Some Generalizations of Commutativity for Linear Transformations on a Finite Dimensional Vector Space Citation Gaines, Fergus John (1966) Some Generalizations of Commutativity for Linear Transformations on a Finite Dimensional Vector Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PBNM-BZ75. https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548 Abstract Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, Ɛ L . Let A i+1 = A i B - BA i , i = 0, 1, 2,…, with A = A o . Let f k (A, B; σ) = A 2K+1 - σ 1 A 2K-1 + σ 2 A 2K-3 -… +(-1) K σ K A 1 where σ = (σ 1 , σ 2 ,…, σ K ), σ i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f n (A, B; σ) = 0 if σ i is the i th elementary symmetric function of (β 4 - β s ) 2 , 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β 4 are the characteristic roots of B. In this thesis we discuss relations involving f k (X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f k (A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X 1 , X 2 …X r belong to the radical of the algebra generated by A and B over F , where X i has the form f 2 (A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L , if the characteristic of F does not divide n and if there exists σ so that f k (A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g k (w; σ) = w 2K+1 - σ 1 w 2K-1 + σ 2 w 2K-3 - …. +(-1) K σ K w over F . We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231]. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Taussky-Todd, Olga Thesis Committee: Unknown, Unknown Defense Date: 4 April 1966 Record Number: CaltechTHESIS:09222015-114033548 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09222015-114033548 DOI: 10.7907/PBNM-BZ75 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9165 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 25 Sep 2015 16:46 Last Modified: 28 Feb 2024 17:24 Thesis Files Preview PDF - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

SOME GENERALIZATIONS OF COMMUTATIVITY FOR LINEAR TRANSFORMATIONS ON A FINITE DIMENSIONAL VECTOR SPACE Thesis by Fergus John Gaines In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasa dena, California 1966 (Submitted April 4 , 1966) -ii..;. ACKNOWLEDGl 'iENTS I wish to thank my thesis advisor , Dr. Olga Taussky Todd, for suggesting much of the area of research discussed in this thesis. Her advice and encouragement have been . invaluable to me. My special thanks go to Dr. E. A. Bender for much helpful criticism and advice, especially in connection with Chapter IV. Dr . E. C. Dade also assisted with some ideas in the earlier stages of my research. Finally 9 I am indebted to the California Institute of Technology and to the National Science Foundation for financial support in the form of teaching and research assistantships. -iiiABSTRACT LetJ:'.. be the algebra of all linear transformations on an n-dimensional vector space V over a field ca=: and let = 0 ,1, 2, ••• , with A= A • . K o Let fk(A , B ; ~) = A2K+1 - '01A2K-1 + ~2A2K-3 - ••• +(- 1 ) bKA1 where 'o = ('o 1 , 'o , •• • ,'oK), 'Oi belong to ~ and K = k(k-1 )/2. 2 A , BE...£. Let Ai·+ 1 = A.B - BA., i i i Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-73~ showed that fn(A,B;'o) = 0 if 'oi is the illi elemen2 tary symmetric function of ( ~r - f3s) , 1 E;; r c::::s ~n, i = 1, 2, ••• ,N, with N = n(n-1)/2, where fr are the characteristic roots of B. In this thesis we discuss relations involving fk(X , Y;"o) where X,Ye:land 1~ k<n. We show: 1 • If °3"' is infinite and if for each Xc:t:. there exists 'o so that fk(A,X;'o) = 0 where 1~ k<n, then A is a scalar transformation. 2. If (Fis a lgebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products x1 x2 ••• Xr belong to the radical of the algebra generated by A and B over'T , where Xi has the form f 2 (A,P(A,B);~), for all polynomials P(x,y). We partially generalize this to the case where the blocks have dimensions~k. 3. If A and B generate..C, if the characteristic of "'.Fdoes not divide n and if there exists 'O so that fk(A,B;"o) = O, for some k with 1 ~k <n , then the characteri stic roots o:f B belong to the . . . '-) = w2K+1 - o w2K-1 + 'o w2K-3 - •• splitting field of gk ( w;v 1 2 • • +(-1 )K 'oKw over 'J='. We use this result to prove a the-0rem - involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull. , 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231]. -iv- TABLE OF CONTENTS Chapter Title Introduction 1 I On theorems of McCoy and Drazin 4 II The Kato-Taussky-Wielandt commutator relation 18 III The t wo-dimensional blocks 33 IV The relation fk(A,B;'b) Bibliography = O, when k<n 42 59 -1- INTRODUCTION Let X be the algebra of all linear transformations on an n-dimensional vector space V over a field -=.F- and let A,B s;l. It is well-known [9f. Jacobson, 4 pp. 120-121] that there exists a basis of V with respect to which the matrices of A and B are in block upper triangular form where corresponding blocks are of the same dimensions, the blocks on the diagonal are square and corresponding diagonal blocks cannot be reduced furt h er by a simu ltaneous similarity. Most of .this thesis is devoted to the problem of determining how certain properties of A and B are reflected in propertie s of the diagonal blocks. In [ 1] McCoy showed t hat , if Tis algebraically closed, all the diagonal blocks are one-dimensional if and only if P(A,B)(AB - BA) is nilpotent for every polynomial P(x,y) in the non-commuting variables x and y with coefficients in~. In Chapter I we prove a generalization of McCoy's theorem when the field cg=- is quite arbitrary. We also generalize a theorem of Drazin which is related to McCoy's result. To a ssist with the investigation of the diagonal · blocks we introduce what we call " KTW commutator expressions" fk(A ,B;n). Let Ai+1 = AiB - BAi , i = 0 ,1, 2, ••• , with A = A. Let k be an integer, 1 ~ k ~n,and K = k(k-1 )/2, 0 then define whe:re o = (0 1 ,-0 2 , ••• ,'t>K) with 'oi c;J= , i = 1,2, ••• ,K. We note f (A , B;o) =AB - BA. The Kato-Taussky-Wielandt (KTW) 1 commutator relation [cf. 1 2] then says that fn(A,B;b) = 0 if oi is the illi elementary symmetric function of (pr - fs)~ 1 ~r..:: s ~n, where Pr are the characteristic roots of B. In Chapter II we prove that, if '!F is infinite and if for -2each X £L there exists 'o so that, fk(A ,X;'o) = 0 for some k with 1~k<n, then A is a scalar transformation. We also show that if J=°is algebraically closed then there exists a b asi s of V with respect to which the matrices of A and B are in block upper triangular form, where the diagonal blocks have dimensions~k ? if, for each polynomial P(x , y), there exists 'o so that fk(A ,P (A ,B);'o) belongs to the radical J of the algebra generated by A and B over "J= • In Chapter III we characterize those A,B for which there exists a basis of V with respect to which the matrices have block upper triangular forms where the diagonal blocks are one- or two-dimensional. The necessary and sufficient condition we give is, essentially, that certain products x 1 x 2 ••• Xr belong to}, where Xi has the form f (A , P(A ,B );'o), for every polynomial P(x , y). 2 Now f 1 (A,B;'o) = AB - BA and the Kato-Taussky-Wielandt commutat or relation says that there exists ~ so that fn(A,B;n) = 0. So in Chapter IV we examine what happens if there exists 'o so that fk(A,B;'o) = 0 f or some k with 1 ~ k <. n. We show that if A and B generate J:, , if the characteristic of ~do es n ot divide n and if there exists 'o so that fk(A,B;'o) = O, for some k with 1:;;;.k<n, then the chara cterist ic roots ~r of B belong to the splitting field o f gk ( w ; 'o) over c:a=', where = . .. Moreover if k = 2 and B has at least two d i stinct characteristic roots then there exists an ordering ~1 ,p , ••• ,pr 2 of the distinct characteristic roots of B so that f1 - P2 = P2 - p3 = ••• = Pr- 1 - Pr satisfies g 2 (w;b) = O. Now f (A ,B;'o) = 0 means AB= BA and t his in turn implies A 1 and B have property P, which means A and B have property L [cf . 16]. Property L demands that xA + yB have all its chara cteristic roots of the form x~ + yp, for all x,yc;F. -3We employ the main theorem of Chapter IV to prove a result which says that if A and B satisfy a certain KTW commutator equation then they have a generalized property L. The results which are, perhaps, of most interest are Theorems 2.10, 2.11, 2.13, 3.1, 4.2, 4.9 and Corollary 4.6. Note: throughout this thesis the symbol O denotes the end of a proof. -4CHAPTER I ON THEOREMS OF McCOY AND DRAZIN. In this chapter we prove some the orems which generalize results of McCoy and Drazin on matrix commutators. Rational methods are used except in part of Theorem 1.9 and in Lemma 1 .12. Let A and B be linear transformations on an n-dimensional vector space V over a field cg:- • To set the stage we introduce the so-called "P-propertyrr and state the main theorem about it. 1 .1 Definition [cf. McCoy , 1] • If J= is algebraically closed then A and B have property P if and only if there exist orderings <Xi , ~, ••• , ~ and p1 , µ2 , ••• , f3n of the characteristic roots of A and B, respectively , so that the characteristic roots of P(A,B) are P(o<i'~i), i = 1,2, ••• ,n, for every polynomial P(x,y) in the non-commuting variables x and y, wi th coefficients in "3=' • Unless otherwise stated, " polynomial P(x , y) " in this thesis will always mean a polynomial in the non-commuting variables x and y with coefficients in "::F • 1 • 2 Theorem [McCoy, 1] • Assume ZF is algebraically closed. Then the following three conditions are equivalent. (i) A and B have property P. (ii) P(A,B)(AB - BA) is nilpotent for every polynomial P(x,y). (iii) There exists a basis of V with respect to which the matrices of A and B are in upper triangular form, i.e. all the elements below the main diagonal s are zeros. We aim to prove a generalization (Theorem 1.9) of this -5- ct:= theorem where we do not assume that the field is . algebraically closed. We shall need the well-known concept given in the following definition. 