Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics TRANSFORMATIONS ON TENSOR PRODUCT SPACES MARVIN DAVID MARCUS AND BENJAMIN NELSON MOYLS Vol. 9, No. 4 August 1959

2 TRANSFORMATIONS ON TENSOR PRODUCT SPACES MARVIN MARCUS AND B. N. MOYLS 1. Introduction* Let U and V be m- and n-dimensional vector spaces over an algebraically closed field F of characteristic 0. Then U V, the tensor product of U and F, is the dual space of the space of all bilinear functionals mapping the cartesian product of U and V into F. If x e U, y e V and w is a bilinear functional, then x Cξ) y is m defined by: x y(w) w(x,y). If e 19,e m and f 19 *,/ w are bases for U and F, respectively, then the e t <g)fj, i 1,, m, j = 1,, n, form a basis for U V. Let M mtλ denote the vector space of m x n matrices over F. Then U V is isomorphic to M m>n under the mapping ψ where ψiβi^fj) = 2? tj, and E tj is the matrix with 1 in the (i, i) position and 0 elsewhere. An element z e Ϊ7 F is said to be of rank k if z = Σ L A i/*, where m x u *",Xjc are linearly independent and so are y lf 9y k. If i? fc = {2; e Z7 FI rank (2;) = k}, then ^(i2 fc ) is the set of matrices of rank k, in M mtn. In view of the isomorphism any linear map T of U <S) V into itself can be considered as a linear map of M m>n into itself. In [2] and [3], Hua and Jacob obtained the structure of any mapping T that preserves the rank of every matrix in M m>n and whose inverse exists and has this property (coherence invariance). (In [3] F is replaced by a division ring, and T is shown to be semi-linear by appealing to the fundamental theorem of projective geometry.) In [4] we obtained the structure of T when m = n, T is linear and T preserves rank 1, 2 and n. Specifically, there exist non-singular matrices M and N such that T(A) - MAN for all A e M nn, or T(A) = MA'N for all A, where A! designates the transpose of A. Frobenius (cf. [1], p. 249) obtained this result when T is a a linear map which preserves the determinant of every A. In [5] it was shown that this result can be obtained by requiring only that T be linear and preserve rank n. In the present paper we show that rank 1 suffices (Theorem 1), or rank 2 with the side condition that T maps no matrix of rank 4 or less into 0 (Theorem 2). Thus our hypothesis will be that T is linear and T(R 1 ) (^R ι. We remark that T may be singular and still its kernel may have a zero intersection with R λ \ e.g., take U = F and 2. Rank one preservers* Throughout this section T will be a linear transformation (l.t.) of U V into U V such that T(R,) c R λ. Here Received March 2, The work of the first author was sponsored by U.S. National Science Foundation Grant G

3 1216 M. MARCUS AND B. N. MOYLS U and V are m- and ^-dimensional vector spaces over F. e m aud f x,/ be fixed bases for U and V, and set Let e lf, (1) ΓίβiΘΛ) = ^ 0 ^, i = 1, «,ra; i = 1,,%. Note that no u 4J or v tj can be zero. We shall show, in case m Φ n that there exist vectors u t and v 3 such that T(e i (g)fj) = u i (ξ v J, and hence that the l.t. T is a tensor product of transformations on U and V separately. In case m = n it will be shown that a slight modification of T is a tensor product. Denote by L(x lf, # { ) the subspace spanned by the vectors ^,, x tf and let />(#!,, x t ) be the dimension of L(x 19, # t ). LEMMA 1. Lei ^,,x ί., w x,,w s be vectors in U, and let y 19, y r, Zi, *, z s be vectors in V. Let (2) Σ(34 l/ι)=σk 3,). 2/ />(Xi,, x r ) r, then y t e Lfe,, z 8 ), i = 1,, r αm similarly if PiVu *,yr) = r, then x t e L(^,, w s ), i = 1,, r. Proof. Suppose that ^(a?^, x r ) r. Let Θ be a linear functional on C/ such that 0(0^) = 1, ^(xj = 0, ί Φ 1, and let α be an arbitrary linear functional on V. For x e U, y e V, define ( 3 ) g(x, y) = θ(x)a(y). Applying (2) to g, we get d = Σ 0(Wj)a(Zj) = a ί l where each 0(^) is a scalar. Since a is arbitrary, ^, and similarly 2/2» > 2/r> a r e contained in L(^,, «s ). The second part of the lemma is proved in the same way. LEMMA 2. If T(Rj) c R lf and T satisfies (1), then for i = 1,, m, either (4) P^,...,u ίn ) = n or ( 5 ) />(w tl,, u in ) = 1 αncί /)(?;,!,., v ln ) = w. Similarly, for j = 1,, w, either (6 ) jθ(w u,, u m; ) = m and ^(v^,, v TO^) = 1, or

