LINEAR TRANSFORMATIONS ON MATRICES by ROGER A L E X A N D E R B.A., University A THESIS THE of B r i t i s h SUBMITTED MASTER FOR OF THE 1957 Columbia, IN PARTIAL REQUIREMENTS in PURVES FULFILMENT DEGREE OF OF ARTS t h e Department of MATHEMATICS We accept required THE this thesis as conforming to the standard. UNIVERSITY O F ' B R I T I S H COLUMBIA April, 1959 ABSTRACT In this formations complex o n M^, w h i c h map concerning of n-square sum matrices f o r some of the r x r p r i n c i p a l what f o l l o w s , we "direct product" use E T(A) over the i s the transformations matrices; those positive preserve integer subdeterminants this sum, to transformations = cUAV trans- of the s t r u c t u r e of to denote to refer matrices to non-singular i s the determination which, linear The f i r s t of the s t r u c t u r e of those transformations In the algebra non-singular second the two p r o b l e m s numbers, a r e c o n s i d e r e d . determination the thesis for a l l A r, of each matrix. and the phrase of the form in M n or where U, T ( A ) = cUA'V V are fixed . The main members result singularity preservers products. The r=l, cases i t i s shown structure. products. which Finally, have that preserve forthcoming results paper b y M. of Mathematics, of M a t r i c e s : Funct i o n s . Marcus proofs no Invariance number. nonk, are direct separately. If significant a r e two Eg, and which that these types of are not counter r=3• will also a n d JR. P u r v e s entitled both i f r > there i t i s shown and t h e i r n i s a complex i s that do n o t g e n e r a l i z e t o t h e c a s e These Algebras and c are discussed i t i s shown direct in M and E ^ - p r e s e r v e r s , E^ p r e s e r v e r s transformations Journal n of the thesis f linear examples of M r=l 2,3 that I f r=2, for a l l A Linear be found in a i n the Canadian Transformations of the Elementary of Symmetric In p r e s e n t i n g this thesis i n partial fulfilment of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make it freely 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 . agree t h a t p e r m i s s i o n f o r e x t e n s i v e I further • copying of t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by t h e 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 understood t h a t c o p y i n g or 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 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 2-6 > /?(Pf permission. ACKNOWLEDGEMENTS Without it i s doubtful appeared. the contribution whether Over a period shown a constant to the elementary me found for at more and s o l v i n g mundane education. the I major would like years ideas student contained he h a s patiently I have and t o keep t o complete with deep an the fact i n the thesis my have of obtaining along Marcus h i s techniques new p r o b l e m s , requisites to express would to explain freely efficient For this, ideas of five mathematical to reveal a not e n t i r e l y the thesis willingness puzzling, posing this of Marvin that are h i s , gratitude t o Dr. Marcus. I thank like the National financial this would t o take Research assistance thesis. given this opportunity Council during to of Canada f o r the w r i t i n g of TABLE OF 1. INTRODUCTION 2. NON-SINGULARITY 3. E -PRESERVERS ^ 45. CONTENTS 1 PRESERVERS 2 6 r Eg, BIBLIOGRAPHY 17 37 1• Introduction Let the of M denote t h e algebra- of n-sguare m a t r i c e s n over complex numbers, and GL^ t h e s e t o f n o n - s i n g u l a r members M . n In t h e f i r s t structure of those p a r t of t h i s linear t h e s i s we d e t e r m i n e t r a n s f o r m a t i o n s - T of M the into M n having the p r o p e r t y In those t h a t T ( G L ) _• GL^. n the second linear p a r t , we d e t e r m i n e t r a n s f o r m a t i o n s of M the s t r u c t u r e of into M n _ 4f p r e s e r v e r of are The thesis: the The r e m a i n i n g i n the f i n a l A » , J>(A), t r ( A ) , (A) o f A, t h e rank cases subdeterminants. of r = 1, 2 or 3 section. following notation w i l l transpose w h i c h , f o r some n ' t h e sum o f t h e r x r p r i n c i p a l t h e members o f M . considered n be u s e d throughout , det (A), denote ± the respectively o f A, t h e t r a c e o f A, t h e e n t r y at of p o s i t i o n ( i , j ) , t h e d e t e r m i n a n t o f A, where A i s a member M . In a d d i t i o n , 0 , I , E . , , denote r e s p e c t i v e l y t h e n , * n' n l j ' n x n zero m a t r i x , i d e n t i t y m a t r i x , t h e n x n m a t r i x w i t h a 1 at p o s i t i o n ( i , j ) and z e r o s of p-tuples of p o s i t i v e of a l l p-tuples We w i l l M n into M n The c o l l e c t i o n i s d e f i n e d as t h e s e t where 1 < o f t e n be l e d t o l i n e a r <••-< i ^ < n . t r a n s f o r m a t i o n s of o f t h e form = cUAV or T ( A ) = cUA»V U, V f i x e d members o f M , and c a s c a l a r . * n' formations The integers Q w = T(A) for elsewhere. we w i l l motivation speak of l o o s e l y as " d i r e c t i s that, f o r the f i r s t Such t r a n s products". alternative, regarding - the members £l] the matrix of V 2> and T of M as n n first transformation then mapping i s itself i s similar to a matrix with from linear of M. to a matrix is the matrix Marcus and J . L . McGregor whose diagonal E-j^z we 1. l / ) . . . 