1 .3 nefinition [Sf. Herstein, 2, p.±J Let ti? be a ring, Man irreducible left'R-module, O(M) =[rt:</?: rM = (o)J. Then the (Jacobson) radical of ~ is () O(M), where this intersection is taken over all irreducible ~-modules M. The followi~g result is well-known • . 1 •4 Theorem [cf. McCoy, 3, pp. 113, 12QJ • The radical of Cl( contains every nil left (or right) ideal of~ • If ~ satisfies the descending chain condition on left ideals, then the radical of~ is nilpotent. In particular, this theorem applies if </?is a finitedimensional algebra over a field. Now let 5{ be the algebra of polynomials P(A,B) in A and B (including I, the identity transformation) and let~ be the radical of~. We denote AB - BA by [A,B]. 1.5 nefinition. A and B have property Q if and only if [A,B] f } · 1. 6 Lemma ~f . McCoy ? 1]. Let C e<f<.. Then Cc j.-if and only if P(A , B)C is nilpotent for every polynomia l P( x,y ). · Proof. If Ce} then P(A?B )C -e:j- f or every polynomial P(x,y). Hence P(A,B)C is nilpotent, by Theorem 1.4. Conversely, if P(A,B)C is nilpotent for every polynomial P(x,y), then the left ideal <f(.c is nil and hence is contained in}, by Theorem 1 • 4. Thus C e j- since I c ~. CJ We note that essentially the same result was proved in McCoy[1] in a somewhat different manner. -61. 7 Lemma. If A and B have property Q and Vis irreducible as a left~ -module , then AB = BA. Proof. [A,B] e }and hence acts as 0 on all irreducible '1\-modules. Hence AB = BA. O The following definition is a natural analog of (iii) of Theorem 1.2. A and B have property T if and only if there exists a basis of V with respect to which the matr-ic es of A and B have the following block forms 1.8 Definition~ ... A1t A22 ... A2t A11 A12 B 11 : ~ l 2 0 O B22 ••• B2t 0 0 • • • >B 1 t and 0 0 ' ... Att ••• Btt respectively, where Aii and Bii are square blocks of dimension ni' i = 1, 2 , ••• , tp and there are zeros below the main block diagonals. Also Ai iBii = BiiAii and the minimum polynomials of Aii and Bii are irreducible over~ , for i = 1,2, ••• ,t . We prove now the promised generalization of Theorem 1.2 . 1.9 Theorem. The following three conditions are equivalent. (i) A and B have property Q. (ii) A and B have property T. (iii) There exist orderings Ol1 , 2 , ••• , cxn and {!>1 , [32 , ••• , Pn of the characteristic roots (which are contained in some extension of ~ ) of A and B, respectively, so that P(A,B) °' -7has characteristic roots P(<Yi'~i), i = 1 ,2, ••• ,n, for every polynomial P(x,y). Proof. We shall prove that (i)=9(ii)~(iii)"""(i). Let A and B have property Q. Let :> v t+1 = (O) be an1\.-composition series for V. In the usual manner [gf. Jacobson, 4, p . 120] choose a basis for V so that the matrices of A and B have the same block forms as the matrices in 1.8, where Aii (resp. Bii) is the matrix of A (resp. B) restricted to the quotient space Vi/Vi+ 1 ' with i = 1,2, ••• , t. Now Vi/Vi+ 1 is irreducible as anR-module. Since A and B have property Q, the matrices Aii and Bii have property Q for i by Lemma 1. 7. = 1,2, ••• ,t. Hence AiiBii = BiiAii Now the minimum polynomial of Aii (resp. Bii) is irreducible over'J=" , for each i = 1,2, ••• ,t . Suppose the minimum polynomial p(x) of Aii is reduc ible for some i. Then p(x) = q(x)r(x) , where q(x) and r(x) are non-constant polynomials with coefficients inc;f". Let W = Vi/Vi+ 1 • Let w1 = {we W: r(Aii)w = o}. Now w1 is clearly a subspace of W and also Aii W1 C if r(Aii)w = 0 then w1 • Now Bii w1 C w1 since, because AiiBii = BiiAii" Thus w1 is an ~-submodule of W. But W is irreducible. Hence w = (0) or w = W. · Now 1 1 w1 I W, since, then, p ( x ) would not be the minimum polynomial of Aii" Hence w1 = (0). But then r(Aii) is nonsingular and, since 0 = q(Aii)r(Aii), we have q(Aii) = 0. This contradicts the fact that p(x) is the minimum polynomial of Aii" Hence p(x) must be irreducible. In a simila r fashion it can be s h own that the minimum polynomi- -8al of Bii is irreducible. (We note that the discussion of this paragraph is contained in Jacobson [4, p. 133] ). Thus A and B have property T, so ( i) => (ii) • Now suppose A and B have p roperty T. We have AiiBii = B11 . . Ai. i. , i = 1 , 2, ••• ,t. Let a c_ be the algebraic closure of J=". Let <f{i be the algebra of all polynomials P(Aii 9 Bii) where P(x,y) is a p olynomial in the non-commuting variables x and y with coefficients in S.-Then [Aii ' Bii] belongs to the radical of </Zi. Now consider Aii and Bii as matrices with elements in S-- and apply the fact that (i) => (ii) in \ ' Hence there exists a non-singular . the present theorem. 1 1 matrix Ui with elements in ~ so that Ui Aiiui and Ui Biiui are both in upper triangular form (i.e. there are only z eros below the main diagonals), i = 1,2, ••• , t. So, if A1 and B1 denote the matrices in 1. 8, there exists a nonsingu lar nxn matrix U with elements in J-- so that u- 1 A U 1 1 and u- B U are both in upper triangular form. The matrix 1 u- 1 A1 U (resp. u- 1 B U) has the characteristic roots 1 ~ , °2' •·. ,cxn (resp. 13 1 , p2 , ••• ,pn) of A (resp . B) on the main diagonal. Thus P(A , B) has characteristic roots P(~i ' ~i) , i = 1 , 2 , ••• , n for every polynomial P(x,y) in the n on-commuting variables x and y with coefficients in Then (iii) follows as an immediate consequence. We now show (iii) :::> (i). Assume (iii). Then P(A,B)[A, B]has only O as a characteristic root , and hence is nilpotent, for every polynomial P(x,y). Thus A and B have property Q by Lemma 1 .6. CJ 5-· 1. 1 0 Corollary Q'acobson, 4, p.13:D • A and B have property T. If AB = BA, then 1.11 Corollary Let A ahd B be nxn matrices with elements in ':F". Then each of the statements (i), (ii) and (iii) in Theorem 1.9 is equivalent to the fact that A and B have prop.e rty P, considered as matrices with elements in -9~, where ~ is the algebraic closure of -=a:- • Proof. If there are orderings °<i , ~, ••• , °'.h and ~ 1 , p2 , ••• ,pn of the characteristic roots of A and B, respectively, so that P(A,B) has characteristic roots P(~,~i), i = 1 , 2, ••• ,n , for each polynomial P(x,y) in the noncommuting variables x and y with coefficients in~ ' then this statement is clearly true for t h ose P(x,y) with coefficients in"F. Hence A and B satisfy (iii) of Theorem 1 • 9. The converse is contained in the proof given above that (ii)~ (iii) in Theorem 1.9. 0 We remark that if "'.:Fis algebraically closed in Theorem 1.9, we get Theorem 1.2. The following lemma gives an example of a pair of linear transformations with property Q. 1.12 Lemma. Le t A and B be linear transformation s on an n~dimensional vector space V over a field ".F , where the characteristic of '3=°does not divide n . If each polyn omial P(A , B) has only one characteristic root then A and B have property Q. Proof . All t h e characteristic roots of [A,B] are equal - equ al c , say . Thus 0 = trace [A , B] = nc. Hence c = O. Now P( A, B)[A , B] has all its characteristic roots equal, for each polynomial P(A , B). Bu t 0 is a characteristic root, since [A , B] is nilpotent. Hence P(A,B) [A,B] is nilpotent and thu s A and B have property Q. 0 McCoy ' s formulation of property Q is that P(A,B)[A,B] is nilpotent for each polynomial P(A,B). We raise the question here whether it is possible to assume P(A,B)[A,B] ; nilpotent for a smaller class of polynomials P(A,B) and still get property Q. Williamson [5] has shmm that if 7F' is the complex numbers and if A is non-derogatory (i.e. -10- if the minimum polynomial of A equals its characteristic polynomial) and if we assume h(A)[A,B] nilpotent for cer ta:i.11 polynomials h(A) of degree :!in - 2, then A and B have property. Q. We should at least like to bound the degrees of the P(A,B). If 'a=" is algebraically closed, then a careful examination of the proof of Drazin et al. [6] of McCoy's theorem (Theorem 1.2) gives the result that if P(A,B)[A,B] is nilpotent for each polynomial P(A,B) of degree ~ n(a + b - 1) - 2 where a and bare the degrees of the minimum polynomials of A and B, respectively, then A and B have property P. This bound seems much too large. We prove the following result for some . small values of n and conjecture its validity in general. Theorem. Let A and B be linear transformations on V of . dimension n(n~2) over a field J='so that P(A,B)[A,B] is nilpotent for each polynomial P(x,y) of degree :$n - 2 then for n = 2,3 the transformations A and B have property Q. Proof. Case 1. n = 2. We assume [A,B] is nilpotent. Hence either [A,B] = 0 or the rational canonical form If [A,B]= O, the result is true. matrix of [A,B] is ~] · 1.13 [g Otherwise replace A and B by matrices, again called A and There should be no conf usion causB, so that [A,B] = ~· [g ed by the new meanings for A and B. We. have A [A,B] = [A,AB] and also B[A,B] = [BA , B]. Hence A[A,B]and B[A,B] are nilpotent (s'ince each matrix has its determinant and its trace equal to zero). If X = (x . . ) then X(A,B] -- ~o x11J1J . 