4 TRANSFORMATIONS ON TENSOR PRODUCT SPACES 1217 ( 7 ) (u ιjy, u mj ) = 1 and (v lj9, v mj ) = m. Proof. Suppose that u icύ and u iβ are independent. Then T(e t <g) (/* + / β )) - (w 4Λ <g) v <Λ ) + (u tβ <g) v^) must be a tensor product w (g) v. By Lemma 1, ^ία5, v tβ e L(v). Since all v i3 Φ 0, L(t; ίfl> ) = L(v iβ ). For y Φ a, β, L(v iy ) = L(t; ίβ ), since w ίy must be independent of at least one of u iΰ6, u iβ. We have shown that if ρ(u ίlf, u in ) > 2, then p^,..., v in ) = 1. Suppose next that ^^^, % 4w ) = 1, viz., u loύ c Λ u ilf c α Φ 0, α = 1,, w. If i,, v in ) <n y be a non-trivial dependence relation. let Σ α*^i«= 0 Then = Σ (c«un (g) -^-O = tt 41 (8) ( Σ α Λ v, β ) = 0, l V / \ / which is impossible by the nature of T. Hence p(u ilf,u in ) = 1 implies ^0*!,, v in ) = n. It follows by a similar argument that if p(v il9 *,v in ) = 1, then jθ(t6 41,, u tn ) = n. Hence either (4) or (5) must hold. The second part of the lemma is proved similarly. We remark that if m < n (or n < m), then (4) (or (7)) cannot hold. LEMMA 3. Either (4) and (7) hold for all ί,j; or (5) and (6) hold for all i, j. Proof. We show first that either (4) or (5) holds uniformly in i. Suppose that for some i and k, 1 < i < k < m, p(u tl,, u in ) n while l (w*i>, ^n) = 1- Then for some a, 1 < a < n, p{u icύy u koc ) 2. For β φ a consider = c(u ίa (g) v tλ ) + (tt ij3 (8) ι; 4β ) + c(^fcα 0 v fcαj ) + (^fcβ (g) v fcj3 ), where c is an arbitrary scalar. By hypothesis and Lemma 2, v ioύ av kai and v ιβ = 6^^ = for δ^fcαί suitable non-zero scalars a and 6, while p(v ka, v kβ ) 2. Thus η = + (αc^ίλ 6^ίβ + c^fcα5 ) (g) ^fcα> + (u kβ (g) v Λ/3 ), and by Lemma 1, ρ(acu ia + bu iβ + CMfcαs* 'Mfcjs) = 1 for all scalars c. Since p{n k0clj u kβ ) = 1, this implies that P(cu ίcύ + u ίβ, u kβ ) = 1 for all c. This is impossible, since p(u ia, u ίβ ) 2. Thus either (4) is true for all i, or (5) is true for all i. A similar argument applies to (6) and (7).

