 23 32 13 31 12 21 2 3 3 1 1 2 Conversely, i f (4) hold , then t r ac e A B = A b +A b + A „ b = A u + A j u . + A M « 1 11 2 22 3 33 I I 2r 2 3 ^ 3 2rs 2 . ^ 2 ^ . 2 A 2 . . . 2 . N 2 t r a c e A B = X b + A b + A 2 b =A M + A K + A U 1 11 ^2 22 3 33 1 1 ' 2 ^ 2 3 ^ 3 ' - 8 - (4) Cont'd t r a c e AB 2= A ( b 2 + b b + b b )+A (b b + b 2 + 1 11 12 21 1 3 31 2 21 12 22 2 + b0,, b,J + A„ (b„ b + b b .+ b ) 23 32! 3 31 13 32 2 3 3 3 "* \ 2 *\ 2 2 = A h _ bnn + A b + b b ( A + A )+ b b 1 11 2 22 3 3 3 12 21 1 2 23 3 3 2 ( A + A ) + b b ( A + A ) 2 3 31 13 3 1 = X u 2 +A u 2 + A u 2 + k (A - A ) + K ( A - A ) (A + A ) l r l 2 / 2 3 ^ 3 1 2 2 3 2 3 + k ( A - A ) (X +A ) 3 1 3 1 = XlJ^l + X 2 / . 2 + A ^ 2 . Hence by theorem 1 , A and B haize property L. 4 (ii) Let A = A i o o o At o 1^ 0 o A 3J where A ^ X , then A and B hatee 1 3 property L if and only if ^ 1 ^ 2 1 1 2 2 ^ 3 3 3 b.,, b + b b = o . 1 3 3 1 2 3 3 2 (iii) Let A = A ^ A , then A and B have property L t 1 3 if and only if ^ 1 ^ 2 " b l l + V >S = b 3 3 ' b 21= ° ' (b b + b b ) (X - A. ) = b b 13 31 23 32 1 3 23 31 - 9 - (iv) Let A= \ x 1 o o ^ i O O .Xj then A and B have property L i f and only i f b = o , j 21 \ b (v) Let A= \ 1 o O 0 \ 1J then A and B have property L i f and only i f b = o , 31 b 3 2 = " b 2 l >. b r t (b - b ) 21 11 33 \ o o o c X = O (vi) Let A= = X I , then, for any B , A and 3 have property L . -10- Case i n , n = 4 : in this case we let B = (bij)> (i»J 1>2,3,4), have eigenvalues/^y '^T^ andpf.4. . There are fourteen cases to be considered, but we shall only state the necessary and sufficient condition for these cases without giving their proofs. A , o o o where A ( / A 2 , A 3 and A^ are (i) Let A = O O D O O A3 O O O O different, then A and B have property L if and only if .j^i =bxl , i = 1,2,3,4, < bij bji( Xi+\j) = o, j 2. 2. V . b i j"b j i (A,i+ A± A j+ A j) = 0, i,j=i ^ H ( i _ , b i J b J ^ b k i " b i i ^ A i ~ °* A , O O O o A., o o (ii) Let A = where A1 \$ and are different. o o A 3 o ^ O O O then A and B have property L if and only if ,M± =b1± , i = 3,4, b13b31+bl4b4l+b23b32+b24b42+b34b43 = °> (b31b13+b32b23+b34b43) (A3-X]J+(b4]b1^ bz(.2b24+b43b34) = 0. (iii) Let A = \ 0 0 0 0 A , 0 0 0 0 A 3 ° 0 0 0 A 3 where A, ^ A3 , then A and -11- B have property L i f and only i f yU1+/*2 = bll+D22 , 2. 