0 x21 and, if this matrix is nilpotent, x 21 = O. Hence if A= (aij) and B = (bij) we get a 21 = b 21 = O and thus the matrices A and B are in upper triangular form. Hence A -11and B have property Q. Case 2. We assume that (xA + yB + zI ) [A,B] is nilp o tent for a l l x, y, z e :Fi . and V is three- dimensional over~. As above assume that A and B are mat r ices s o that [A,B] is in r at ional canonical f o rm. Hence ~ci0 1 0 1OJ or 0 0 0 If [A,B] = O, the result is true. [A,B] = 0 or (.A,B] = rn IT00 0,0 1100 Assume 10 OJ1 0 0 • If X = ( xij ) , then X[A , B] = [~ ::~ :~~ X3~ 0 X31 • Now let X = xA + yB + zI, where A= ( a 1. J. ) and B = (b . . ). 1 lJ . Then X[.A,B] is nil potent, and thus [x 21 x 22 is nilp o tent. X31 X3~ Thus we have J x a 21 + yb21 xa22 + yb22 + z [ xa31 + yb31 xa32 + yb32 nilpotent. Thus both the trac e and the determinant are zero. Hence a = -a 2 1 and b = -b • Al s o 21 32 32 This gives a = b 31 = 0 and a 2 1 = b 21 = O. Thus A and B 31 are both in upper triangul ar f o rm, and p r operty Q fo l lows immediately. If [A,B] = [g 6 gl, then ~J 0 0 -12- = X[A,B] [~ :~~ ~] where X = (x .. ) = xA + yB + zI with A= (a .. ) and B = (b . . }. J.J J.J J.J Then X [A, If! nilpotent gives x = 0, which leads to a = O 21 21 and b = O. If both a and b are zero then A and B 31 31 21 are in block upper triangular form, and we can apply Case 1 to the 2x2 blocks to prove the result. So assume a Then l et b = ha • The (2,1) 31 31 31 element of [A,B] is ha a - b a and this is zero. 31 23 23 31 Hence b = ha 23 , since a 31 I O. Thus 23 IO. A = and B If a = O, then, by means of a permutation similarity, 23 simultaneously interchange the first and second rows and the first and second columns of A and B. The matrices obtained are in block lower triangular form and, once again, we can apply Case 1 to the 2x2 blocks to prove the result. So assume a 23 IO. a11b11 + ha13a31 The (1,1) element of [A,~gives - b1 1 a 1 1 - b 1 3a31 = o. Hence b13 = ha13' since a 31 [A,B] gives I o. The (2,2) element of a22b22 + a23b32 - b22a22 - ha23a32 = 0 and hence b = ha32 , since a 23 I 0. 32 gives The (3,1) element -13a31 b11 + ha33a31 - ha31a11 - b33a31 and hence b 11 - b = h(a 11 - a 33 ), since a 31 33 (2,3) element gives and hence b 22 - b = h(a 22 - a 33 ). 33 gives =O ,£ O. The The (1,3) element which is the same as which gives b 12 = ha 12 on applying the equation derived from the (3 , 1) element, and using the fact that a 23 Then the (1,2) element of [A,B] is IO. which equals a 1 2 [h(a 11 - a 22 ) expression equals (b 11 - b 22 )]_. This last which is zero, by the identities derived from the (3,1) and (2 , 3) elements. But the (1,2) element of [A,B] is 1. This contradiction shows that at least one of a and a 23 31 is zero and thus the result is true, by the arguments given above for these cases. O -14Drazin [7] has proved a generalization of McCoy ' s theorem (Theorem 1.2) when the field'3=°is algebraically closed of characteristic zero. We shall now generalize Drazin's results to an arbitrary field~(insofar as they do generalize). Our proofs, of course, are completely different. 1.14 Definition. where the union is over all linear transformations ck in ek. We remark that there are at most 2k- 1 linear transformations in ek, k :;::::.1. 1 .15 Definition. [Drazin, 7] • A and B have property Qk for some k~ 1 if and only if P(A,B)Ck is nilpotent for every Ck in ek and for all polynomials P(x,y). It is clear , by Lemma 1.6, that A and B have property Qk if and only if e k c }• Property Q1 is thus property Q. 1.16 Definition. [Drazin, fr. A and Bare quasi kcommutative for some k~1 if and only if Ck= fof. We shall need the following lemma which is due to Jacobson. 1 .17 Lemma.[8] Let C>l- be an associative algebra over a field~ and let x,y ce>t.. Assume that x commutes with [x,yJ_ and that x is algebraic over ":Fwith minimum polynomial of degree r. Then if "3=' has characteristic O or p ~r, the commutator[x,y] is nilpotent. Proof. Jacobson stated this theorem for characteristic ·O -15and there is a slight error in his proof (a fact which was pointed out by Drazin [unpublished]). We present a repaired version of Jacobson's proof. Let m(z) be the minimum polynomial of x over c.F. Then m(x) = O. The mapping x - x'= [x,y] is a derivation. Hence. m' (x)x' = O. Assume that k m(k)(x)(x•) 2 - 1 Hence m(k+ 1 )(x)(x 9 ) 2 k = o, for some k~1. k + m(k)(x)((x•) 2 - 1 )' = Multiply on the left by (x') [x,x~ = O. Hence 2k-1 m( k+1 ) (x) (x' )2 o. . and use the fact that k+1 -1 = o. Thus, by induction, k m(k)(x)(x 9 ) 2 - 1 = 0 fork= 1,2, ••• Now m(r)(x) = r!. • Hence r r r!(x•) 2 - 1 = 0 and (x•) 2 - 1 = O, since the characteri s tic of c.:F' does not di vi de r ! • Cl We apply this lemma in proving the following result. 1 .18 Lemma . Let "'J= have characteristic 0 or p :>.n = dim. V. Suppose A and B are quasi k-commutative for some k~1 and -16that V is irreducible as an 'R -module, then AB = BA. Proof. The proof is by induction on k. Assume ek = ~O~. If k = 1, then AB= BA and the result is proved. Assume the lemma is true for some k~1. Suppose £k+ 1 = {O~, k~1. Then ACk = CkA and BCk = CkB for all Ck in ek. Now each Ck in ek is of the form [A,Ck_ 1 ] or [B,Ck_ 1 for some Ck_ 1 in e k- 1 • Suppose X = [A, Ck_ 1 ~ 0 for some Ck_ 1 in k- 1 • The~ Xis nilpotent by Lemma 1.17. We see that v =XV is 1 an invariant subspace of V for both A and B, since A and B commute with X. Hence either = V or 1 = (0), since V 1 is an irreducibleq\-module. But v -f (0), since X ~ O. 1 Hence v = V and thus X is nonsingular. But X is nil1 potent and this gives a contradiction. Thus we must have [A,Ck_ 1 ] = o. By similar reasoning [B,Ck_ 1] = o. Thus ek = f and, by the induction hypothesis, this means AB ·;,;; BA. 0 J J v -e v 01 The theorem we now prove is our generalization of Drazin's main theorem[?]. 1 .1 9 Theorem. IfJhas characteristic 0 or p >n, then A and B have property Qk for some k~1 if and only if they have property T. Proof. It is clear (as in the proof of Theorem 1.9) that, if A and B have property T, we can find a basis of V so that the matrix of each P(A,B)Ck' with respect to that basis, has zero blocks on and belov1 the main block diag onal. Hence P(A,B)Ck is nilpotent for all polynomials P(x,y) and for each Ck in ek. Thus A and B have property = 1 '2' ••• Conversely, let A and B have property Qk for some k~1. Use the composition series argument of Theorem 1.9. Qk' k Then Aii and Bii of that theorem are quasi k-commutative here, and, since Vi/Vi+ 1 is an irreducible ~-module, we actually have AiiBii = BiiAii' i = 1,2, ••• ,t, by Lemma 1.18. -17The minimum polynomial of AJ_l. .. (resp~. BJ.1. .. ) is irreducible as in Theorem 1.9, i = 1,2, ••• ,t. Hence A and B have property T. o 1. 20 Corollary. If T has characteristic 0 or p:;:.. n then the properties Qk are equivalent, k = 1 ,2, ••• 1.21 Corollary. If '"F has characteristic 0 or p~n, and if A and Bare quasi k-commutative for some k~ 1 , then they have property Q. Finally we shall give a counter-example to Theorem 1 • 1 9, when the conditions on the characteristic of °3=' are not satisfied. In order to construct the example we shall need the following well-known result. 1. 22 Theorem. [shoda, 9; Albert and Miickenhoupt, 1~. Let z be an nxn matrix with elements in a field T. Then there exist nxn matrices X and Y, with elements in CF, so that z = [X, Y], if and only if trace z = o. 1 .23 Example. Let -.F have characteristic p ;;.o and let I be the pxp identity matrix. Then trace I = 0 and hence there exist pxp matrices A and B with elements in J= so that I= [A,BJ, by Theorem 1.22. Then A and Bare que.si , 2-commutative, but they do not have property Q, since e.g. [A , B] is not nilpotent. For n;;:::.p we constru ct a counter-example by "filling out" the above matrices with zeros. -18CHAPTER II THE KATO-TAUSSKY-WIELANDT COMMUTATOR RELATION · In the first chapter we considered a generalization of matrix commutativity (property Q) and extended known results by considering an arbitrary field J=". From here on we shall seek to generalize property Q itself, and thus to generalize the notion of commutativity. The kind of . generalizations we want are stated as Problem 2.1. So let A and B be linear transformations on an ndimensional vector space V over a fields:-. Again let 'R be the algebra generated by A and B and ~the radical of~ • By applying the composition series argument cited in Theorem 1.9 to Vas an~-module we get the matrices A11 A12 0 A22 A1t ... A2t (*) and 0 0 ... Att 0 0 of A and B with respect to a suitable basis of V where A .. J.J. and Bii (which are square b locks of dimension ni). cannot be reduced further by a simultaneous similarity, i = 1, 2, • • •• ,t. When '3= is algebraically closed McCoy's theorem (Theorem 1.2) characterizes those A,B for which n l. = 1, i = 1 ,2,. .. Actually, for McCoy's theorem to hold, it is only necessary to assume that c:Fcon tains the characteristic roots of A and B. We now raise the following general question. 