1 l / n follows canonical trace, [ 2 ] we vectors U ( 2 entries complete ) - -(n-1) column U^ ^,...,U^ ^ 3 n by a of A. result J(A) i s similar tr(A). the If J(A) similarity •J n _ x vector Then with a l lentries Normalize so t h a t and the set of the ( i , i )entry of vectors UE-^U* zero. the proof by d e f i n i t i o n . form n ) i s orthonormal. i s not have to construct are orthogonal. u.,u._ w h i c h 11 i 2 We from . zero Set k-component and determine to non- a l ldiagonal entries are n proceed as f o l l o w s . i s the K a different n and has non-zero D<*> - J J that matrices Let J(A) denote the Jordan J(A) i s diagonal is lemmas zero. If and product non-singular. of M transform as i n i s the direct non-singular i s a member Proof. where i n three If A different vectors preservers establish matrices 2.1. A of T column U. We Lemma dimensional representation Non-singularity singular 2 2 - If A by induction. is in M The c a s e n=l ^ and J ( A ) i s d i a g o n a l - 3 - with zero trace from zero. and column two non-zero all diag form eigenvalues B obtained there diagonal zero (l,V) j(A) diag (l,V - 1 such ) matrix. Then, 1 has product diagonal (A,, V B V - 1 ) trans- entries. c a n assume o f J(A) i s 1 and t h e submatrix t h e zero VBV i s a similarity we least By t h e that = diag of J(A), J(A) i s not d i a g o n a l , j ( A ) has a t i s zero. The of A and has a l l non-zero entry since i s a V'"£ entries. i s different by d e l e t i n g t h e f i r s t i f i t s trace i s the ( l , l )entry If not the ( l , l )entry o f J(A) i s . n o t hypotheses non-zero where X c a n assume The s u b m a t r i x row induction we t h e (1,2) B, d e f i n e d as b e f o r e , we h a v e as above i s a V in M n such that the product, P = diag has of a l l non-zero (l,V) j(A) diag e n t r i e s on t h e d i a g o n a l , - 1 ) i f the ( l , l ) entry P i s not zero. If (1,1) t h e (1,1) e n t r y entry o f VBV 1 i s zero, there non-zero represents diagonal Lemma 1° diagonal entries. diag n a similarity _ 1 * b 1 J Then ) P diag transform as t h e that \i f l i s taken the product ( i f 1 , I ^ ) o f A and has a l l non-zero entries. 2.2; — — — — — — /0 (U, I and b ^ i s a U such U has (^V F o r any A £ 0 i n t h e M n there i s a Z in M n - 4 - such that A + Z h a s no e i g e n v a l u e s eigenvalues Proof. P lemma ^AP h a s a l l d i a g o n a l Define X = x.. ij entries 1 = - (P 1 + _ 1 itself t o 1. 2.3. I t i s then We some In A i > j equal to 1 while i=l,...,n, easily P ^AP + X h a s none verified Any T w h i c h p r e s e r v e s have of which are that Z = PXP"" 1 non-singularityi s that i f x, other a subset - [T(I)]" - A) = 0 words, T(A)] = 0 of the d i s t i n c t have that 1 T(A T ( A ) = 0. + x. eigenvalues eigenvalues no e i g e n v a l u e s [T(I)]- f o rthat the distinct i s a n A /= 0 s u c h + Z, Z 1 then det(xl there zero. property. det[xl are from that non-singular. Proof. for different AP).. ' i j (P ^ A P ) ^ the required Lemma such rt i < j X has n e i g e n v a l u e s eigenvalues has i s a P in M i=l,....n x. . = 0 equal there (x,.)as f o l l o w s . x..= Then with the o f Z. By t h e a b o v e — — - i n common of o f A. Choose i n common. Z) = [ T ( I ) ] " We 1 [T(l)]~ T(A) Now 1 suppose a Z such have T(Z). that - 5 - From this, equation eigenvalues of [ T ( l ) ] eigenvalues of A eigenvalues of [ T ( l ) ] eigenvalues o f Z. the lemma Theorem then is + Z. of M such and lim B ^ r — > <*» . By we T(Z) are a subset 1 This that distinct the of the contradicts the choice distinct distinct of Z and n o n - s i n g u l a r i t y and T ( l ) = I set of n d i s t i n c t s e t of e i g e n v a l u e s . eigenvalues I f T(A) does then not have =T(A). the continuity of T - 1 , as B ^ — T ( A ) , T - 1 ( B ^ ^ )-> A . r a s r -—> oo = B 5> e . v . T ( A ) r e.v. T ~ ( B ) • the eigenvalues singular of the distinct a l l eigenvalues. e.v. If have the x have Corollary also that (r) e i g e n v a l u e s , l e t [B } be a s e q u e n c e o f members (r) that B has n d i s t i n c t eigenvalues f o r a l l r -. n d i s t i n c t and B u t we I f T(A) has a A h a s t h e same Then T(Z) are a subset 1 If T preserves T preserves n can conclude proven. 2.1. Proof. we 1 r -> e . v . A of T(A) are the eigenvalues o f A. 2.1. T preserves U, V such 1 or n o n - s i n g u l a r i t y , then that either T(A) = UAV T(A) = UA»V there exist non- - Proof. [T(l)] there If T preserves T 1 preserves exists that [T(I)] 3 • —i this 1 fora l l A in M n fora l l A in M n b y m u l t i p l y i n g on t h e l e f t s e c t i o n we transformations r x r principal a group. first T on which there the trace with zeros i f r > preserve of every are linear but which i s a p r o j e c t i o n onto a l l matrices show t h a t subdeterminants If r=l, preserve example of 1 by T ( l ) . . linear which = UAtT Preservers In form T(A) follows 3 °f [ 3 ] either 1 the corollary the _ 1 By T h e o r e m [ T ( I ) ] " * T ( A ) = UA'U" and E non-singularity, the transformation eigenvalues. a U such or 6 - have t h e sum o f member o f transformations no the subspace a t some 2 the inverse. of fixed An consisting set of o f f - d i a g o n a l positions. If of then 3.1. * we n If r > ~ Suppose shall Lemma diagonal 2.1 A entries. r r (A) t o d e n o t e (A) = E T ( A ) f o r a l l A r v r T(A) = 0 , r E and E ( A ) = E ( T E ( A + X) = E ( T ( A r write t h e sum subdeterminants. 2 and E T i s non-singular Proof. By i s in M the r x r principal Lemma " '' (1) A A r 0. _ 1 (A)). Then + X)) = E ( T ( X ) ) i s similar in M n r to a matrix = with E (X). r a l l non-zero - 7 - Let X = ( x . . ) be d e f i n e d a s x,, = x i=l,...,r-l x. . = 0 ii x. . = ij i= r,•..