The optimal version of Hua s fundamental theorem of geometry of matrices

Transcription

1 The optimal version of Hua s fundamental theorem of geometry of matrices Peter Šemrl Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 SI-1 Ljubljana Slovenia peter.semrl@fmf.uni-lj.si Abstract Hua s fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all m n matrices over a division ring D which preserve adjacency in both directions. Motivated by several applications we study a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings we solve all three problems simultaneously, thus obtaining the optimal version of Hua s theorem. In the case of general division rings we get such an optimal result only for square matrices and give examples showing that it cannot be extended to the non-square case. AMS classification: 15A3, 51A5. Keywords: rank, adjacency preserving map, matrix over a division ring, geometry of matrices. The author was supported by a grant from ARRS, Slovenia 1

2 1 Introduction Let D be a division ring and m, n positive integers. By M m n (D) we denote the set of all m n matrices over D. If m = n we write M n (D) = M n n (D). For an arbitrary pair A, B M m n (D) we define d(a, B) = rank (A B). We call d the arithmetic distance. Matrices A, B M m n (D) are said to be adjacent if d(a, B) = 1. If A M m n (D), then t A denotes the transpose of A. In the series of papers [4] - [11] Hua initiated the study of bijective maps on various spaces of matrices preserving adjacency in both directions. Let V be a space of matrices. Recall that a map φ : V V preserves adjacency in both directions if for every pair A, B V the matrices φ(a) and φ(b) are adjacent if and only if A and B are adjacent. We say that a map φ : V V preserves adjacency (in one direction only) if φ(a) and φ(b) are adjacent whenever A, B V are adjacent. Hua s fundamental theorem of the geometry of rectangular matrices (see [25]) states that for every bijective map φ : M m n (D) M m n (D), m, n 2, preserving adjacency in both directions there exist invertible matrices T M m (D), S M n (D), a matrix R M m n (D), and an automorphism τ of the division ring D such that φ(a) = T A τ S + R, A M m n (D). (1) Here, A τ = [a ij ] τ = [τ(a ij )] is a matrix obtained from A by applying τ entrywise. In the square case m = n we have the additional possibility φ(a) = T t (A σ )S + R, A M n (D), (2) where T, S, R are matrices in M n (D) with T, S invertible, and σ : D D is an anti-automorphism. Clearly, the converse statement is true as well, that is, any map of the form (1) or (2) is bijective and preserves adjacency in both directions. Composing the map φ with a translation affects neither the assumptions, nor the conclusion of Hua s theorem. Thus, there is no loss of generality in assuming that φ() =. Then clearly, R =. It is a remarkable fact that after this harmless normalization the additive (semilinear in the case when D is a field) character of φ is not an assumption but a conclusion. This beautiful result has many applications different from the original Hua s motivation related to complex analysis and Siegel s symplectic geometry. Let us mention here two of them that are especially important to us. There is a vast literature on linear preservers (see [16]) dating back to 1897 when Frobenious [3] described the general form of linear maps on square matrices that preserve determinant. As explained by Marcus [17], most of linear preserver problems can be reduced to the problem of characterizing linear maps that preserve matrices of rank one. Of course, linear preservers of rank one preserve adjacency, and therefore, most of linear preserver results can be deduced from Hua s theorem. 2

3 When reducing a linear preserver problem to the problem of rank one preservers and then to Hua s theorem, we end up with a result on maps on matrices with no linearity assumption. Therefore it is not surprising that Hua s theorem has been already proved to be a useful tool in the new research area concerning general (non-linear) preservers. It turns out that the fundamental theorem of geometry of Grassmann spaces [2] follows from Hua s theorem as well (see [25]). Hence, improving Hua s theorem one may expect to be able to also improve Chow s theorem [2] on the adjacency preserving maps on Grassmann spaces. Motivated by applications we will be interested in possible improvements of Hua s theorem. The first natural question is whether the assumption that adjacency is preserved in both directions can be replaced by the weaker assumption that it is preserved in one direction only and still get the same conclusion. This question had been opened for a long time and has finally been answered in the affirmative in [13]. Next, one can ask if it is possible to relax the bijectivity assumption. The first guess might be that Hua s theorem remains valid without bijectivity assumption with a minor modification that τ appearing in (1) is a nonzero endomorphism of D (not necessarily surjective), while σ appearing in (2) is a nonzero anti-endomorphism. Quite surprisingly it turned out that the validity of this conjecture depends on the underlying field. It was proved in [19] that it is true for real matrices and wrong for complex matrices. And the last problem is whether we can describe maps preserving adjacency (in both directions) acting between spaces of matrices of different sizes. Let us mention here Hua s fundamental theorem for complex hermitian matrices. Denote by H n the space of all n n complex hermitian matrices. The fundamental theorem of geometry of hermitian matrices states that every bijective map φ : H n H n preserving adjacency in both directions and satisfying φ() = is a congruence transformation possibly composed with the transposition and possibly multiplied by 1. Here, again we can ask for possible improvements in all three above mentioned directions. Huang and the author have answered all three questions simultaneously in the paper [12] by obtaining the following optimal result. Let m, n be integers with m 2 and φ : H m H n a map preserving adjacency (in one direction only; note that no surjectivity or injectivity is assumed and that m may be different from n) and satisfying φ() = (this is, of course, a harmless normalization). Then either φ is the standard embedding of H m into H n composed with the congruence transformation on H n possibly composed with the transposition and possibly multiplied by 1; or φ is of a very special degenerate form, that is, its range is contained in a linear span of some rank one hermitian matrix. This result has already been proved to be useful including some applications in mathematical physics [23, 24]. It is clear that the problem of finding the optimal version of Hua s fundamental theorem of geometry of rectangular matrices is much more complicated than the corresponding problem for hermitian matrices. Classical Hua s results 3