2.1 Problem. How can property Q be generalized so as to characterize those A and B for which the matrices in (*) -19have diagonal blocks Aii and Bii of specific dimensions, i = 1,2,. · ~.,t? In this chapter and the next we partially answer this question (cf. Theorems 2.13 and 3.1). If we wish to have A11 and B11 , say, of dimension k, then somehow we must introduce a relation satisfied identically by kxk matrices. This leads us to a commutator relation which was proved by Kato and Taussky [11] in the two-dimensional case, and by Taussky and Wielandt [12] in the general case. Let A and B be linear transformations on V. Let Ai+ 1 =[Ai,B] for i = O, 1,2, ••• , where A0 = A. Let K = = k(k - 1)/2, where k is a positive integer and let 'o 1 ,b 2 , • • • • , 'oK be K arbitrary elements inc;:-. 2.2 Definition. fk(A,B;'o) = A2K+1 - 'o1A2K-1 + 'o2A2K-3 - ••• +(- 1 )1S,KA1 2.3 Theorem [11,12]. if bi' i = 1,2, ••• ,N, are the elementary symmetric func~ 2 . tions of ( f> r - ps ) , 1 ~ r < s ~n, where Pr are the characteristic roots of B, i = 1,2, ••• ,n , and N = n(n - 1)/2. Note. 2 2 2 'o1 = 1E>{<s61.n (f3r - f>s) , 'o2 = 'L_(f3r - t3s) (Pt - f3u) etc., where the second sum is taken over 1-.:;;; r < s ~n, 1 ~ t <u E;.n and (r,s)c:::(t,u), and this last ordering is lexicographic. We shall call expressions of the type fk(A,B;o ) -20- "Kato-Taussky-Wielandt commutator expressions" or "KTW commutator expressions", for short. Theorem 2.3 is then the Kato-Taussky-Wielandt commutator relation. We remark that Theorem 2.3 is independent of A. We can express the result in a somewhat different form. Let TB be the linear transformation on the space of all linear transfor mations on V defined by TB(X) = [X,B], then Theorem 2.3 becomes 2.4 Theorem [Taussky and Wielandt, 12; Khan, 13]. We do not need to assume that (J3r - Ps) 2~ CS:, 1 ~ r < s ~n, in either Theorem 2. 3 or Theorem 2. 4. We note that f 1 (A , B;"o) = [A,B] and thus, if there exist elements "o 1 ,"o 2 , ••• ,-oK in3="so that fk(A,B;'o) = 0 for some k with 1:::;; k <n, we might suppose this to be a "good" generalization of commutativity, in that it might give a partial answer to the question raised in 2.1. However the condition turns out to be a bit too weak for this. This is borne out by Example 2.6. 2.5 Definition. on V then 2.6 Example. assume · "3 £~. If x1 ,x 2 , ••• ,Xk are linear transformations Let '3'='.have characteristic f 2 or 3 and Let A and B be matrices, -21and -(1+:)/2 (1-=)/2] (1-x)/2 -(1+x)/2 where x = 1/'13. Now f 2 (A , B ; 'o ) = [A , B , B , B] - 'o 1 [A , B] • We have A2 - B 2 = I and this gives [A,B , B , B] = 4[A,B]. Hence f 2(A,B ; b) = O, where ~ 1 = 4. Incidentally, we also have f (B , A;b) = O, where "o 1 = 4. But even with both 2 f (A , B;"O) and f (B , A;"o) zero , we cannot transform A and 2 2 B to t h e form ( -x· ) b y a simultaneous similarity , (even if we extend the f i eld 3="') where the diagonal blocks A. . and . 11 Bii have dimensions~ 2. This comes from the f act that A and B have n e ith er a row characteristic vector nor a column characteristic vector in common. Since A and B are symmetric , it is only necessary to verify this last statement for row vectors. The characteristic vectors of A are (a , O, -a) wher e a £7F , corresponding to the characteri stic value -1 ; and (a , b,a) where a , be::F, correspon ding to the characteristic value +1. Clearly B cannot have (a,O , -a) as a characteristic vector. Suppose B has a characteristic vector of the form (a,b,a) . Hence (a , b,a)B = ± 1 (a,b , a). Thus (x(2a + b), (a - b)(1 + x)/2, -(a - b)(1 - x)/~ = .:t.Ca,b , a). Hence x(2a + b) =+a orb= -a(2x + 1)/x. Also (a b)(1 + x) = + 2b orb= a(1 + x)/(1 + x + 2). -22these equations contradict each other Hence A and B have no characteristic But since x = 1//3, unless a = b = O. vectors in common. a We shall, in Chapter IV, discuss a result concerning linear transformations A and B which satisfy fk(A,B;'b) = 0 for some k with 1 S:::k<n. But the result in this chapter (Theorem 2.13) which sheds some light on Problem 2.1 involves KTW expressions of the type fk(A,P(A,B);b), where P(x,y) is a polynomial in x and y. Before we can prove any results about KTW commutator expressions we need some preliminary results. The following we ll-kno~m theorem will be of use to us. 2. 7 Theorem [cf. : Jacobson, 14, p.11 ~. Let ~be an infinite field and g(x ,x 2 , ••• ,xr) a non-zero polynomial . 1 in the polynomial domain::F'[x 1 ,x , ••• ,xr]' where the xi 2 are algebraically independent, then there exist elements c 1 ,c 2 , ••• ,cr in j='so that g(c 1 ,c 2 , ••• ,cr) IO. We need this to prove the next lemma •. 2.8 Lemma. Let ".:Fbe an infinite field and n a positive integer, then it is possible to choose x ,x , •• • ,xn _in ·: F 12 2 so that the n(n - 1 )/2 elements (xi - xj) , with 1 ~ i <. j ~n, are all distinct and non-zero. Proof. Let n.rc:v. _ y.)2 _ (yr _ y s )2J. ~ i J where the product is taken over 1 ~ i < j ~n and 1 ~ r < s ~n with (i,j) I (r,s). Since g(y ,y 2 , ••• ,yn) i O, by 1 Theorem 2.7 there exist x ,x , ••• ,xn E'3=so that · 1 2 g(x 1 ,x 2 , ••• ,xn) This proves the result. Q IO. -232. 9 Lemma. LetJ=°be any field and X an nxn matrix with elements in 'S"so that, for every non-singular matrix U with elements in~, the matrix u- 1 xu is diagonal, then X is a scalar matrix. Let U = u1 if n = 2 and U = u1 @ In_ 2 otherwise, where In_ 2 is the (n-2)x(n-2) identity matrix. Then o••• o 0 1 u- xu = • 0 0 and, since this matrix must be diagonal, we get x 1 = x 2 • By a simultaneous permutation of rows and columns (using a permutation similarity) we can replace x 2 by any xi for i~3. Hence x 1 =xi' i = 2,3, ••• and thus X = x 1 I. 0 The result we prove next will be applied to prove a theorem about KTW commu tator expressions, but it is of some interest in itself. Theorem. Let d=°be an infinite field and X an nxn matrix with elements in J= • If X is not a scalar matrix , then there exists a non-singular matrix U with e lements in :Fso that u- 1 xu has none of its elements zero. Proof. Let X = (xij) be the given matrix. We shall subject X to a s u ccession of similarity transformations to put it in the required form. To avoid cumbersome notation, after each similarity we shall still refer to the new matrix as X = (xij). Since Xis not a scalar matrix, by Lemma 2.8 we may assume it has at least one offdiagonal element which is non-zero (transform X by a 2.10 -24similarity transformation., if necessary). By a simultaneous permutation of rows and columns (using a permutation similarity) put this element in the (1,2) place of Thus we may assume X = ( xij) with x 12 I 0. Let S =I+ yEij' with i f j, where I is the nxn identi~y matrix, Eij has 1 in the (i,j) place and zeros . -1 elsewhere and y e'3=. Then ·s = I - yE 1J .. and hence X. s- 1 xs = x + yXEij - yEijx - y 2EijXEij" Thus the effect of the similarity s- 1 xs is t o add y times the ith column of X to the jth column, subtract y times the jth row from the i th row and change the element x .. to 2 1J xij + y(xii - xjj) - y xji. Call such a similarity transformation an elementary similarity. Now consider the matrix X with x I O. By means of 12 an elementary similarity add yi times the second column of X to the ith column, for each i f 2 , where yi = 0 if x 1 i l 0 · and y.1 = 1 if x 1 1. = o. We have possibly changed some rows of X, but , in any case , we now have a new matrix X = (x .. ) . 1J with x 11 ,x 12 , ••• ,x 1 n a ll non-zero. We shall prove , by indu.c tion, that we can transform X by a similarity so t hat the first n - 1 rows of the matrix obtained contain no zeros. If n = 2, we have already proved this statement. If n~· 2, let us assume that we have succeeded in transforming X by a similarity so that the first k rows contain no zeros, where 1 ~·k <n - 1 • So now we assume X = ( xij ) where the first k rows contain no zeros. If xk+ 1 ,k+ 2 I 0 we can proceed as in the next paragraph. Otherwise, by means of an elementary similarity subtract y times the kth row fr om the (k+1)st row. This adds y times the (k+1)st column to the kth column and changes the value of the element in the (k+1,k) place. Let -25k g(y) = y n(xik + yxi,k+1) • . J.=1 Since g(y) i O, by Theorem 2.7 we can choose y fj= so that g( y) is not zero (because cg: is an infinite field). We thus get a matrix X = ( x .. ) whose first k rows contain no J. J zeros and for whi ch xk+ 1 ,k+ 2 I O. For each i I k + 2 add yi times the (k + 2)nd column to the illi , by means of an elementary similarity. This operation subtracts yi times the illi row fr om the (k + 2)nd row and changes the value of the element in the (k + 2,i) place. So for each 1 I k + 2 let = -M(x .. + y.x. k 2). ~=i Jl. J. J, + Since g (yi) i O, choose yi t: c;f" so that g(yi) I O. Thus, by induction , we have shown that the given matrix can be transformed by a similarity transformation so that the first :n - 1 rows of the matrix obtained contain no zeros. So now we have a matrix X = (xij) with zeros, perhaps, only in the last row. By means of an elementary similarity subtract y times the (n - 1)st row from the nlli row. This adds y times the nlli column to the (n - 1)st column and changes xn,n-1 to xn,n-1+y(xnn-xn-1,n-1)-y2xn-1,n· Let where the prime means that the term containing i = n - 1 is omitted. Since g(y) i 0 we can choose y ~:Fso that g(y) I O. Thus we have obtained an X = (xij) where none of the elements is zero, and this proves the theorem. 