-n ' ' 0 i < j x. . = -(P IJ where By r— P "*"AP h a s a l l d i a g o n a l (l)E (P"" AP must entries entries be z e r o . y i > j i j J entries different Therefore we from zero. the coefficient of c a n show assuming diagonal n-r+1 t h e same that n or 2 < e n t r i e s which sum e x c e p t that a " B S Then includes B replaces = by c h o o s i n g X diagonal To to zero. show T 1 Thus E_(Y) It follows = E T(T transformations form a group. such a group We E - 1 r easily 4» f (X. Then, B, a n d from = E T r lemma preserve This a n d T has an inverse. : this now p r o c e e d f o rr > b e t h e sum o f a and excludes be z e r o , (Y)) from which l e t e n t r i e s o f P "''AP a r e e q u a l , A must preserves r , l e t a,B be a n y 0 a = 8, a n d a l l d i a g o n a l equal linear Thus, (n-r) + 1 t h e sum o f a n y ( n - r ) + 1 n-r+1 < s and o f P "**AP. e n t r i e s o f P -^AP. diagonal have i s t h e sum o f t h e l a s t o f P "*"AP i s z e r o . Now, two This on t h e d i a g o n a l suitably, we AP) , . 1 x Sp + X ) = 0. 1 _ 1 _ 1 (Y). that t h e s e to f E , f o r some to determine r > 2, the s t r u c t u r e of i s the content of - Theorem 3.1. If E E for (A) = T(A) X in some a l l A r M such n By for some that in M r > 4/ Lemma UV a deg(det(xA UAV for a l l A in M T(A) = UA»V for a l l A in M = e I where we rank Let A £ B)) < 1 of 0 rank be a for can has eigenvalues i t cannot that A eigenvalue than deg T exist U, V n n which preserves first which to B A most in one follows member in M four of of M i f and n E^, lemmas E^, preserves matrices at 1 at Jordan rank n . for 1. Then only i f superdiagonal If A entry. immediately. There position ( l , l )being form. non-zero have more ( l , l )p o s i t i o n . superdiagonal. As n-1, 277") . Our fixed is zero the at A this i n the A assume eigenvalues greater r < A 1. We on 1 any zero where a transformation 1 then the (mod is non-singular. linear + r 6 =a. 0 know t h a t rank be there = maps m a t r i c e s Proof. , then 4 < T(A) 2, r > 3.2. of that either Lemma 3 « 1 show t h a t such It and to - J. or is for 3 at mean is If A If A one. cannot (1,2) also would than be the of has has a 1 and on a rank a l l non- Assume impossible (2,2); that is this the without 1 of elseA 1. has det(xA only one non-zero + < for B) 1 a l l entry B. i t follows 1 easily is - In t h e other has a t most one the s e t of n o n - z e r o A i s i n Jordan 9 - direction, non-zero we first eigenvalue. e i g e n v a l u e s A,, form with Suppose A , ...,A,. e i g e n v a l u e A.. 1 Choose B d i a g o n a l w i t h (i.. ij). f (ij/ij) Then j=l#"««#k and d e t ( x A , ++ l's 0*s k > in x, 1, det(xA and so A has We diagonal TT B) B) = entries such If there form of that 1. Let one 0 and zero. there with that in position j at positions (\. x) - ( TT // X. )x j=l j j non-zero eigenvalue show t h a t k Let i be Q at p o s i t i o n the B = n > 2 and (b,.) and be X* largest (i ,i Q (1,2-) of t h e e i g e n v a l u e s X,0, 1 a l l super- i s a 1 at p o s i t i o n i s a zero of a m a t r i x rank a t most are has w o u l d have d e g r e e g r e a t e r t h a n assume X t now integer n=2, + B) and A elsewhere. j=l If show t h a t A +l). Jordan is clearly defined as follows b . . = ii 0 b. , = 1 i£ 11 l o b,, i =i 1 = 1 i o i£ o i +1 o +1 +1,1 ' o = 0 elsewhere. Then det(xA If X t must be a zero 0, at + B) in order = -A.x - t h a t deg ( i , i +l). x. det(xA Repeating + B) this < 1, there procedure, we can if 7V. * show t h a t 1. that 10 - are no l*s above the diagonal, 0. Now is there - a s s u m e A, = Once there again, is a 1 0 and let at that i be the the position (1,2) entry largest integer ( i ,'i +l) . o' o If i B = (b..) such > 2 as b.. ii = 0 i = l , 2 , i +1,i . ' * o ' o b ^ = 1 elsewhere b. . 1+1,1 o b A o v define of 21 the diagonal. =1 . ' on o - 1 = 0 elsewhere off the diagonal. Then det(xA In 2 < i < cannot this n-1, be a way, can 1 (2,3) = shown implies that the Lemma 3.3 and A £ for a l l B 0 a in Jordan thus A Let r an M , 3 that and fixed x . 2 in positions be ] / I A n A 3 show t h a t there ) . deg(det(xA form i s of member then To (i,i+l), set d i a g ( E have entry, = eliminated. We zero B) a l l l*s be at B B + of rank integer of M^. has at A + B)) has < at 1 for a l l most one non- 1. such If most that deg one 3 < r < E (xA + r non-zero n, B) < 1 eigenvalue, - Proof. Assume A A,, , . . . .A, . 1 ' n Then 11 - i s i n Jordan form with Let B = diag (z...... z ). \ 2 ' n f E (xA + B) = r TT S /. . \,_ „ w= ( 1 . . . . . 1 We have r ;eQ r k=l with k S x k — t =0 i s t h e sum t members and S over TIV a l l subsets set. + z. ) k k • TTox* —> a e s " S t ^ w i s t h e empty q (xA.. k=l TT (XA.. + z. ) - *y c v where eigenvalues Bew-S? of w = We ( i c a n now l t . . . , i ) r write r t=0 and for t > a - t y Z w£Q T for 2, any c h o i c e This we have x rn ^ weQ y * S,<Zt — TT v TT z = ^ X ^ ve<2„ .„ r^-tn From symmetric this = o P . "TT TT t - z c a n be c o n s i d e r e d z. ;^ a s f o l l o w s : s z BEW-S a PEW- w of z.,....z I n sum rn aeS . S. <ZW t - ^ we pev P "N ^ - Z B ( ^ as a p o l y n o m i a l TT x ; TT y * N s <S(Q - v ) t nn v can conclude i n the aeS. a 1 7 that the t - t h elementary f u n c t i o n o f a n y n - ( r - t ) o f t h e A.^,...,A. n i s zero. - 12 - In particular, the second of any n-(r-2) o f t h e A»'s i s z e r o . are equal and that k they are equal n'-(r-2) < = n-(r-2), we We Then, can choose r > a set of i . 3 i f a l l t h e A,*s 3 and A, £ A.^ c setting eigenvalues d for j=l,...,k-l. 3 have 0 = E ( A . ,A.. ,...,A, ^ l 9 c 1 x k _ ) = A. E ( A , . i c i i 1 + E 1 0 ,...,A. i k 1 ) 1 k - l = E (A,,,A,. ...A. ) = A-.E^A.. , ...,A., l k - l a ± i i ) 9 x 1 1 k + E (A. - 1 , ... ,A. ? 