5 1218 M. MARCUS AND B. N. MOYLS If (4) and (6) hold for all i and j, then there exist non-zero scalars c tj such that v tj c tj v U9 i = 1,, m, j = 1., n. For a j9 b scalars, consider K m \ "Ί / m w Σ «ιβ,) <8) (/i - δ/ 2 ) = ( Σ «Ai^ii - δ Σ α Let ^, *,2 m and w lf,w m be the m-column vectors which are respectively the representations of u n,, u ml and u 129, w m2 with respect to the basis e 19 *,e m. Let C be the m-square matrix whose columns are c n z lf, c TO i m and let W be the m-square matrix whose columns are c n w 19, c m2 w m. Then with respect to the basis e ί9, e m the vector Σί^i α «c ίi^ίi ~ δ Σί^i <χ i c ί2 /^i2 has the representation (C bw)a where a is the column m-tuple (a 19, α TO ). Now C and W are non-singular since fk M n> #» u^i) = i (^i2> % ^2) = w, so choose b to be an eigenvalue of W~ X C and choose a to be the corresponding eigenvector. Then (C bw)a = 0 and hence there exist scalars a 19 *,a m not all 0 and 6 such that a contradiction since T{R^) i2 lβ Hence (4) and (6) cannot hold for all i and j. Similarly both (5) and (7) cannot hold for all i and j. This completes the proof of the lemma. In view of the remark preceding this lemma, (5) and (6) must hold when m Φ n. THEOREM 1. Let U and V be m- and n-dimensional vector spaces respectively. Let T be a linear transformation on Z7(x) V which maps elements of rank one into elements of rank one. Let T x be the l.t. of V U into Z7 V which maps y 0 x onto x(g)y. If m n 9 let φ be any non-singular l.t. of U onto V. Then if m Φ n, there exist nonsingular l.t.'s A and B on U and V, respectively, such that T A 0 B. If m n, there exist non-singular A and B such that either ' T = 4 ΰ or T = T x {φa 0 φ^b). Proof. By (1) and Lemma 3, T(e t 0/j) = u i3 0 v ij9 i = 1,, m, j = 1, -, w, where either (5) and (6) hold or (4) and (7) hold. Suppose first that the former is the case; in particular, p(u lly,% <n ) = 1 for i 1,, m and p(v υ,, v wi ) = 1 for j = 1,, w. Then there exist non-zero scalars s ijf t tj such that w 4i = s^u^ and v^ = t ij v υ. Thus (8) Γ( where u % u il9 v } = ^, and c^ = s t /^. For i 2,, n 9

6 TRANSFORMATIONS ON TENSOR PRODUCT SPACES 1219 must be a direct product a? (g) w. By (6) and Lemma 1, Σ?=AΛ = ^ί ΣJ=I C IJ^J f r some constant d t. By (5), c tj = d^. Hence (9) T(e i f J ) = x i y j, where x t d^ and y 3 c υ Vj. Since the {a?j and {I/J} are each linearly independent sets, there non-singular linear transformations A and i? such that x t = Ae t and y 3 = /,. Then Γ = A (g) JB. When m n, (4) and (7) may hold in particular, P(vu,, ^ι») = 1 and ρ(u 1Jf, u nj ) = 1 for i,,/ = 1,, n. As in the preceding case, there exist linearly independent sets x 19, x w and 2/1, *,Vn such that (10) 7(^0/,) = ^!/,. There exist non-singular transformations A and B oί U and V, respectively, such that Ae t φ'^yt and JB/^ = φx jt ί, j = 1,,n. Thus ΓΓ'Γίe* Λ) - ^Ae 4 (g) ^~ 1 5/,. Q.E.D. In matrix language we have the following. COROLLARY. Let T be a l.t. on the space M nn of n-square matrices. If the set of rank one matrices is invariant under T, then there exist non-singular matrices A and B such that either T(X) = AXB for all X e M nn or T(X) = AX'B for all X e M nn. 3 Rank two preservers. In this section T will be a l.t. of Z7 ξ) V such that T(R 2 ) c R 2. We shall show that under certain conditions T(R X ) c R lm LEMMA 4. If W is a subspace of U(&V such that, for some integer r, 1 < r < min (m, n), (11) dimw^ > mn r max (m, n) + 1, Proof. Suppose that m = max (m,n). The products β«(g)/,, i = 1,, m, i = 1,, r, are linearly independent and span a space T^ of dimension mr. Furthermore, W 1 c Uϊ-A T h e n dim(w 1 ΠT7) = dim TΓi + dim PF - dim (W 1 U TF) > mr + (mw - rm + 1) - mn = 1. The result follows, since WΊ n W c U? i=i^j Π W. LEMMA 5. // Γ(Λ a ) c Γ(JK a ) c i? 2, ίfee^ Γ^) c #> u