4 characterize bijective maps from a certain space of matrices onto itself preserving adjacency in both directions. While in the hermitian case we were able to find the optimal result by improving Hua s theorem in all three directions simultaneously (removing the bijectivity assumption, assuming that adjacency is preserved in one direction only, and considering maps between matrix spaces of different sizes), we have seen above that when considering the corresponding problem on the space of rectangular matrices we enter difficulties already when trying to improve it in only one of the three possible directions. Namely, for some division rings it is possible to omit the bijectivity assumption in Hua s theorem and still get the same conclusion, but not for all. In the third secion we will present several new examples showing that this is not the only trouble we have when searching for the optimal version of Hua s theorem for rectangular matrices. Let m, n, p, q be positive integers with p m and q n, τ : D D a nonzero endomorphism, and T M p (D) and S M q (D) invertible matrices. Then the map φ : M m n (D) M p q (D) defined by A τ φ(a) = T S (3) preserves adjacency. Similarly, if m, n, p, q are positive integers with p n and q m, σ : D D a nonzero anti-endomorphism, and T M p (D) and S M q (D) invertible matrices, then φ : M m n (D) M p q (D) defined by [ t ] (A φ(a) = T σ ) S (4) preserves adjacency as well. We will call any map that is of one of the above two forms a standard adjacency preserving map. Having in mind the optimal version of Hua s theorem for hermitian matrices it is natural to ask whether each adjacency preserving map between M m n (D) and M p q (D) is either standard or of some rather simple degenerate form that can be easily described. As we shall show in the third section, maps φ : M m n (D) M p q (D) which preserve adjacency in one direction only can have a wild behaviour that cannot be easily described. Thus, an additional assumption is required if we want to have a reasonable result. As we want to have an optimal result we do not want to assume that matrices in the domain are of the same size as those in the codomain, and moreover, we do not want to assume that adjacency is preserved in both directions. Standard adjacency preserving maps are not surjective in general. They are injective, but the counterexamples will show that the injectivity assumption is not strong enough to exclude the possibility of a wild behaviour of adjacency preserving maps. Hence, we are looking for a certain weak form of the surjectivity assumption which is not artificial, is satisfied by standard maps, and guarantees that the general 4

5 form of adjacency preserving maps satisfying this assumption can be easily described. Moreover, such an assumption must be as weak as possible so that our theorem can be considered as the optimal one. In order to find such an assumption we observe that adjacency preserving maps are contractions with respect to the arithmetic distance d. More precisely, assume that φ : M m n (D) M p q (D) preserves adjacency, that is, for every pair A, B M m n (D) we have d(a, B) = 1 d(φ(a), φ(b)) = 1. Using the facts (see the next section) that d satisfies the triangle inequality and that for every positive integer r and every pair A, B M m n (D) we have d(a, B) = r if and only if there exists a chain of matrices A = A, A 1,..., A r = B such that the pairs A, A 1, and A 1, A 2, and..., and A r 1, A r are all adjacent we easily see that φ is a contraction, that is d(φ(a), φ(b)) d(a, B), A, B M m n (D). In particular, d(φ(a), φ(b)) min{m, n} for all A, B M m n (D). We believe that the most natural candidate for the additional assumption that we are looking for is the condition that there exists at least one pair of matrices A, B M m n (D) such that d(φ(a ), φ(b )) = min{m, n}. (5) Of course, standard maps φ : M m n (D) M p q (D) satisfy this rather weak assumption. Our first main result will describe the general form of adjacency preserving maps φ : M n (D) M p q (D), n 3, having the property that there exists at least one pair of matrices A, B M n (D) such that d(φ(a ), φ(b )) = n. It turns out that such maps can have a certain degenerate form. But even if they are not degenerate, they might be far away from being standard. Nevertheless, the description of all possibile forms will still be quite simple. In the non-square case, that is, the case when the domain of the map φ is the space of all m n matrices with m possibly different from n, we need to restrict to matrices over EAS division rings. For such matrices we will prove the desired optimal result stating that all adjacency preserving maps satisfying (5) are either standard, or of a certain degenerate form. The next section is devoted to notation and basic definitions. Then we will present several examples of adjacency preserving maps, some of them quite complicated. Having these examples it will be easy to understand the necessity of the assumption (5) in the statement of our main results. At the same time these examples will show that our results are indeed optimal. In particular, we will show that in the non-square case the behaviour of adjacency preserving maps satisfying (5) can be very wild in the absence of the EAS assumption on 5