1 A then ) - (A.. , . .. ,A. * We function Assume t h a t since i . £ c, 1 symmetric Now to zero. n. A.. ,...,A,. where l . k - l x elementary ) k-1 have (A. - A, ,) E.. (A.. , . . . ,A.. ) = 0 - c d 1 i, ' ' l , ' 1 If A,^, r > i£ c , d 3.1. Now 3 then n k-1 < are equal we k-1 n-2 to zero and t h e e i g e n v a l u e s by t h e argument o f Lemma can w r i t e A, A, , = 0 e d and i t follows that A can have a t most one non-zero eigenvalue. If r = 3 we have that the second symmetric function A. of any ( n - l ) o f t h e A., i s z e r o . j E . (A,, , • • ., A,. , A.. . , . . . ,A, ) i i * j - 1 j +!' ' n L e t E.(A.,) i j i= l,2. ' We first denote show - A, j E ^ (A, j ) j=l,...,n is 13 - zero. A A. E (A. ,... A. ) 2 1 / = Tv-.E^A,..) + n E (A..) 2 = A.. E . (A.. ) J 1 J Summing these n n E equations 2 ^ 1 ' * *" = 2 2 ^ 1 ' *' * E A Since n > 2 E (A. ,...,A, ) *~ l Setting j=l,...,n. = 0 and A,.E.(A..) = n X j 0, j A.. j=l we have A. J = A.. s J 2 A,. (Tv.. J J Thus the non-zero were more function than of completes Lemma 3.4. integer for such a l l B Proof. rank any 1, in M We A A n+3 can has principal function. case that 0 non-zero n-1 Let = e i g e n v a l u e s of one any the s) of t h e A, s fixed r < n + 3 . i f and a t most A one only deg If there second not member Then i f A be symmetric zero. This non-zero E^(xA of + B) „ a n d r an n+3 d e g ( E ( ' x A + B) < 1 i s of i s i n Jordan subdeterminant Thus would T a assume equal. eigenvalue the 0 be r < are r=3. that £ A + < 1. B) M r rank form. entry. (xA of 1. If A i s of It follows i s a t most a that linear - 14 - We now p r o c e e d by i n d u c t i o n on t h e " o n l y S ( l ) i s t r u e by lemma 3 - 2 . of t h e lemma, S ( n ) . be E r r in M ,. n+4 i f * part Let A We have an r s u c h t h a t 4 < r < n + 4 — (xA + B) i s l i n e a r ' • in v x and — f o r each B i n M .,. n+4 If = n + 4# A i s o f rank 1 by lemma 3 « 2 . If zero and r < n + 4# by lemma 3 - 3 A has a t most eigenvalue the (2.3) A.. In A t a k e A. t o be i n p o s i t i o n ( l , l ) t o be e ( = l o r 0 ) . entry positions (3/2) and r - 3 positions ( i , i ) i > 3, l * splaced with one non- zeros L e t B have a 1 a t i n any o f t h e d i a g o n a l i n a l l remaining positions. Then E (xA + B) = -A.- £• x r I f A, ± 0 and £ = 0 , row and column 2 o r A. = £ = 0 , we have t h a t o f A have o n l y entries. B to those matrices with column, we s e e t h a t t h e h y p o t h e s e s o f S(n+3) Therefore zero zero the second I f we restrict e n t r i e s i n t h e s e c o n d row and the submatrix of A o b t a i n e d are s a t i s f i e d . by d e l e t i n g row and column 2 o f A i s o f rank 1 and hence A i s a l s o . I f A. = 0 , we show f i r s t at ( 1 , 2 ) where £ ^ i s t h e e n t r y Let B be d e f i n e d t h a t A cannot and the entry as f o l l o w s : b h 21 i ± b.. = b n+4,n+3 =1 = 0 3 < i < r-2 elsewhere. have £ ^ = £,-, = 1 at (n+3,n+4). - + B) = e^c^x Then E ( x A r Then t h e i n d u c t i o n the 15 - and we pan assume argument applies ~ °- as b e f o r e t o g i v e r a n k of A as 1. Lemma 3 . 5 . I f f o r some r such that 4 < r < n E T ( A ) = E (A) f o r a l l A i n M t h e n m a t r i c e s o f r a n k r r n a r e mapped t o m a t r i c e s o f r a n k Proof. L e t A be o f r a n k 1. lemma, d e g ( E ( x A + B) < 1 E T(A) = E (A), deg E^(xT(A) r r r non-singular, T(A) i s of rank Lemma 3 . 6 . for this Proof. Let 1 by T. Then by t h e p r e c e d i n g f o ra l l B i n ^ . + T ( B ) ) < 1. r such t h a t 4 < r < n n A be o f rank r . We can w r i t e i =l where C. i s o f r a n k T exists Since T i s 1. r Since Since h o l d s f o r a l l members o f M , and t h u s n' I f f o r some a l l A in M 1 1 = 1, • • .r,. 1 Then r r i =l i =l and p r e s e r v e s E we have - We of this rank are now in section. we have By T write, has for position to prove c o r o l l a r y 2.1, the since main T theorem preserves either or Since a - 16 an T(A) = T(A) = UA'V. V inverse a l l B in UAV U,V are non-singular and we can M n E (U(VBV~ )V) = 1 r Setting UV = P our E^VBV" ) condition E (PB) = r If will = or tr(C (P) I) on of be the C (B), r r r - r different entry E (B) tr(C (PB)) t I) from we can zero. C^CB) the 0 is 1 matrix of which set we the P = e l 6 I, For of theorem r6 that the r B such entries entries subdeterminant equal to 1. - conclude 0 ( i , j ) entry that are the at (j,i) zero. except is for B those (j,i) position of Then I) C (B)) C (P) = r r * 0. I which implies [4] ==0(27r) . later 3»1 we = a l l zero r contradiction C (B)) choose tr((C (P) By r a l l other with diagonal tr(C (B)) assume Then and E ^ (B) becomes or (C^CP) - = 1 as reference we state this part of the proof - 17 - Lemma 3.7. Suppose E (UAV) = E r where UV r = e 4• l 6 i s fixed I E In and M n used E 12 + for E E ° l s f section, , f o r some Then r a n we a linear 4. r > determined For r = l,2,3, In t h e c a s e s lemma i s not true. k 2 a n d E 3«4 3^ 12 E + necessary t r a n s f o r m a t i o n of 4 fails. F o r r=3, 34 n. conditions that for r > clear. r < 3" sufficient ing and 2 < the preceding .preserve in M r6=0(2ir). where E^, (A) f o r a l l A r E 34^ + B ^ the r=l,2, reason- this i s The 4 x 4 i s l i n e a matrix r a l l B i n M. . 4 The there question are linear r=l,2,3# subsets now arises i s whether transformations which and which We that are not d i r e c t are going to look preserve E f o r some r products. f o r such of t h e s e t of l i n e a r or not transformations transformations on M i n two . The n first of these which arise example is here i s the set of l i n e a r from permutations Hadamard That of entries i s the interchange the set of l i n e a r product transformations o f two transformations of every matrix with (c, . a .. ) The consisting some i s C : ( a . , ) -—> of matrices. entries. C S fixed An second of the matrix. - for a l l ( a . ..) i n M C ( c , .) h e r e . 