7 1220 M. MARCUS AND B. N. MOYLS Proof. Suppose x x^y x e R lt and choose x 2 (&y 2 e R 1 such that p(x lf x 2 ) = ρ(y lf y 2 ) = 2. Then a = st^ (g) j/j + ίγ(α; 2 (g) t/ 2 ) e i? 2 for all non-zero scalars s, t. Now suppose that T(x x (g) y x ) = ΣJ-I^J v j, where p(u lf,%,,) = ^(Vi,, v p ) = p, and that T(x 2 (g)?/ 2 ) = Σ?-i^ w,, where pfe,, 3 Q ) = /o(wi,, w q ) g. Let % p+1,, u m be a completion of ^i> *» ^p to a basis for J7. It follows that for some vectors h 3 e V, j P = 1,..., m. Then + Σ% (g) thj + Σ, + th } ) Since a e R 2, it follows by Lemma 1 that p{sv ι + th u ι >,sv p + t 'K) < 2 fc The vectors s^ + tλ lf, sv p + th p are linearly independent when s = 1 and ί = 0. By continuity, they remain independent for small values of t. Hence p < 2 and T{x 1^y 1 ) e R x U i? 2. THEOREM 2. // T(R 2 ) c i? 2 αticί 0 Proof. Suppose x x (g) y x 6 ^ and 7(0?! 2/0 0 Λi By Lemma 5, T(x 1 (8) ί/0 e # 2, since 0 0 T{R,). Thus Tί^! (g) 2/0 = (u, ^ + (^ (g) v 2 ), where jo^, u 2 ) = io('?; 1, v 2 ) = 2. Let x 1?, x m and 7/!,, y n be bases for U and F respectively. Then for st Φ 0 (12) st(x 1 2/ x ) + ίγ(α? 4 ^) e R,U R 2 for ΐ = 1,, m, j = 1,, n. At this point it seems simpler to regard the images T{x t (g) y 3 ) as elements of M mn. It is clear that there is no loss in generality in taking Let i and j be fixed for this discussion, and let A = T(Xi (g) ^). Let a lf,α n be the m-dimensional vectors which are the columns of A, and let ε ft be the unit vector with 1 in the &th position. It follows from (12) that (13) p(sε 1 + ta ly se 2 + ta 2, ta ZJ, ta n ) = 2 for si =^ 0. The Grassmann products (14) (se ι + taj Λ (se 2 + ία 2 ) Λ ta*, 3 < fc < w

8 TRANSFORMATIONS ON TENSOR PRODUCT SPACES 1221 must be zero for st Φ 0. In the expansion of (14) the coefficient of s 2 t is 0; that is, ε 1 Λ ε 2 Λ α* 0. Thus the matrix A has non-zero entries only in the first two rows and columns. It follows immediately that the dimension of the range of T < 2(m + n) 4. Hence the dimension of the kernel of T > mn 2(m + n) + 4 > mn 4 max (m, n) + 1. By Lemma 4, there exists an element of Uί=i whose image is zero. This contradicts the hypothesis; hence T{R^) c R lm We see then that the form of T satisfying Theorem 2 is given in the conclusions of Theorem 1. REMARK. We feel that the hypothesis 0 $ T(\J* jml R 3 ) of Theorem 2 should not be necessary, but we have not been able to prove the theorem without it. More generally, we conjecture that T(R k ) c R k for some fixed fc, 1 < k < n, should suffice to prove that T is essentially a tensor product. REFERENCES 1. E. B. Dynkin, Maximal subgroups of the classical groups, A.M.S. Translations, Series 2, 6 (1957), (Original: Trudy Moskov. Mat. Obsc. 1 (1952), ) 2. L. K. Hua, A theorem on matrices and its application to Grassmann space, Science Reports of the National Tsing Hua University, Ser. A, 5 (1948), H. G. Jacob, Coherence invariant mappings on Kronecker products, Amer. J. Math. 77 (1955), Marvin Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Can. J. Math. 11 (1959) Marvin Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Can. J. Math. 11 (1959), UNIVERSITY OF BRITISH COLUMBIA