0 -26We shall now prove our first result involving KTW commutator expressions. It supports the idea that we can use the vanishing of a KTW commutator expression as a generalization of commutativity. The theorem we prove generalizes the fact that the center of the algebra of linear transformations on a finite-dimensional vector space over a field is the scalar transformations. 2.11 Theorem. Let V be an n-dimensional vector space over an infinite field 7F , A a linear transformation on V and k an integer with 1 ~ k < n . Suppose that for each linear transformation X on V there exist elements ~ ,~ , ••• ,~K in7.F, where K = k(k - 1)/2, so that 1 2 then A is a scalar transformation. Proof. Let B be the matrix of A with respect to some basis of V. Then for each nxn matri·x Y with elements in '3=" there exist elements o 1 ,~ 2 , ••• '~K in "3::' so that Let Y = diag(y 1 , y 2 , ••• ,yn) where the yi e ':Fare such t h at (yi - yj)2 are distinct and non-zero f1/B~Y;'o) = O. for 1~i-<j~n (by Lemma 2.8). bijyji(yK - 'o1yK-1 + ••• If B = (bij), we get +(-1)~K) = 0 2 where y = Y:.. a nd y . . = y . - y . • Since there are N = Jl. Jl. J J. 2 = n(n - 1 )/2 distinct non-zero values for yji with 1 ~i<j~n and N>K, we get bij = O for some i, j. Thus the matrix of A with respect to any basis of V has at least one element zero. Hence A is a scalar transformation, by Theorem 2.10. C We use the following known result in proving one of -27the main theorems of this chapter. 2.12 Theorem ["Burnside's theorem", cf.4, p.276]. IfS\' is an irreducible algebra of linear transformations on a finite-dimensional vector space V over an algebraically closed field, then~ is the complete algebra of linear transformations on V. The next theorem is a generalization of part of McCoy's theorem on property Q (Theorem 1 .2) and it gives some information about a solution to Problem 2.1. 2.13 Theorem. Let A and B be linear transformations on an n-dimensi onal vector space V over an algebraically closed field °.F. Let k be an integer with 1~ k< n. Suppose that for each polynomial P(x,y), there exist ~ ,~ , •• 1 2 •• ,~K in"'a=", where K = . k(k-1)/2, so that then there exists a basis of V with respect to which the mat rices of A and B have the forms (*), where fk(A .. ,P(A .. ,B . . ) ;'t>) = 0 and dim.A .. = dim.B ..E;;; k. ll 11 ll ll ll Proof. By the usual composition series argument we can find a basis of V so that the matrices of A a nd B have the forms (*) where Aii and Bii cannot be reduced any further by a simultaneous similarity and fk(A .. ,P(A .. ,B .. );~) = O ll ll ll with i = 1,2 , ••• ,t. We claim that dim.Aii = dim.Bii~k for i = 1 ,2, ••• ,t. For suppose dim.Aii = ni>k for some i. Since Aii and Bii cannot be reduced by the same simil,ari ty, _· the algebra ~ . of polynomials in A .. and B . . is . . 1 -· ~ l.1. 1.1 irred.ucible and hence by "Burnside's theorem" (Theorem 2.12), q\i is the complete algebra of nixni matrices ' with elements in~. Hence, given any nixni matrix X with elements in c::F' , we can find K elements 'o 1 ,'o , •• 2 -28•• ,'oK in ca=- so that fk(Aii'X;'o) = O. Since k<ni this means Aii is a scalar matrix, by Theorem 2.10. But then Bii can be reduced to upper triangular form by a simila r~y _ (since '3= is algebraically closed) and this leaves Aii unchanged. This contradicts the fact that Aii and Bii cannot be reduced by the same similarity transformation. Thus ni~k and this completes the proof of the theorem. Cl :i Remark. This theorem does not characterize those A,B for which the matrices in(*) have dim.Aii(resp. dim.Bii) ~k. The exact conditions on A and B for this characterization would appear to be quite complicated. However, in Chapter III we give necessary and sufficient conditions on A and B that the matrices in (*) have dim.Aii = dim.Bii :::;; 2 • We close this chapter with some remarks on the KatoTaussky-Wielandt commutator relation. When V is twodimensional the relation is [A,B,B,B] - ({3 1 - p2 ) 2 [A,B] = O or TB(T~ - (~1 - P2)2I) = 0 where ;B 1 and J3 2 are the characteristic roots of B and TB is the linear transformation defined on the space of all linear transformations on V by TB(X) = [x,B]. The KatoTaussky-Wielandt commutator relation is not the most general commutator relation between A and B in the two-dimen- · s iona l ~ ca~e. We prove the following generalization. 2.14 Theorem. Let A and B be linear transformations on a two-dimensional vector space V over a field c:a=:and · let the linear transformation TX be defined by Tx(Y) = [Y,X] -29where X and Y are linear transformations on V. = Then 0 where x = trace Atrace B + 2detA + 2detB - 2det(A + B)#. Proof. Replace A and B by matrices, again called A and B. We let A = (a .. ) and B = (b . . ) • Then J.J 0 -a12 -a21 °12 TA = a12 0 0 J.J a21 0 0 a21 °'2 1 -a12 a12 -a21 0 ' TB = 0 -b12 .,.b21 /3>1 2 b21 0 0 b21 b12 0 µ21 -b12 0 b12 -b21 0 where 0( = a . . - a .. and ~ .. = b .. - b . . • We are usi'i'lg i. ii JJ riJ 11 JJ the factJthat , if X is any :squa r e matrix, then TX = = Xt ® I - I ® X, where " t " means transpose and " @ " is the tensor product sign [cf. 13]. Let I be the 4x4 identity matrix, then = # Th e notation "detX" means "the determinant of the linear trans formation X". 0 I !'<\ I I a12~21 + a21b12 - x - a1 2 P12 a21 P21 - a12b21 - a21b12 -cx; 2b21 2a21 b1 2 + °12P12 - x - 2a21b21 C><J 2b21 °21b12 - 2a12b12 2a12b21 + ~1P21 - x -~1b12 - a12b21 - a21b12 a1 2~12 - a21P21 a12b21 + a21b12 - x -31Let y = 2a 21 b 12 + 2a 12 b 21 + °'1 2 1312 - x. = Then 0 -b12Y b21y 0 -b21Y b12y f>12Y 0 0 0 b12y f>21 y -b21Y b21y -b12Y where, in computing, we use the facts that f312 = - f3 21 • Thus y = 0 gives = 0 ~ 2 = -~1 and o. This happens if Now traceAtraceB + 2detA + 2detB - 2det(A+B) = (a11 + a22)(b11 + b22) + 2 <a11a22 - a21a12) + + 2 (b11b22 - b21b1 2) - 2 <a11a22 - a21a12) + 2 <a11b22 - 2 (b11b22 b21b1 2) a21b12) + - 2 (b11a22 - b21a12) = (a11 - a22)(b11 - b22) + 2 a21b12 + 2 a12b21 = x. This completes the proof of the theorem. 0 2.15 Corollary. Let A, Band C be linear transformations on a two-dimensional vector space over a field. Then [C , B , A, B] where x x [Q , B] = 0 = traceAtraceB + 2detA + 2detB - 2det(A + B). -32The Kato-Taussky-Wielandt commutator relation for the two-dimensional case follows from this corollary on putting A = B. We note that we do not use the generalized form of the Kato-Taussky-Wielandt commutator relation (Theorem 2.14) in this thesis. We also remark that we have not succeeded in generalizing Theorem 2.14 when the dimension of V is greater than 2. -33CHAPTER III THE TWO-DIMENSIONAL BLOCKS In Theorem 2.13 we gave a condition on A and B sufficient to guarantee that dim.A . . = dim.B . . ~ k in ( *). ll 11 However this does not characterize matrices of the form (*) with dim.Ai~ k, i = 1 , 2, ••• , t. In this chapter we characterize those linear transformations A and B on V for The which dim.Aii = dimBi~ 2 in (*) for i = 1,2 , ••• ,t. characterization is in terms of two-dimensional KTW commutator expressions. Again we have linear transformations A and B on an n-dimensional vector space V over a field':F. f< is' the algebra generated by A and B over a=-' and ~ is the radical of~. We have the following main theorem. Theorem. Let :f' be algebraically closed. Then the following statements are equivalent. (a) For each polynomial P(x,y) th.ere exist an integer r = r(A,B,P(A,B)) and distinct elements h 1 ,h 2 , ••• ,hrc:<=.:fso that 3. 1 for every permutation n(1), TT(2), ••• ,n(r) of 1 ,2, ••• ,r, where = U,P,P,P] - h s [A,P'], s = 1,2, ••• ,r and P = P(A,B). (b) There exists a basis of V with respect to which the matrices of A.and B have the forms A11 A12 0 A22 ... A1 t -34B11 B12 • • • B1t 0 B22 B2t . .. A2t and (*) • 0 0 .. .