13 ^ is C 1J a member formations E^: Here preserve n of is a . We any S w h i c h the trace. ( ^j)h E^: Lemma c a s U.l. linear now on the set of pairs at a subset preserve leaves involves entries on t h e m a i n f o r any C two l T entries of S . such order have moves two that the entries leaves Eg w h e r e ir^ 7Tg i s a permutation a n d 7T^, entries, located positions, entries. on a d i a g o n a l zero matrix at ( i , j ) , entries that SD be zero. these on t h e m a t r i x a In diagonal Thus S E,. + E,.. that located positions. ( j , i ) t o some entries. with 7T^ . entries l o c a t e d p o s i t i o n s and then fixed D on t h e d i a g o n a l . i t i s necessary l ' s at symmetrically symmetrically or S preserve will that the 31 i j In ( c , ,) 1 j diagonal = ^-^^^ located of the diagonal the effect S only, i t i s necessary s and r e m a i n i n g a permutation Consider Eg of S n 1. Eg then entries the e f f e c t S preserve with holds located on M that E^, Eg. of symmetrically symmetrically Mote to characterize the trans- entries T h e same l * s and t h e r e m a i n i n g matrix proceed of diagonal Consider that ( c , ,) i n M . ij' n of the set of symmetrically interchanges order v transformation If S preserves a permutation Proof. fixed a l l diagonal is two a n d some C and S which fixed 18 - other either s ( .jj E + Thus pair ^j ±~) S of interchanges - We following can w r i t e - 19 any m a t r i x (a,.) = A i n M as t h e 1 j n sum A = D + y P, . 3 <_- a. i where < D = diag(a j ...,a 1 1 # ) nn ^ 1 1 P. . = lj ( a . .E, , + a . ,E . . ) Then <-—^ ij. SA = SD < 3 i sp.;. i The effect < o f S on P., j i s known. We have either S ( a . , E . . + a,.E..) = a,.E.-, + a . . E . ij i j ji ji i j kZ j I -ik f or S (a . . E . . + a..E.,) = a , . E, „ + a, J . , ij i j j i] i j i kl 13 -Zk We can c o n s i d e r t h e e f f e c t following order. ^ i S takes o r n o t , d e p e n d i n g on S. \ < J S P ij a s ' l' 2^ w r ^ i j ) P i < j to take the p a i r (k,-?) , (-2, k) and t h e n to a^, First of S on P,. place of e n t r i e s interchanges Then we a^ i n the at and can w r i t e where T 2 is a permutation - on the s e t of p a i r s interchanges depending 20 - of symmetrically or leaves on t h e e f f e c t fixed located entries th.e e n t r i e s o f S on P ^ j - Then, effect o f S on t h e d i a g o n a l effect o f S on t h e o f f - d i a g o n a l e n t r i e s in entries at (i,j),(j,i) since the i s independent we and have of the for a l lA M n S(A) Lemma Then 4.2. c.. = c,. ij jl c . , = — 1, ii Proof. A Suppose = V^^W^K C:(a^) and e i t h e r 1 =>(c^ja^^) p r e s e r v e s c , . = 1, , 1 i=l,...,n 1 Eg. or i=l,....n. ' L e t i , j be f i x e d i n t e g e r s between 1 and n. Define as f o l l o w s : a..=x.. n i l a . . = x., i j i j a . , = x . , J J JJ a. . = 0 kl Then elsewhere, E _ ( A ) = x..x,. - x..x.. and E „ ( c ( A ) ) 2 i iJJ i jj i 2 V = c..c.,x,.x.. n JJ n JJ V c..c..x..x... ij j ii j j i expression i i gives S e t t i n g these identically (c..c,, This a..=x,. Ji J i JJ - zero 1)x,,x.. us c , . c , , i i JJ i i JJ equal f o r any n we have the following x,.,x,.,x..,x.,. j j i j j i + ( l - c..c..)x,.x,, i J J i = 1 f o r any i f i . IJ J i = 0 - - As n > 2 we have the 21 - equalities c . , c , , = 1 i i 33 for any and that either now c, , c . , = k k 33 1 c.. ii product I This ir(i) s i n c e by l*s at the p o s i t i o n s s i n c e row L e t N be (i,7r(i)) j consists Let N have off diagonal positions. at (i,i)(j,j) f s at and I X by I 7T^, ir^,ir^ exhibiting, non- matrices l e m m a 3-7 a by direct ( j , j )to position (i,j) and On s u c h t h a t 7T ( j ) a permutation of at (j,i) (i^,ij). = j and matrix with T h e n ir^U' i s for i=l,...,n. S u p p o s e ir^ m a p s e n t r i e s l at 7T o f l , . . . , n i for a l l i £ j. singular i=l,...,n. preserves non-singularity. i s a permutation ^ 1 i s done S u p p o s e TT^ m a p s t h e e n t r y There o r c . , = +1, I a r e mapped t o s i n g u l a r i s sufficient preserving I c , , = c , . = c,, the transformations N which This ,7r imply i=l,...,n, products. singular matrices iri» 2 3' These = -1, show t h a t not d i r e c t 7r =1 i ,j ,k distinct. We are c . . c. , n kk zeros. (i,j) and (j,i) zeros the diagonal l * s elsewhere. Then to (k,J?) i n the remaining l e t N have ^ s C^,k). zeros singular and N i s non-singular. S u p p o s e 7T leaves fixed 1 those interchanges at (k,^), entries C^,k). at (i,j), (j,i) and I t i s not d i f f i c u l t to - find a non-singular N 22 - s u c h t h a t 7T.. N i s s i n g u l a r . 