2. * r* y*L+/«2 = t ) l k b k l + L b 2 k b k 2 > . At^? = t I b l k b k i b i l + L ^ b 2 k b k i ' b 1 2 ' b 1 3 b 3 1 + b l 4 b 4 l + b 2 3 b 3 2 + b 2 4 b 4 2 = °' ( i v ) Let A = A , o o o 0 A , o o O o A , 0 O O O A4 have property L i f and only i f , where A, A^, then A and B ^ 4 = bA4 , 3 1L "°kA.k = 0 , K= 1 (V) Let A - A , l o o o A , I o ° o A , I o o o X , . , then A and B have propertyLL i f and only i f *>4i = 0 , b 3 1 + b 4 2 = 0 ' b 2 1 + b 3 2 f D 4 3 = 0 , H ( b2k bkl+ b3k bk2+ b4kbk3) = 0 , b2k Dki bil+ b3k bk£ bi2+ b4kbkibi3) = o, • ( b i r b 4 4 ) b 3 1 - b 4 3 b 2 £ = 0 . ( v i ) Let A = i f and only i f ° 3 1 = 0 , f X , 1 0 o 1 o X , i o o o X , o O O O X| -12- , then A and B have property L b 2 1 + b 3 2 - 0 ' t, ( b 2 k b k l + b 3 k b k 2 ) = 0 ' IL t L ( b 2 k b k i b i l + b 3 k b k i b i 2 ) = 0 ' K-=.| "«. - I ( v i i ) Let A = ' X , I o o o X | o o o o X , | o o o X , then A and B have property L i f and only i f b 2 1 + b 4 3 = 0 » < b 2 k b k l + b 4 k b k 3 ) = 0 ' [ ~2J ( b 2 k b k i b i l + b 4 k b k i b i 3 ) = 0 ' b 2 3 b 4 r b 4 3 b 2 1 = 0 A , I o o "j 0 A , o o ( v i i i ) Let A = i f and only i f o o X , o 6 O O X , , then A and B have property L '21 0 , L b 2 k b k l = 0 > . I I L b 2 k b k i b i l = 0 . - 1 3 - ( i x ) Let A = A, I o D o A, o o o o A 3 o O o o A^., , then A and B have property L i f and only i f ^3 = b 3 3 ' / ^ = b 4 4 h 2i = 0 , 2 2 2 v* 2 2 2 3 ( L ^ £ 3 ^ 3 3 ) ^ 3 - ^ 1 ) + 3(L, b 4 k b k 4 _ b 4 4 ) ( ^ 4 - ^ 1 ) _ - 2(b 1 3h 3 1+b 2 3b 3 2 )(A 1 -A 3 ) - 2(hl4b4l+b24bZf2)(A1-A4f+ + 8 \ L to2kbki+ 4 ( ^ 3 b 2 3 b 3 1 + V 2 4 b 4 l ) = °> ' ( L t b 3 k b k i b i 3 - b 3 3 ) ( A l - A 3 ) ' + < L t ; b 4 k b k i b i 4 - b 4 4 ) ( \ - ^ 4 ) - K - I i.= I b 2 k b k i b i l = °« (x) Let A = A , I o o o A , o 0 O O A/3 o , 0 0 f> A3 . have property L i f and only i f ,where X, ^  A 3 , then A and B < b 2 1 = 0 ' +^4 = b33+ b44 ' L b 2 k b k l = ( b l 3 b 3 i + b i 4 b 4 l + b 2 3 b 3 2 + b 2 4 b 4 2 ) ( \ " ^ 3 ) ' ^ 3 V*4 =b33+ 2 b34 b43+ b44 > L b ' & b k i b i l = ( - | ^ b 3 k b k i b i 3 + t b 4 k b k i b i 4 - b 3 3 - 4 b 3 3 b 3 4 b 4 4 + + b 3 3b44+b3 3b44-b4iji -4b34b44b4 3) ( X 1 - A 3 ) . t.it=i ( x i ) Let A = A | I o o D A , O O O O A3 I _ O o t> A3, have property L i f and only i f , where X. £ A,, then A and B -14- b 2 k b k l + f_i b 4 k b k 3 = bg! = 0 , b 4 3 = 0 , ( L b 3 k b k 3 + L b 4 k b k 4 ) ( ^ 5 - ^ 1) + H K = l K ^ l k.