~ . ••• Att • • 0 . .. Btt r espectively , where Aii and Bii are either 1x1 or 2x2 matrices which cannot be reduced further by a simultaneous similarity, i = 1 , 2, ••• ,t. Proof. Assume (b) holds. Let A' and B' denote the matrices in (b) (since we do not refer to transposes in this theorem, the notation is unambiguous). The block diagonal of P(A ' ,B ' ) has blocks P(Aii ' Bii), i = 1,2, ••• ,t. Let Aj, Bj be the 2x2 blocks on the diagonals of A', B•, respectively, j = 1 , 2 , ••• ,q. Let x 1 . and x 2 . be the J J 2 characteristic roots of P(Aj,Bj). Form (x 1 j - x 2 j) , . j = 1 ,2, ••• ,q and let h 1 , h 2 , ••• ,hr be the distinct ele ments among these. If we let P. = P(A.,B . ), then we have J [A . , P. , P . , P. J J J J J J J (x 1 J. - x 2 J. ) 2 [A J. , P.J J = O. If we form X~ = [A ' , P ' , P ' , PQ - hs [A' , P '] for s = 1 , 2, ••• , r, where p t = P(A' , B• ), we see that XTI( 1 )Xn( 2 )•••Xrr(r) has zero blocks on and below the main diagonal, for each permutation rr(1) , rr(2), ••• ,rr(r) of 1,2, • •• ,r. This is also true of Q(A 9 ,B')XIT( 1 )Xn( 2 ) ••• Xn(r ) for all polynomials Q(x,y). Thus Q(A,B)XTT( 1 )Xrr( 2 ) ~ •• xrr(r) is nilpotent for all Q(A,B) and hence X11( 1 )XTT( 2 ) •• • Xn(r) €} by Lemma 1 .6. Hence (a) holds. It is clear , from this part of the proof, why we must take a product of Xs's , instead of a single one. Conversely, let (a) hold. By the usual argument, there exists a basis of V with respect to which the matrices of A and B have the forms (*),where Al...l. and Bl.l. .. cannot be reduced any further by a simultaneous similarity, -35i = 1,2, ••• ,t. Let Aii and Bii be nixni matrices. We wish to show that ni ~ 2. So assume, for some i, that ni~2. Under this hypothesis we shall show that Aii is actually a scalar matrix. As in Theorem 2.13 this means Aii and Bii can be reduced by a simultaneous similarity, which is a contradiction. Hence ni~2 for i = 1,2, ••• ,t. We proceed to show that ni-::>2 implies Aii is a scalar = = matrix. For simplicity, let Aii = C, Bii D and P = P(C,D). Then, if Ys = [C , P,P,P] - hs[C,P], we have YTI( 1 )Yn( 2 ) ••• Yn(r) = 0 for all permutations n(1),n(2), •• •• ,n(r) of 1,2, ••• ,r. · Since C and D cannot be reduced by the same similarity transformation, the algebra of all polynomials in C and D with coefficients in '3=' is irreducible. Hence, by "Burn- side's theorem" (Theorem 2.12), this algebra is the complete algebra of nixni matrices with elements in '3="". Thus, if X is any nixni matrix with elements in::F" , for some integer r (depending on X) there exist distinct elements h 1 ,h2 , ••• ,hr s T so that, if xs = [c ,x,x,x] - hs [9 ,x], then Xn( 1 )Xn( 2 )•••Xn(r) = 0 for all permutations of 1,2, ••• ,r. Without loss of generality, assume C is in Jordan canonical form. m C = jl=-1© C j where C j = O<j I j + E j , j = 1 , 2, ••• , m, where I. is an identity matrix and E. is a matrix of the J J . same dimension as Ij, with 1•s on the superdiagonal and . zeros elsewhere, i.e. the Cj are Jordan blocks. We wish to show C is a scalar matrix. This we do in three stages. Stage 1. Each Jordan block C j has dimension~ 2. Proof. Suppose some block, c1 say, has dimension;::::.3. Then -36~ 0 01 = 1 0 ••• 0 °"I 1 0 • • • 0 • • 0 ••• 0<1 0 We may ignore the diagonal, since we shall be taking com- . Thus we consider E 1 • mutators. Let i.e. the leading 3x3 diagonal block of E • Since we may 1 consider any nixni matrix X in the expression [9,X,X,X] - hs[c ,x] , we shall now consider only those matrices X which have arbitrary 3x3 blocks Z in the place corresponding to the block E and zeros elsewhere. Thus we may restrict ourselves to 3x3 matrices. Let z = [o o 1 0 0 1 ~J then [E ,Z] = [E,Z,Z] = ~ [ 0 00 1 ' 0 0 -1 [-~ ~ -~J~l 0 -1 and -37[E, Z, Z, Z] = [! _: -J~l. Hence = [E, z , z , z] - hs [E, z] - = [-h 1s 0 ~ ~ -h J ~1 -( 1-h:) which is in block lower triangular form . We have Zn( 1 )zn( 2 ) ••• zrr(r) = o, for each permutation of 1,2, ••• ,r. Hence some h is zero, say h 1 = O. Let 6 Hence H = 1 = 1, if r (nhs)H 1 , otherwise. s=2 r since H 1 =[ 1 This is a contradiction. = .1,2, ••• ,m. r?2. 1] and -1 - 1 Hence ns=2hs -/= o, : dim.CJ~ 2, j = Stag e 2. C is a diagonal matrix. Proof. Suppose for some j that dim.Cj = 2. Let Cj = = ~· By simultaneous permutations of rows and [6 columns, if necessary, we may assume Cj is not the leading block on the diagonal of C (since dim.a = ni-:;;::::.3). Thus -38- since dim.0 :::=;;2, 1 1 ff OOJ = 0. 1 0)0. JJ 0 0( 1 (3 1 0 0 0 {!> 0 0 or 0 0 C( = 1,2, • • • ,m. = G 0 0 ~ 1 0 0 0 Ol Let r:: ~J. L~ ~ o If O . t+'i 0 . is 4x4, it equals J - 1 \;!.:/ J [3 1 0 0 0 0 G 0 which is in block upper triangular form. We shall be multiplying 0 by matrices which have zeros everywhere except in the place corresponding to G, so again we restrict ourselves to 3x3 matrices . z = Let [~ ~ ~] where x is chosen in "".F- so that x I 0 and x I [G,Z] and = ·[o op-cxj 0 x 0 0 0 -x (3 - x (p -ex) · ' (9-,z,~ = 0 -2x 2 0(. Then -39- . 3x - 2x ([3 -~ ((3-oV-XJ 0 0 0 0 2 Qt,z,z,zJ = • Hence Z8 = W,z,z,Z] - h 8 [9-,z] = [~ r and thus ( qhs )xr = o. = 0 zn(1)Zn ( 2) • • • zn(r) Now means some hs is zero , say h1 = o. ~hus z1 = G 2 3X -2 (~-oVX 0 0 This ~-~1 0 • 0 The ( 1 ,3 ) element of z 1 z 2 • • • zr is ( ~-o<-x ) ex r-1 , where c = = 1 if r = 1 and c = h 2h 3 •• • hr IO, otherwise . But this gives a contradiction, since x I 0 and x I p-o<. Henc e C = diag ( °1 , ~ , .•. , e><n . ) • J. Stage 3. C is a sc a lar matrix. Proof . Since we may put any three of °'1, ~, ••• ,oh. in the ]. fir s t thr e e places on the diagonal of C (by simultaneous row and column p e rmutations), we may as we l l assume that C = diag(~,~'°3) . Let z = [~ ~ ~]· 1 0 0 We have, on letting <X_j_ - °'j = O<ij , -40- [c,zJ = [~, 0 ~3] 0 0 ~3 and [c, Z, Z, Z] [9, Z, ZJ = , = [a 5~3 0 [ 2~3 0 °31] 0 °32 , °23~~ 3 0 2°)1 °'.31 . J ~2 0 2~3~3°'J 3 5<><-;1 0 • Hence z s = [c, z, z, Z] - hs [C, Z] = [ ~2 (5-hs)°31 The product of an even number of z s 's has the form and the product of an odd number has the form [: 0 0 0 ~ :J 0 0 0 ~J ' where * denotes an element not necessarily equal to zero. The (1,3) element (if r is odd) or the (1,1) element (if r is even) of the product z1 z2 ••• zr gives c~3 = O, where r JJ ( c = 5-hs) • h 1 (say) = 5. . If no hs = 5 we get <Xi Thus = ex,. Otherwise = The (2,1) (if r is odd) or (2,3) (if r is even) element of z1 z2 ... Zr gives c<Xi 2 <Xir-1 3 = O where c = 1 if r = 1 and c = ~J ( 5-hs) otherwise. Hence <><t = °2 or <Xi = ex,. Now this last statement is true whether or not some h 8 = 5. If ~ = . ~, put ~ in the ( 1 , 1) place (by a simultaneous 2 -41row and column permutation) and repeat the argument. Similarly 1 or C¥2 = a-3 i.e. 0<1 = °'2 = °'3. a-1 = Q'2 implies 0(1 = ~ = ~. Hence C is a scalar matrix i.e. Aii is a scalar matrix. Thus ni~2 and the result is proved. O Thus ~ = C¥ We make some remark s about the above theorem. Note, in the course of the proof, we did not use all the products X11 ( 1 )Xn( 2 )•••Xn(r)• It seems likely that it should be possible to consider even fewer of these products than were used in the cour s e of the proof; we do not have an example which contradicts this surmise. It should be noted that the proof that (a) ==;>(b) does not gu arantee that the constants h 1 , h 2 , ••• ,hr are of the form (x - x 2 ) 2 where x 1 , x 2 are characteristic roots of 1 P(A,B). It .is clear, for example, that if A and B have property Q, then the constants h 1 ,h2 , ••• ,hr may be quite arbitrary. Finally, it seems likely that a similar theorem should characterize those matrices (* ) with dim.Aii~k, if we replace the two-dimensional KTW commutator expressions by k - dimensional ones. But we have made no progress with this problem. -42CHAPTER IV THE RELATION f k (A, B ;'"o) = 0 , WHEN k <n Example 2.6 indicates that the relation fk(A,B;"t>) = O, · with 1~k<n is not a " good" generalization of commutativity in that it does not help us to solve Problem 2.1. However, it seems reasonable to ask what the relation fk(A , B ; b) = 0 does imply. We might hope to prove that if 'O = ·'.1 (t> 1 ,'0 2 , ••• ,"OK) then t> 1 ;o 2 , ••• ,'OK are elementary sym2 metric functions of some of (~r - Ps) , 1~r,s:s:;;n, where fr are the characteristic roots of B. When k = 2 we do get a result like this (cf. 4.6)but, in general, the result we obtain is not quite this strong (cf. Theorem 4.2). We begin the investigation with a definition. 4.1 De finition. Let fk(X,Y;"t>) be a KTW commutator expression, with '"O = ('o 1 ,-o 2 , ••• ,"oK) where K = k(k - 1 )/2. Then = ... The main theorem in this chapter is 4. 2 Theorem. Let X. be the algebra of linear transform~~­ _tion on an n-dimensional vector space V over a field '3=' and let A, BeZ. Suppose A and B generate~ i.e. every linear transformation in£.. has the form P(A , B) where P(x,y) is a polynomial. If the characteristic of :F does not divide n and if there exist "0 1 ,'0 2 , ••• ,'oK in <J=' so that where 1 ~k<n and "O = (u 1 ,'0 2 , •••"'OK), then the characteristic roots of B belong to the splitting field of gk(w;"o) over~. :...43_ We shall prove · this theorem presently, but first we introduce some well-known facts from the Theory of Graphs which we shall employ in the course of the proof. A general reference for the graph theoretical material we use is Varga [15]. 