4 1 7T^ i n t e r c h a n g e s t h e e n t r i e s a t (.1,2) ( 2 , l ) i n M„ or M 3 For example, and holds fixed those E + E„„ + E, „ 4J i s non-singular £j 0 0 We suppose proceed to at (4,3) (3,4). construct 7T. E 1 and an Then N in M Then, none set mutation 2 and P equal rows of IT o f following equalities = k i = 1 j = k j 1. + E.. ^ + E ^ l,...,n leaves such identity Similarly we can find We now show that the 7T» 1 where We R have = PQ, that maps tr % such fixed 4 x 4 non-singular singular. the find E fixed. derived 21 per- 1 and j , Then, by setting permuting t o 7T we V + a will the have + that = E 1 2 + E 2 1 • E 3 • 4 E ^ . transformation : A — ? R7T, ( A ) R ' 1 a non-singular interchanges holds + E can 7T i n t e r c h a n g e s according QPir^P.Q. We k numbers " 12 a Q = E^, . matrix matrix ^ i V + that remaining to the permutation the occur: i = E ^ i , and of the + E„. + <• 1 f o r n > 4. n Suppose = E., 14 singular. is E entries at matrix It follows the (3/4) C such matrix to entries (4,3)• that a at singular (1,2) Then we 7T* d i a g ( C , I t h a t 7T^diag(C, I _ ^ ) n ( 2 , l ) and can n matrix. ^) find a i s non- i s singular. ' - If find, any o f t h e e q u a l i t i e s as b e f o r e , an R s u c h IT interchanges entries above to f i n d that above the modified hold we can transformation 1 at (1,2) the entries at listed : A—>R77\ ( A ) R » 1 1 the 23 - (3»l) (l»3)- We a non-singular ( 2 , l ) and h o l d s and can proceed N such fixed e x a c t l y as 7r'N i s s i n g u l a r . that 1 Thus and we have which A direct linear transformations are not d i r e c t C the matrix Then d e t A = A." product, preserves : = ^ with find l sense E 12 + E 2 3 + A E A. £ 1 3-7/ lemmas will linear in i s defined as 31 C i s not a i f a direct the absolute S which 3»1« ^* ^ = where t o lemma an E ^ - p r e s e r v i n g of theorem and i s not a ( a . .) e M d e t C ( A ) = 1. i t preserves transformations E^ elsewhere. n f o r according E^ °21 = 1 The f o l l o w i n g t h r e e cannot the 1 ^* diag(A,I _^) A preserves 5> ( c . a . . ) = E^ as f o l l o w s : ( a . .) c.. C maps C which is-given c^2 preserve products. transformation product S which value be u s e d direct product of the t o show t h a t transformation i s not a d i r e c t determinant product amongst in the we - 4.3. Lemma If A elements are r A Since determinant M zero, E Proof. e then = ( r r We also sum l's r r > i s of rank k > and r E (a r < r (A) < — 1 = I. of A has integer (det(X )) d over 2 r A 2 a l l r x r subdeterminants. j|C (A)|| = tr C 2 r (AA*) » t r C (A) = E r we a ) 2 = have 0 C (A*) r = r (a ,...,a ) r 1 n 2 E^(A) 2 < ( ). n So we continue: 2 ) The i f A J Z.—i.r = k < r ( 1 only r take remaining have 2 A the submatrix (X ) extends j>Jd (A) If and have E the n i f and x det where - for n > ) n every we has 24 ( * inequality of = E (a ,a ) 2 < 2 r )k" (tr(AA*)) X r = (l) is justified ( \ by ( \ )k~ E^(a ,...,a ) r 2 )k- ||A|| r theorem 2 2 r 5.2 of [5]« Since l|A|| we = n < ( * r have (2) E (A) r We (i) 2 r k of c o n s i d e r two = n. A are A of )k" r n r cases: i s a permutation modulus one. matrix and a l l eigenvalues L e t Tv., , . . . ,X, 1' ' n be the - 25 - eigenvalues o f A. We E which states modulus is one one. the r have that (A,, , . . . ,A. ) = 1' ' n ( r that a sum of ( ^ is ( ^ )• We show t h i s Set m = summands. ( ^ Since ) n ) complex numbers implies j would each summand ) a n d l e t a_. + i B . . , j=l,...,m a, a. < 1. j=l,...,m; J some each of < be 1 for jo imply m —i / „ a, < n. * J j=l Thus a . = 1 a n d B. J Now A. , c E r = 0, j=l,...,m. J we show A.^ b e a p a i r a l l the eigenvalues of eigenvalues. (A., , . . . ,A, ) we l ' n will have ^c^i two In t h e e x p a n s i o n Let of terms "" ' i X 1 1 are equal. i* r - l 2 ' 3=1/'.-/r-l J A, ,A,. • • - A,, d l I r-l N Since the eigenvalues equal t o o n e we have A, c Also, is A i s normal, diagonal. by A a n d t h e two t e r m s a r e cancellation = A. , d s o we have a unitary L e t A, b e t h e e i g e n v a l u e UAU" Since are not zero consists only 1 = \l of l T A s a n d O's U such o f A. = A.I that Then UAU"" 1 - (ii) If k < If k k > r = 1. jJ(A) = = I f E^ (A) so, from )k" Lemma 4.4. of be described or < on the as left that ( * ( » for a l l A e M S(A) of e n t r i e s at ( i , j ) and = ( * that Thus we by P and so P T that e l e m e n t s of S by consider N The where n > — any 4, pair ( j , i ) are either ) we must e n t r i e s on can have S ( l ) = I the main m o d i f y S by diagonal pre- and r e s p e c t i v e l y , where P i s a the transformation ——?-PS(A)P» diagonal modification n position implies diagonal. A the ) ) E^S(l) permutation m a t r i x chosen We < fixed. This post-multiplying leaves r follows. located In o r d e r lemma 4-3« remain n 0 symmetrically Proof. r If E„S(A) = E ( A ) J i interchanged by contra- (2) r can A = I which 2 E (A) S = n, 1. ( I and - n. k = 1 then dicts 26 fixed. We denote vr. the 0 A e f f e c t of T on the « diacf(0 _ , J ) n 2 wh e r e J = 1° matrix: such a - We wish 27 - t o show 7TN = N o We assume 7TN ( i ) 7TN Either ( k , p and or o * 1 a t some o has a 1 (k,.|) s u c h (k,,f) a t some (i)letD £ M _ n elsewhere such 2 S i n c e 7r l e a v e s d i a g o n a l rows o f 7 r d i a g ( D , j ) we can c o n c l u d e fixed can c o n c l u d e the e n t r i e s Now and zeros entries fixed, we at (n-l;n) We call 1 o f non- = 0 ( i ) cannot interchanges occur. (ii). or leaves (n,n-l). consider a matrix i s a P such (k,k), Therefore can e l i m i n a t e a l t e r n a t i v e t h a t ir e i t h e r elsewhere. a 1 at t h e number i s a t most 2. TN There and d i a g ( D , j ) = -1 3 that alternative In a s i m i l a r manner we Thus we (k < / ) Then E^ 7T d i a g ( D , j ) and that that k > be d i a g o n a l w i t h on t h e d i a g o n a l . E zero o (n,n-l). In n-3 O's has a £ N t (n-l,n) ( i i ) irN (kj) o o t N with this 1 that PN P» = N., o 1 l T s a t ( i , j ) and ( j , i ) matrix N^ and assume - 28 - Then TT(PN or o P») PN P»7r(PN P»)P o v This £ inequality with »7rN » follows: o alternatives The t o N If there o "P»7r(PN P )P«». irN^ - Lemma 4.5. — £ N to the P»7r(PN P')P o or P» ' l e a d s us r e p l a c e d by o = (i) ( i i ) conclusion w o 1 exist U,V ' in M such n that S(A) = UAV for a l l A in M S(A) = UA»V for a l l A in M either n or n then U,V are permutation