= l = y^3 1^4 ) ( ^ 3 - ^ 1 ) > 4 4 ( L ^ A l - L b 4 k b k 3 ) ( X l - A 3 ) + 2 b 2 3 b 4 l - - ( b 1 3 b 3 1 + b 2 3 b 3 2 + b l 4 b 4 l + b 2 4 b 4 2 ) ( ^ 1 - ^ 3 ) = 0 > * 4 4 ( . L _ i b 3 k b k i b i 3 + H b 4 k b k i b i 4 ) ( A 3 - A X ) + b 2 k b k i b i l + V + £ b 4 k b k : i b i 3 = ( y ^ + X ) ^ A ) • A, I o o » X , 96 X 4 , then A and B have ( x i i ) Let A = O A j 0 o o o X , o property L i f and only i f b 21 = 0 , yW-4 = b 4 4 , ( f_i b4k bk4" b44 ) ( X 4 - \ ) + I l b 2 k b k l = 0, ( l l b 4 k b k 4 - b 4 4 ) ( ^ 4 + ^ l ) + b 2 4 b 4 l = 0 , 4 ^ 4 ( . L L b 4 k b k i b i 4 - b 4 4 ) ( ^ 4 - A - L ) + b 2 k b k i b i i = 0. ( x i i i ) Let A = have property L. y K=H \ o o o o X o 0 o o X * 0 0 0 X = Al » then, f o r any B, A and B_. -15- o o X , o ( x i v ) Let A = o * o \ property L i f and only i f A, I o ° o A, I o , A,£ A^ , then A and B have < b 2 l + b 3 2 = 0 > >*4 = , ( I l b 4 k b k 4 - b 4 4 ) ( * 4 - A l ) + I L b 2 k b k l + L b 3 k b k 2 = 0 > K = l K = l K = t ( L b 4 k b k 4 ~ b 4 4 ) ( ^ 4 - \ ) - - % ( b 2 4 b 4 l + b 3 4 b 4 2 ) ( \ ~ \ ) + b 3 4 b 4 i = 0, < ( b 4 k b k i b i 4 - b 4 4 ) ( A 4 _ A 1 ) + b 2 k b k l b 1 1 + + / b 3 k b k i b i 2 = 0 4 . The Necessary Condition on B such that A and B have  property L. Theorem 2. Let A = diag ( A-,, Ag? A r) .'where / A 1 1 0 • • • 0 N A 1 =1 0 A 1 2- 0 0 0 A i s . and A i j = (u^ 1 o" '0 s] 0 u. 1*• ' 0 1 / V 0 0 O'^'U, Here A i s of order n, A, i s of order a., and Ai . i s of order m±y where m1]L + m i 2 + • • •+ mis_ = , na+n2+-..+ n r = n , and u-^jUg, • • •, u p are d i f f e r e n t eigenvalues of A. -16- And l e t B.= B l l B12 B 2 1 B22 B l r ) B 2r B r l B r 2 B, rr, be an n-square matrix w i t h eigenvalues v-^Vg, '">^ n> .where B i i = ( b ^ X ^ ) i s of order a^. I f A and B have property L, then r v _ r b 1 y v - r b^) r v - r ^ r r bR , + r b(D . ., + . . H . r b<x> = 0, ( k = 1,2, • • -,^-1 ) , Vr) ;_7l.i J J " k f-H. m 2 1 + J ' m 2 1 + J _ k ^—' no- moo +J» no" i noo +J" i J.K*. 2 2 s, 2 2 s, - 0, ( k = 1,2, • ",a -1 ) , + • • •+ r b ( r ) jsk*-i r r r s = 0, ( k = 1,2, *' ' . n ^ - l ). Proof: There &s no l o s s of g e n e r a l i t y i n assuming that the orders of ( j = 1, 2, • • •, ) ore so arranged such that m i l i m i 2 - S s . ( i = 1,2, • •' ,r ) -17- One can show e a s i l y by i n d u c t i o n that A k 0 0 A k 0 0 N and A. . = and A k j • • • • 0 • • • A k / Is U i - 1 ( 0 < ( 0 * • • 0 • • • 0 0 \ 0 • where A^ = ' 4 « o ,f2 0 0 ^ k-2 k v . . k - l 0 1 0 ••• 0 l s i ( T ) u <*-i> ui ^ o u. u i / l^k<.m. .-1 k I. . - ,k-m. .+ 1 ^  k u i m. .