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10 PACIFIC JOURNAL OF MATHEMATICS DAVID GILBARG Stanford University Stanford, California R. A. BEAUMONT University of Washington Seattle 5, Washington EDITORS A. L. WHITEMAN University of Southern California Los Angeles 7, California L. J. PAIGE University of California Los Angeles 24, California E. F. BECKENBACH C. E. BURGESS E. HEWITT A. HORN ASSOCIATE EDITORS V. GANAPATHY IYER R. D. JAMES M. S. KNEBELMAN L. NACHBIN I. NIVEN T. G. OSTROM H. L. ROYDEN M. M. SCHIFFER E. G. STRAUS G. SZEKERES F. WOLF K. YOSIDA UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA OREGON STATE COLLEGE UNIVERSITY OF OREGON OSAKA UNIVERSITY UNIVERSITY OF SOUTHERN CALIFORNIA SUPPORTING INSTITUTIONS STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE COLLEGE UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY CALIFORNIA RESEARCH CORPORATION HUGHES AIRCRAFT COMPANY SPACE TECHNOLOGY LABORATORIES Mathematical papers intended for publication in the Pacific Journal of Mathematics should be typewritten (double spaced), and the author should keep a complete copy. Manuscripts may be sent to any one of the four editors. All other communications to the editors should be addressed to the managing editor, L. J. Paige at the University of California, Los Angeles 24, California. 50 reprints per author of each article are furnished free of charge; additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is published quarterly, in March, June, September, and December. The price per volume (4 numbers) is $12.00; single issues, $3.50. Back numbers are available. Special price to individual faculty members of supporting institutions and to individual members of the American Mathematical Society: $4.00 per volume; single issues, $1.25. Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 2120 Oxford Street, Berkeley 4, California. Printed at Kokusai Bunken Insatsusha (International Academic Printing Co., Ltd.), No. 6, 2-chome, Fujimi-cho, Chiyoda-ku, Tokyo, Japan. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies.

11 Pacific Journal of Mathematics Vol. 9, No. 4 August, 1959 Frank Herbert Brownell, III, A note on Kato s uniqueness criterion for Schrödinger operator self-adjoint extensions Edmond Darrell Cashwell and C. J. Everett, The ring of number-theoretic functions Heinz Otto Cordes, On continuation of boundary values for partial differential operators Philip C. Curtis, Jr., n-parameter families and best approximation Uri Fixman, Problems in spectral operators I. S. Gál, Uniformizable spaces with a unique structure John Mitchell Gary, Higher dimensional cyclic elements Richard P. Gosselin, On Diophantine approximation and trigonometric polynomials Gilbert Helmberg, Generating sets of elements in compact groups Daniel R. Hughes and John Griggs Thompson, The H-problem and the structure of H-groups James Patrick Jans, Projective injective modules Samuel Karlin and James L. McGregor, Coincidence properties of birth and death processes Samuel Karlin and James L. McGregor, Coincidence probabilities J. L. Kelley, Measures on Boolean algebras John G. Kemeny, Generalized random variables Donald G. Malm, Concerning the cohomology ring of a sphere bundle Marvin David Marcus and Benjamin Nelson Moyls, Transformations on tensor product spaces Charles Alan McCarthy, The nilpotent part of a spectral operator Kotaro Oikawa, On a criterion for the weakness of an ideal boundary component Barrett O Neill, An algebraic criterion for immersion Murray Harold Protter, Vibration of a nonhomogeneous membrane Victor Lenard Shapiro, Intrinsic operators in three-space Morgan Ward, Tests for primality based on Sylvester s cyclotomic numbers L. E. Ward, A fixed point theorem for chained spaces Alfred B. Willcox, Šilov type C algebras over a connected locally compact abelian group Jacob Feldman, Correction to Equivalence and perpendicularity of Gaussian processes