Proof. matrices. C o n s i d e r i n g members of M described i n the i n t r o d u c t i o n we representation o f S as direct of V product f and U must be an Then e v e r y e n t r y of element, which, V i s the one row same. or Thus U = A,P and and i n t u r n , i m p l i e s every Q contains two from the Thus one direct matrix. or non-zero 0 s T and l T s . product Since zero. the this element of elements i n of V and 1 V = A. "*"Q where P i s a p e r m u t a t i o n only the i n v e r s e of non-zero as matrix a permutation U must be column so w o u l d t h e column v e c t o r s matrix. entry d i f f e r e n t If U contained 2 must have t h e a permutation Suppose V c o n t a i n e d non-zero n as n U. matrix direct product - of V no rows o r columns o f z e r o s . or column 29 - and U i s n o n - s i n g u l a r , T Q i s non-singular I f Q h a d two l * s i n any row so w o u l d t h e d i r e c t Q i s a permutation and c o n t a i n s product of V a n d U. T matrix. If E^S(A) = E ^ A ) f o r a l l A i n M Theorem 4.1. Thus n + 2 then either (i) or S ( A ) = PAQ ( i i ) S(A) = PA»Q where P, Q a r e p e r m u t a t i o n Proof. n=l , a By By i n d u c t i o n on t h e s t a t e m e n t E^(A) f o r A i n linear transformation corollary which preserves 2.1 we have a U, V i n or If Thus S becomes the determinant. such that either = UAV S ( A ) = UA»V. f o l l o w s from As lemma 4«5 that t h e members we can w r i t e , of M diag E (TTC) Since fixed f o r any C i n ^ 7T 3 U, V a r e p e r m u t a t i o n matrices. i n lemma 4«4 we u s e t h e m o d i f i c a t i o n , 7T, o f S i n showing t h e i n d u c t i v e s t e p . of o f t h e lemma. i s the determinant. S(A) It matrices. and p r e s e r v e s n + 2 diagonal (0,C) = diag 3 = E (diag(0,C)) = (0,TTC) = E^tT E (c) 3 elements E „ , by lemma , = E (diag(0,irC)) 3 1T l e a v e s diag(0,C)) 4«4 - 30 - By the induction hypotheses, 7TC or = PCQ 7TC = P C Q . However, s i n c e v l e a v e s diagonal we have 7TC = C o r 7TC = C and elements f i x e d . t Similarly P = Q = I we can w r i t e IT d i a g ( D , 0 ) = d i a g ( 7 r D,0) and c o n c l u d e 7rD = D o r 7rD = D . for notational NOW s e t m = n+2 and, T convenience only, such that a - . = a. = 0. ml lm follows: as C : C Ds 12 d 3 = 2 ° C =0 i j d d.. = a. . d., ii = a.. i i = 2 a 3 l e t A be any member Define i +l j+1 - a 3 C = (c,,) 1 j of M m and D = ( d . . ) Ij i,j=l,...,m (i,j)£(l,2) 3 j=l,...,m i=l,...,m ' ' d.. = 0 elsewhere, i j Now (0, C ) + diag A = diag (D, 0) and 7TA = IT d i a g ( 0 , C ) = diag By lemma 4«3 leaves e (0,7T C ) + d i a g C = D* . ~ (TT D , 0 ) . e i t h e r 7T i n t e r c h a n g e s know t h a t them f i x e d . °12' 21' 7TD w + TT d i a g ( D , 0 ) In t h e f o r m e r a n c * s l n c e case, a s i n c e TT TT i n t e r c h a n g e s d 2 3 3 2 ,a 2 3 or interchanges and 31 - - Therefore 7TA = d i a g ( 0 , C » ) + d i a g ( D » , 0 ) = A» in t h e same way t h e s e c o n d a l t e r n a t i v e TA = A. With the exception (l,m) case the e f f e c t with of the e n t r i e s of w i s determined. the test matrix E 1 0 12 I f 7T does n o t i n t e r c h a n g e and interchange a change interchanges interchange the entries • We eliminate this ., ., + E , „. m+1,1 m+1,2 n a t (1,2) and ( 2 , l ) a t ( l , m + l ) and ( m + l , l ) we 1 t o 0. the entries conclude a t ( m , l ) and + E„ + E 2,m+l the entries the e n t r i e s i n E^ f r o m l e a d s us t o A similar change o c c u r s have i f ir a t (1,2) and ( 2 , 1 ) and does n o t a t ( l , m + l ) and ( m + l , l ) . Thus we can c o n c l u d e that either 1TA = A 7TA = A' or and from t h e d e f i n i t i o n o f TT t h e t h e o r e m We now s t a t e Theorem 4.2. =s are and p r o v e I f E3 _ C ( A ) = E3( A ) f o r a l l A i n Mn t h e n 0 d i a g o n a l U,V i n M C(A) We w i l l integers i s proven. so t h a t = UAV for a l l A in M u s e Q^n t o d e n o t e t h e s e t o f 3 ~ " t P l s u w = (i^ there i ^ ) where 1 < i ^< i 2 < i ^< n e n of p o s i t i v e 32 - is to t o be subspace zero the e n t r i e s of t h e 3 positions w = the (i., J- i _i ), Lemma 4.6. 3 3 0 exist h : A w . i n d u c e d on L n . The 2.1 R L w by t o M„ • J> such that 3n there = det hC(A) 0 that > h" (U h(A)V ) W W 1 have d e t h ( A ) transformation : h(A) > (h)C(A) and p r e s e r v e s d e t e r m i n a n t . there exist U , V W W *r U : h(A) such Thus, that h(A)V W which equal to the Then f o r a l l w £ Q W i s linear corollary from a Hadamard t r a n s f o r m a t i o n E R setting submatrix determined S i n c e C p r e s e r v e s E^ we a l l A in L from complementary isomorphism diagonal U V i n M. s u c h ^ w w 3 Proof. by the f o r a l l A in M C for A in 3 principal x L e t C be E_C(A) = E A resulting of e a c h and 0 j> of - W implies C(A) = h _ 1 (U h(A)V W We have y e t t o p r o v e to do this following we that restrict our ). W U , V are d i a g o n a l . attention t o R and In o r d e r employ the notation R U : h(A) = U. W W = (a..) V = 7>(r..a..) V. i,j=l,2,3. - We h a v e , where U^"^ 33 - i s t h e i - t h column o f U, Since R : E. . >• r . . E . . 1 1 1 1 11 we have v..U ii - r..e n ( i ) v..U > Ij ( i ) Now r . . £ 0 as C i s n o n - s i n g u l a r ii holds u, . = 0 hi h £ i v.j j * i . f o r any i = 0 j * i and so y This =0, (i so U, V a r e d i a g o n a l . We a r e now i n a p o s i t i o n t o p r o v e t h e o r e m 4*2. It i s sufficient t o show ( c . . ) i s o f r a n k 1. I j every row. F o r , then ' row o f ( c ^ ) i s a m u l t i p l e o f some row, s a y t h e i Q I f we r e - l a b e l t h e e n t r i e s o f t h e i - t h row a s o d, = c, , j =1,...,n o J J then c . . = c. d. U l j i /= i o where c. i £ i i s t h e m u l t i p l i e r . 