-1. Now t r a c e (A B) ' A * 0 o ki — t r a c e 0 0 k T 0 0 r A r B l l B12 B 2 1 B22 B r l B r 2 = t r a c e ( A j B i : L ) + tra c e ( A k B 2 2 ) + t r a c e (A^B-^) + +...+ tra c e (A rB^ r) ( k = 1,2,...n-l ) where B i l = ( b [ ^ ) ( i = 1,2, " T ) i s of order n±. l r 2r B r r / k T -18- Furthermore, t r a c e ( A p 1 1 ) = = t r a c e A l l U 0 A k 2 0 0 = t r a c e ( A 11 0 0 1 S i . b ( i ) b ( i ) . b 2 1 b22 ••• b ( 1} b ^ 1 ) . . . - v V n i 2 ft,* 1) ... A1} 1 m i l m. -, m., i l i l ) + In. I >2n. a±al< + t r a c e ( A 12 m i l + l , m . 1 + l m . 1 + l m i l +m. 2 b U ) . . . b ( i ) . m i l + f f l i 2 m i l + 1 m i l + m l 2 m i l + m i 2 + . .. + rb(D ...b^) n N n i " m l 8 1 + 1 ' n l - , n l 8 + 1 " i ^ i s . * 1 n i + t r a c e ( A I S : n i ni-ai. +1 i s ... n. n. J W ; , J^ i > « . - , | = k -19- 1 H ^ n + J m.-. + j - l ^ l ^ i + + L m , n + j m , 1 + j + + ( r b ( l ) • . ) u k + j t t ^ i s . + ^ i ^ i s . + J ' 1 / r v^(i) w k , k-1 • i f r l 1 " n i - m i 8 1 + J» " i ^ i s * - ^ - ( i t,*1) v u k + c r b^) + F b ^ ) 1 i s i 1 i s i +... + (r *{-\K + r b ( i ) • ,,+••• + L b ( i ) . ..) , where / . h i 1 ? . + . , , = 0, when k = m, , and k = 1,2,' j^K+i t+J' t*.j-k i j n-1 , i = 1, 2,« • • r . Since A and B have property L , by Theorem 1 we have the f o l l o w i n g c o n d i t i o n : ^  M r Trace (B) = L ^ V + L.*\V + ... + J - l r J X j = l L k - - = L - — •** Trace ( A B )  > Trace ( A,B,, ) ( C V 1 ) )uk + ( F b ( i ) i + + -20- 0 I 1 n i - m i B + ' ni-*lS + J ~ k = A L v + uk< L v ) + ... + „*( 2> ) ( k = 1, 2,» • • n-1 ). The determinant of the o o e f f c i e n t s of the above n simultaneous l i n e a r equation i s as f o l l o \ ^ s : t D = 1 0 u x (J) .. uf (^)u (2) 0... 1 o . • o • • • n , - l ' ' * ( - J • • ' .. 0 . . 0 0 • n - 1 n 2 1 1 0 • • • u-.n-1 n-1 u. n-1 • • »0 • • 0 •. 0 n r -1 <4 -1>" n l n 2 , * 0 . n 1 n 3 • • (u ., -u ) v r-1 r ; n r - l n r Hence the s o l u t i o n s i s f The value of D was found by L. Schendal In 1891. See the book " The Theory of Determinants i n the H i s t o r i c a l Order of Development ", V o l . 4 ,p . l 7 8 - l 8 o . -21- = 0 , ( k z 1,2, • - ; n l - l ) , 7 b(. 2\ + r b< 2) +. + r b ^ ) 2 2s, = 0 , ( k = 1,2, • • • , n 2 - l ) , I> ( r ) n r " a i r s + J n ~ m r s + = 0 , ( k = 1,2, ...,n -1 ) . The theorem i s proved. BIBLIOGRAPHY 1. T. S. Motzkin and 0.- Taussky, P a i r s of matrices w i t h property L, Trans. Amer. S o c , v o l . 73, 1952, pp. 108- 114. 

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