4.3 Definition. Let X be an nxn matrix with elements in a fieldc:T, then X is P-irreducible if it cannot be transformed by a permutation similarity to the form where Y and W are square matrices. Note. The " P" iri " P-irre ducible" stands for "permutation". Our " P-irreducible " is the same as the " irreducible" or " indecomposable" of the Perron-Frobenius theory of nonnegative matrices [cf. 15]. We associate a directed graph G(X) with an nxn matrix G(X) consists of vertices X = (xi j ) in the followi ng way. numbered 1,2 , ••• , n and there is an edg e from i to j, i.e. i--:>j, if and only if xij The following well-known theorem is stated in Varga [ 15]. We include a proof since there does not seem to be one in the literature. IO. 4.4 Theorem. Let X be an nxn matrix with elements in a field .cg::--. Then X is P-irreducible if and only i f G(X) is . strongly connec ted. Proof. If P is a permutation matrix then G(P- 1 XP) is obtained by rela beling the vertices of G(X). Suppose G(X) is strongly connected. Then X is P-irreducible. For suppose otherwise; then there exists a permutation matrix P so that Then there strongly connected. ·-.. _ .. - Suppose it is not. exists a vertex i to ~~·~ c ~ at least one other vertex is not By renumbering the vertices we get i = 1. connected. Let 2,3, ••• ,r be the vert~ces which are connected to 1 by some path in G(X) (again by renumbering). It is clear that r::>2, since we cannot have xj 1 = O, j = 2,3, ••• ,n, because this would contradict the fact that X is P-irreducible. It is also clear that r<::::.n, since we have assumed G(X) is not strongly connected. These renumberings of the vertices of -1 . G( X) correspond to a permutation similarity P XP of X. We claim that P- 1 xP has the form (# ) above, because there does not e x ist a path in G(P- 1 XP) from j to i where re:: j "n and 1:;;; i<.r. Since , if j can be connected to i, then j can be connected to 1, because i can be connected to 1. This is a contradiction of the fact that 2,3, ••• ,r are the only vertices that can be connected to 1 by a path in G(X). Hence Xis P-reducible (i.e. not P-irreducible). This contradicts the initial assumption. Hence G(X) is strongly connected. Q 4.5 Proof of Theorem 4.2. Firstly, we dispose of two easy -45cases. If k = 1, then AB= EA and since A and B generate Xthis means n = 1 and the theorem is then obvious. The second easy case is when B has only one characteristic root f3 • Then traceB = npt:-Y. Hence f3e~, since the characteristic of '3=" does not divide n. So the result is trivial in this case also. So assume that there exist '0 1 , '0 2 , ••• ,'OK in -=:F so that fk(A,B;'O) = 0 where 1-<k<n and where B has at least two distinct characteristic roots. Replace A and B by matrices, again called A and B and extend 7F to a field ~ which contains the characteristic roots of B. Let U be a non-singular matrix with elements in ~so t_hat N = u- 1 BU is the Jordan canonical form of B with N = $ B. where i=1 J. - 'i: Bi is a direct sum of Jordan blocks all of which have the same characteristic root pi and f>i -l /3j when i I j. Let -1 A1 = U AU, then Let A1 = (Aij) be the partition of A1 corresponding to that of N = r L i=1 (±)Bi. 1. If Aij gk(w;'o) = O. 2. We shall prove the following statements. Io, then Pi - pj satisfies the equation A1 = (Aij) is P-irreducible as a block matrix. If we assume Statements 1 and 2 we can complete the proof of the theorem in short order. For let G(A 1 ) be the graph of A1 considered as a block matrix, i.e. i~j if and only if A .. I O. Then Statement 2 and a modification of J.J Theorem 4.4 (for block matrices) imply that G(A 1 ) is strongly connected. Thus, if pi' pj are distinct char- -46acteristic roots of B , there exists a sequence i,i 1 ,i 2 , •• •• ,iu,j so that Pi - ~i1' ~i1 - Pi2'···'Piu - rj satisfy gk(w;'o) = o. Let 1\.. be the splitting field of gk(w;'o) over '3=". Thus ~. r i - B . r J e,. )+ ••• +(8. - B.)e:'.1t. = (r13i. - r6 i.. ) + (af . 1· 1 - ri. riu rJ . 2 1 Let nj be the multiplicity of fj as a characteristic root of B, j = 1,2, ••• ,r . Then r ) j;;; r n. A. Jrl. - ~.e.X . J:?i n.JIJ ) Hence n~i - traceB&j{. Thus fiE-:l{, i = 1,2, ••• ,r, since . the charac teristic of j{, does not divide n. It remains to prove Statements 1 and 2 to complete the proof of the theorem. Proof of Statement 1. We employ the relation fk(A 1 ,N;n) = = O. Suppose Aij ~ O. Let ast be the " firsttt non-zero element . of Aij in the following sense: if the lower lefthand corner element of Aij is non-zero, let this be ast; otherwise let ast be a non-zero element of Aij so that auv = 0 if u;;;ai:s and v ~t and (u,v) ~ (s, t) where, of · course, we only consider those elements auv in Aij" Thus 0 Aij = * 0 * 0 ast 0 0 * 0 ... 0 • 0 where * d enotes elements which make up the rest of A .. • J. J -47If ast is the (a,b) element of Aij' we shall calculate the . (a,b) element of the (i,j) block of fk(A 1 ,N;b). To simplify calculations assume that Bj has characteristic fj root 0 and Bi has characteristic root (3 ij = Pi (Subtract fjI from N. Since we take commutators, this operation does not affect the end result of the calcula ti ons • . ). The matrix fk(A 1 ,N;~) is a linear combination of matrices of the type [A 1 ,N,N, ••• ,N]. The (i,j) block of [A 1 ,:rl] is AijBj - B1 Aij• The (i,j) block of [A 1 ,N,N, ••• ,N] only involves Aij' Bi and Bj; it consists of a linear combination of matrices of the type B~A . . B~, . J.. 1.J J where c + d is the number of times the commutator operation is applied in [A 1 ,N,N, ••• ,N']. The (a,b) element of B~AijB~ is obtaine~ by multiplying the blli column of AijB~ by the alli row of Bi. Those elements in the blli column of A . . B~ from the ath element down are all that matter here. J..J J But these elements are zeros, except when d = O, since Bj has zeros on and below the main diagonal. Thus the (a,b) . element of [A 1 ,N,N , ••• , N], where the commutator operation is performed m times , is (-1)mp~jast• Since the monomials in gk(w;~) are of odd degree, the equation fk(A 1 ,N;~) = 0 gives gk(f3ij ;'o)ast = 0 and hence gk(f>ij ;~) = 0 since ast I 0. Thus we have shown that if Aij I O, then ~i - f3j satisfies the equation gk(w;'o) = o. Proof of Statement 2. We now show that A1 = (A .. ) is J..J P-irreducible as a block matrix. For suppose there exists a block permutation matrix P, .Parti t ioned c onformaliy,_ ~r;\ th A = (Aij), so that P- 1 A P has the form . -- - '--·--:. ·. .. 1 1 -48A11 A1m A1 ,m+1 ... A1r • Am1 Amm 0 ... 0 0 0 Amr Am,m+1 Am+1 ,m+1 • • • Am+1,r where m<r, then A1 and N may be reduced by a simultaneous · similarity , since the block permutation matrix P simply permutes the blocks on the diagonal of N. Thus the algebra of matrices of the form P(A 1 ,N) is reducible, where P(x,y) is a polynomial in the non-commuting variables x and y with coefficients in~· But the matric e s A and B generate the complete algebra of nxn matrices with coef~ ficients in ::F. Hence A1 .and N generate the complete algebra of nxn matrices with coefficients in ~which is irreducible. This contradicts the assumption that A1 = = (A J...J ) is not P-irreducible as a block matrix. Thus A1 is P-irreducible as a block matrix. c:r 4.6 Corollary. Let A and B satisfy the conditions of Theorem 4.2 with k = 2 and let B have at least two distinct characteristic roo t s. Then there exists an ordering ~ ,r , ••• ,pr of the distinct characteristic roots of B so 1 2 that 13 1 - 13 2 = ~ 2 - 3 = ••• = Pr_ 1 - Pr satisfies w 2 p - 'o1 = 0. Proof. Consider the matrix A1 . We have = (A J...J ) defined in 4.5. We claim that A1 cannot have more than two off ~diagonal -49blocks in each row or column which are non-zero. Suppose, for example, that Ai j ' Aik and AiL are non-zero off-diagonal blocks, where j,k and~ are distinct. Then 2 ~i - ~j' Pi - fk and Pi - f~ all satisfy x - ~ = o, by Statement 1 of 4.5. Hence at least two of Pj' ~k' r~ must be equal, and this contradicts the fact that the p's 1 are distinct. We also note that if Aij and Aik (resp.Aji and Aki) are non-zero off-diagonal blocks, where j I k, then an argument similar to the one just given shows that if A8 i (resp. Ai 6 ) is a non-zero off-diagonal block then s = j or s = k. Let G(A ) be the graph of A considered as a block mat1 1 rix. We shall write i..-vj if i~j or j~i. The dis cuss ion of the last _paragraph shows that if i .-v j and i ........ k, where i,j , k are distinct, then i ~ 1 implies that 1 must be either j or k. We claim that by renumbering the vertices of G(A ) we get the subgraph 1 ------· ... r-1 2 3 1 where i""i+1, for i = 1,2, ••• ,r-1. r For let = 1 2 3 s-1 s be a maximal "pathn in G(A ) (on renumbering vertices) 1 where i"""'i+1, for i = 1,2, ••·• , s-1 and suppose s I r . If j is a vertex then n e ither j......., -1 nor j..-vs . can hold, since f-<. is maximal. Since G(A 1 ) is strongly connected there exists an internal vertex i e f"- and a vertex j I-'- so that i.......,j. But i..vi+1 and i.-vi-1, and since j I i-1 and . Thus G(A ) contains the j I i+1 , this is a contradiction. 