1 o I f we choose U = diag(c ,c 1 2 < ...,c,_ V = diag(d ,d ,...,d ) 1 it follows that 2 for a l l A in M C(A) = UAV. n n l l l,c, + 1 ,..,,c ) n - 34 - In 2 x 2 o r d e r t o show ( c . . ) i s o f r a n k submatrix matrix i s singular. ; 2 can a 3 x 3 principal and ( ^Pj)« y a four 2 or l e s s . as a p a r t . b s e t {a^ a , B ^ ., B ] c o n t a i n s e i t h e r integers find We d e s i g n a t e t h e 2 x 2 s u b - rows a^,a,> and columns P ^ r P ^ involving The 1 we show e v e r y distinct I f t h e l a t t e r h o l d s we a r e a s s u r e d we y Suppose columns o f t h i s submatrix which i n c l u d e s = (y^ y^ y^) principal (cx^p\) i s t h e s e t o f rows submatrix. By t h e p r e c e d i n g lemma we know C where : Jy *h" [U 1 Y h(J )V ] 3 y i s the n x n matrix with l * s at p o s i t i o n s principal submatrix by y determined and z e r o s of the elsewhere; and Uy From t h i s = diag(u = diag(v i t follows c i Q a P 2 and t h u s If l p = sets v ,v^). x / 2 f o r some i ^ i Z U , J V , x 2 1 a l 2 c ^ i 2 j^ j 2 =u.v. l 2 D P C 3 X = D a B 2 J U . 2 V , i 2 j 2 (cx.^p\) i s s i n g u l a r . fa^ a ,B^,P } # 2 2 # contains four we c o n s i d e r t h e two p r i n c i p a l two 2 =u.v, l l ~ a a C that u ,u^) 1 # of d i s t i n c t distinct integers submatrices derived i n t e g e r s p, = [ a ^ , a , B ^ } , 2 from t h e V = [a.^,a B } 2/ 2 - 35 - By t h e p r e c e d i n g lemma, we C C where J : V d i a ^ ( v l 1 v 2 = u , n x C = n 2*V and f o r some n^ n ^ c m a B 2 f o r some Jv. l l U , a V , 2 X c l a »J # 2 1 i 1 1 = Xu . i i = 2 t t v. l l a C J = u' 1 v' m n j L o a B that l x U . V , 2 l X X 2 D 1 C R v* v,» ) . Vl a 3 V i t follows c this > h~ [U . h ( J ) V ] u» u» ) V this ) y 3 = diag(v» From 3 u ) = diag(u» = — • _ . - ! [ h ( J : J • = diag(u^ Uy From 3 have 2 = u* n c 2 2 s e t of e q u a l i t i e s v' m a_a c 2 a a 2 = u 1 n ; L = 1 v' T U T n V* g i t i s easy t o e s t a b l i s h that V i and t h u s (o.^Bj) i s s i n g u l a r . of Theorem 4 « 2 . This completes the proof - Theorems 4»1 and 4«2, 3.7 f a l l o w us t o c o n c l u d e C and S w h i c h p r e s e r v e 3«1. The q u e s t i o n which p r e s e r v e 36 - i n conjunction with that the only are direct remains: E^ d i r e c t lemma transformations products Are a l l l i n e a r as i n Theorem transformations products? C o n c e r n i n g Eg p r e s e r v e r s , we were a b l e t o e x h i b i t types of l i n e a r were n o t d i r e c t these transformations products. transformations exhaust the class Eg and w h i c h I t i s n o t known w h e t h e r o r n o t and d i r e c t of l i n e a r which p r e s e r v e d two product transformations transformations which p r e s e r v e E_. BIBLIOGRAPHY 1. M. M a r c u s A l l linear operators leaving the u n i t a r y group i n v a r i a n t , t o a p p e a r i n Duke Math. J . 2. M. M a r c u s and J . L . McGregor E x t r e m a l p r o p e r t i e s of Hermitian matrices. C a n a d i a n J . Math., v o l . 8, 1956, p p . 524-531. 3« M. M a r c u s and B.N. M o y l s L i n e a r t r a n s f o r m a t i o n s on a l g e b r a s of m a t r i c e s . C a n a d i a n J . Math., v o l . 11, 1959, P P . 61-66. 4« J . Williamson 5- G.H. H a r d y , J.E. Littlewood, and G. P o l y a M a t r i c e s whose s - t h compounds are equal. B u l l e t i n of t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y , v o l . 39, •1933, P P . 108-111. I n e g u a l i t i e s . 2nd e d . , Cambridge U n i v e r s i t y P r e s s , 1952.
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Linear transformations on matrices. Purves, Roger Alexander 1959
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Title | Linear transformations on matrices. |
Creator |
Purves, Roger Alexander |
Publisher | University of British Columbia |
Date Issued | 1959 |
Description | In this thesis two problems concerning linear transformations on Mn, the algebra of n-square matrices over the complex numbers, are considered. The first is the determination of the structure of those transformations which map non-singular matrices to non-singular matrices; the second is the determination of the structure of those transformations which, for some positive integer r, preserve the sum of the r x r principal subdeterminants of each matrix. In what follows, we use E to denote this sum, and the phrase "direct product" to refer to transformations of the form T(A) = cUAV for all A in Mn or T(A) = cUA'V for all A in Mn where U, V are fixed members of Mn and c is a complex number. The main result of the thesis is that both non-singularity preservers and Er-preservers, if r ≥ 4, are direct products. The cases r=1,2,3 are discussed separately. If r=1, it is shown that E₁ preservers have no significant structure. If r=2, it is shown that there are two types of linear transformations which preserve E₂, and which are not direct products. Finally, it is shown that these counter examples do not generalize to the case r=3. These results and their proofs will also be found in a forthcoming paper by M. Marcus and JR. Purves in the Canadian Journal of Mathematics, entitled Linear Transformations of Algebras of Matrices: Invariance of the Elementary Symmetric Functions. |
Subject |
Matrices Transformations (Mathematics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080636 |
URI | http://hdl.handle.net/2429/39915 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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