1 required subgraph. Hence, on renumbering the distinct ff'v, 1 characteristic roots f i of B, we see that f 1 - p2 , (-> 2 - p 3 , ... -50- 2 ' Pr-1 - ~r satisfy w - ~i - ~i+1 o1 = o. = Pi+1 - ~i+2' Now we must have i = 1 ,2, ••• , since pi"'." Pi+ 1 = ri+ 2 - Pi+ 1 implies that pi =Pi+ 2 • This is impossible, since the Pi's are distinct. C 4.7 Remarks on Theorem 4.2. Example 2.6 illustrates Theorem 4.2 in a trivial fashion. Later on in this chapter we shali give a non-trivial example (Example 4.12) which illustrates Theorem 4.2, Corollary 4.6 and a theorem we have yet to state (Theorem 4.9). We note that Theorem 4.2 iays that if ~ is a characteristic root of B, then it satisfies an equation of degree at most k(k - 1)! over 'S". At first sight this statement does not look too promising, but if k is "small" compared to n it says something about the reducibility of the characteristic polynomial of B over J=". We shall return to this fact in a moment when we discuss a generalization of the so-called "L-property". We note one more fact about Theorem 4.2: it is, that the condition on the characteristic of~ is ne.c essary. We give a counterexample to the theorem later (Example 4.13) where this condition is not satisfied • . Assume for the moment that the characteristic roots of A and B belong to~. We recall that A and B have property P if there exist orderings 0<1 , ot2 , ••• , cxn and g3 1 , 13 , •• 2 •• ,~n of the characteristic roots of A and B respectively so that P(A,B) has characteristic roots P(~i'~i)' i = 1,2, • •• ,n, for each polynomial P(z,w). Now a weak form of property P is property L (a term due to Kac) which demands that property P only hold for linear polynomials P(z,w) = = xz + yw, where x,ye~[Motzkin and Taussky, 1~. Property L then says that the characteristic polynomial p(x,y,z) of xA + yB splits into linear factors, i.e. -51n p(x,y,z) = r-T(xQj_ + yp1 - z) i=1 where ~ and f3i E ;f', i = 1, 2, ••• ,n. We shall use Theorem 4.2 to prove a result about a pair of linear transformations A and B which implies they have a property very much like that of property L (instead of only linear fac~ tors in p(x,y,z) we get both linear and quadratic factors). To prove the result we need a preliminary lemma. We denote by j:"°[1c,y] the integral domain of polynomials P(x,y) and by':F(x,y) the quotient field of this integral domain. (We assume, of course, that the indeterminates commute, in contradiction to the convention on page 4). 4.8 Lemma. Let p(x,y,z) be a homogeneous polynomial in x, y and z with coefficients in a field et. Suppose p(x,y,z) = t . l.= k· p.l. 1 l. where each pi is an irreducible polynomial in z over j=(x,y), then each pi is a homogeneous polynomial in x,y and z with coefficients inc.F". Proof. d=[x,y] is a Gaussian domain (unique factorization domain) [cf. Jacobson 14, p.126]. Since a Gaussian domain is integrally closed Q..bid. p.184:) , the coefficients of the powers of z in pi must be polynomials_ in x and y. Suppose pi is not homogeneous in x,y and z. Define M(q) (resp. m(q)) to be the maximum (resp. minimum) degree of the monomials in a polynomial q. .Now M(pi )"> m(pi) and M (resp. m) has the property that M(qr) = M(q) + M(r) (m(qr) = m(q) + m(r)) , for polynomials q and r. Hence M(p),:::::,. m(p) where p = p(x,y,z). But this is false. Hence M(pi) = m(pi) and thus pi is homogeneous. 4. 9 Theorem. Let 1:, be the algebra of linear transform:;;.- -52ti on on an n-dimensional vector space V over an infinite field CJ=whose characteristic does not divide n. Assume A and B e;I:. are such that A and B generate£.. and suppose that for each x,y £ d=' there exists 'o 1 t: S:- so that where u = (~ ). Then the characteristic polynomial 1 p(x , y,z) of xA + yB splits into linear and quadratic homogeneous factors in x,y and z with coefficients inO::. Proof. We may assume that n~3, since the result is trivial otherwise. Replace A and B by matrices. Let Then xA + yB = X. If [A,:x]= 0 then AB= BA and this implies n = 1. So the result is trivial in this case. So assume [A,X] I o. Then the equation f 2(A,X;'o) = 0 says that ~ 1 is a rational function of x and y. Replace x and y by two algebraically independent indeterminates, again called. x and y, respec t:tvely. ~ Then the equation f 2 (A,X;'o) = 0 still holds, by Theorem 2. 7, since '3=" is infinite. A and X clearly generate the algebra of rucn matrices with elements in 'dix,y), 3 . so we may apply Theorem 4.2. Now g 2 (w;'o) = w - ~ 1 w. So each characteristic root of X = xA + yB satisfies an equation of degree at most 2 over ~(x,y). The theorem then follows on applying Lemma 4.8. 4.10 Corollary. Let A and B be linear transformations on an n-dimensional vector space V over an algebraically closed field C:S:whose characteristic is either zero or greater than n. If, for each x,y ec:a=:, there exists 'o 1 ~ so that r -53:... f 2 (A,xA + yB;o) E }- where } is the radical· of the algebra generated by A and B overc:F, then the characteristic polynomial p(x,y,z) of xA + yB splits into linear and quadratic homogeneous factors in x,y and z with coefficients in~. Proof. Apply the usual composition seri e s argument to V to get matrices for A and B in block upper triangular form. "Burnside's theorem~' (Theorem 2 .12) shows that simul taneously .: irreducible blocks Aii and Bii (cf. (*) on p.18) generate a complete matrix algebra, and the condition on the characteristic guarantees that it does not divide the dimension of any diagonal block. The result follows immediately on applying Theorem 4.9 to the diagonal blocks. The above results do not guarantee that p(x,y,z) actually has a linear factor·, but part of the . next result does. 4.11 Theorem. Let A and B be nxn matrices with elements in a fields whose characteristic is either zero or greater than n. Suppose A and B generate the complete algebra of nxn matrices and let x,y be algebraically independent indeterminates. If ther~ exists 'o ~ -=.F(x,y) so that 1 where~= (~ 1 ), then the characteristic polynomial p(x,y,z) of xA + yB splits into linear and quadratic homogeneous factors in x,y and · z with coefficients in~. Moreover if xA + yB has an odd number of distinct characteristic root a then p(x,y,z) has at least one linear factor. Proof. The first part of this theorem is almost a repetition of Theorem 4.9 and is proved in the same manner· (Note: the field 'fF need not be infinite here). So assume -54xA + yB has r distinct characteristic roots, where r is odd. If r = 1 , the result is trivial. So assume r ~. By Corollary 4.6 there exists an ordering z 1 ,z , ••• ,zr of the 2 distinct characteristic roots of xA + yB so that 2 z 1 - z 2 = z 2 - z 3 = ••• = zr_ 1 - zr satisfies w - 'ol = o. Now the irreducible factors of p(x,y,z) are separable, since the characteristic of'3=°is either zero or greater than n. Hence Now z i - z i +1 = ~ , i = 1 , 2 , ••• , r-1 • Hence :&, zi = r{zr +( (r-1 )/2)~J. Thus zr + ((r-1)/2)J~ 1 ES'(x,y). Now (r-1)/2 = s i s an integer, since r is odd and zr + ((r-1)/2)~~ 1 = z • r-s Hence zr-s£d=°(x,y). By Lemma 4.8 zr-s = XOI'+ Yf5Where ex, f3 £ :F" • c Finally we give two examples to illustrate the results of this chapter. 4.12 Example. This example illustrates Theorems 4.9 and 4.11 (and~ fortiori Theorem 4.2 and Corollary 4.6). Let :Tbe a field whose characteristic I 2 or 3. Let = [~ ~ ~Jo 1 1 -55If Eij is the 3x3 matrix with 1 in the (i,j) place and zeros elsewhere, then ( 1/2 )(A 2 - 3A + 2I) , = E 22 = -A 2 + 4A - 3I, = E11(B - I), E12 E1 1 = I~ E22 - E33' E32 = E33(B - I), E2 1 = (B - I )E11 ' = (B - I)E33' E13 = E1 2E23' E3 1 = E32E21· Hence A and B gen erate the complete algebra of 3x3 matrices with elements in~. Let ·x,y be two algebraically inde~ · pendent indeterminates overc.f and let X = xA + yB. Clea rly A and X generate the algebra of all 3x3 matrices with elements in T{x,y). We have x [A,X] = and = [~ xA + yB -y 0 y = OJ , -~ y 2x+y y [7 J y0 ' 3x+y [-2y2 [A,X,X] = -~y -xy 0 - xy -2y 3 -x 2 y [A,x, x, x] 0 = [ 2y3~x2y 0 2y3+x2y = -2y03 -x 2 y 0 -~yJ 2y2 J (x2 + 2y 2 )[A,X]. Henc e f 2(A,xA + yB; (x 2+2y 2 )) p(x,y,z) = det( X - z I) = x+y-z y 0 = o. y 0 y 2x+y-z y 3x+y-z -56Expand the determinant by the second row to give p(x,y,z) = (2x + y - z)(z 2 - (4x + 2y)z + 3x2 + 4xy - y 2 ) Hence the characteristic roots of X (cf. Theorem 4.9). are = 2x + y, z2 -z 1 = 2x + y + Jx 2 + 2y2 , z 3 = 2x + y - Jx 2 + 2y2 • Clearly = (cf. Corollary 4.6). = We see that Since X = xA + yB has an odd number of (cf. Theorem 4.2). distinct characteristic roots, p(x,y,z) has a linear factor (cf. Theorem 4.11). 4.13 Example. The example we give here is a counterexample to Theorems 4.2 and 4.9 when the condition on the characteristic of 0: is not satisfied. Let "=.Fhave characteristic 3 and let · Then E13 = A2, E12 = E13B E33 = BE13 2 E11 + E13B' - E23' = E13B E23 E22 = 2 = - E13B' A - E12' I - E11 - E33• -57- = = E21 = Hence A and B generate the complete algebra of 3x3 matrices with elements in c:y. As usual let x ,y be indeterminates and X = xA + yB. Then A and X generate the complete algebra of 3x3 matrices with elements in~(x,y). We have x and = ~ x -y 0 -y -YJ [A,X,X] = [A,X] x+~ , [xy = 0 0 , -y y OJ 0 • -xy 0 0 0 [~ -:] 0 y -xy Hence [A,X,X,X] = = O[A,XJ. 0 Therefore f 2 (A,xA + yB;u) = o, where~ = (0) • . Now = and p(x,y,z) = det(xA + yB - zI) -z y y x-y = x2y . 3 = -z -y x+y -y -z z • Now x2y - z3 = ((x2y) 1/3 - z)3 and hence the characteristic roots of xA + yB do not belong to the splitting field of g 2 (w;~) over J='(x,y) (cf. Theorem 4 .2). Also x 2 y - z 3 has no linear factors over -58- "J= (x, y) (cf. Theorem 4.9). -59BIBLIOGRAPHY 1 N. H. McCoy, On the characteristic roots of matric polynomials, Amer. Math. Soc. Bull., 42(1936), 592-600. 2 I. N. Herstein, .Theory of Rings, University of Chicago Mathematics Lecture Notes, Chicago 1961. 3 N. H. McCoy, The Theory of Rings, Macmillan, New York, 1964. 4 N. Jacobson, Lectures in Abstract Algebra, Volume II, _Linear .A~lgebra, Van Nostrand, Princeton, 1953. 5 J. Williamson, The simultaneous reduction of two matrices to triangle form, Amer. J. Math., 57(1935)~ 281-293. 6 M. P. Drazin, J. W. Dungey and K. W. Gruenberg, Some theorems on commutative matrices, J. London Math. Soc., 26(1951), 221-228. 7 M. P. 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