6 the underlying division ring. And finally, the last section will be devoted to the proofs. When dealing with such a classical problem it is clear that the proofs depend a lot on the techniques developed in the past. However, we will deal with adjacency preserving maps under much weaker conditions than in any of the previous works on this topic, and also the description of such maps in this more general setting will differ a lot from the known results. It is therefore not surprising that many new ideas will be needed to prove our main theorems. 2 Notation and basic definitions Let us recall the definition of the rank of an m n matrix A with entries in a division ring D. We will always consider D n, the set of all 1 n matrices, as a left vector space over D. Correspondingly, we have the right vector space of all m 1 matrices t D m. We first take the row space of A, that is the left vector subspace of D n generated by the rows of A, and define the row rank of A to be the dimension of this subspace. Similarly, the column rank of A is the dimension of the right vector space generated by the columns of A. This space is called the column space of A. It turns out that these two ranks are equal for every matrix over D and this common value is called the rank of a matrix. Assume that rank A = r. Then there exist invertible matrices T M m (D) and S M n (D) such that Ir T AS =. (6) Here, I r is the r r identity matrix and the zeroes stand for zero matrices of the appropriate sizes. Let r be a positive integer, 1 r min{m, n}. Then we denote by M r m n(d) the set of all matrices A M m n (D) of rank r. Of course, we write shortly M r n(d) = M r n n(d). In general the rank of a matrix A need not be equal to the rank of its transpose t A. However, if τ : D D is a nonzero anti-endomorphism (that is, τ is additive and τ(λµ) = τ(µ)τ(λ), λ, µ D) of D, then rank A = rank t (A τ ). Here, A τ = [a ij ] τ = [τ(a ij )] is a matrix obtained from A by applying τ entrywise. Rank satisfies the triangle inequality, that is, rank (A+B) rank A+rank B for every pair A, B M m n (D) [14, p.46, Exercise 2]. Therefore, the set of matrices M m n (D) equipped with the arithmetic distance d defined by d(a, B) = rank (A B), A, B M m n (D), is a metric space. Matrices A, B M m n (D) are said to be adjacent if d(a, B) = 1. Let a D n and t b t D m be any nonzero vectors. Then t ba = ( t b)a is a matrix of rank one. Every matrix of rank one can be written in this form. It is easy to verify that two rank one matrices t ba and t dc, t ba t dc, are adjacent if and only if a and c are linearly dependent or t b and t d are linearly dependent. 6

7 As usual, the symbol E ij, 1 i m, 1 j n, will stand for a matrix having all entries zero except the (i, j)-entry which is equal to 1. For a nonzero x D n and a nonzero t y t D m we denote by R(x) and L( t y) the subsets of M m n (D) defined by R(x) = { t ux : t u t D m } and L( t y) = { t yv : v D n }. Clearly, all the elements of these two sets are of rank at most one. Moreover, any two distinct elements from R(x) are adjacent. And the same is true for L( t y). The elements of the standard basis of the left vector space D n ( the right vector space t D m ) will be denoted by e 1,..., e n ( t f 1,..., t f m ). Hence, E ij = t f i e j, 1 i m, 1 j n. Later on we will deal simultaneously with rectangular matrices of different sizes, say with matrices from M m n (D) and M p q (D). The same symbol E ij will be used to denote the matrix unit in M m n (D) as well as the matrix unit in M p q (D). As always we will identify m n matrices with linear transformations mapping D m into D n. Namely, each m n matrix A gives rise to a linear operator defined by x xa, x D m. The rank of the matrix A is equal to the dimension of the image Im A of the corresponding operator A. The kernel of an operator A is defined as Ker A = {x D m : xa = }. It is the set of all vectors x D m satisfying x( t y) = for every t y from the column space of A. We have m = rank A + dim Ker A. We will call a division ring D an EAS division ring if every nonzero endomorphism τ : D D is automatically surjective. The field of real numbers and the field of rational numbers are well-known to be EAS. Obviously, every finite field is EAS. The same is true for the division ring of quaternions (see, for example [2]), while the complex field is not an EAS field [15]. Let D be an EAS division ring. It is then easy to verify that also each nonzero anti-endomorphism of D is bijective (just note that the square of a nonzero anti-endomorphism is a nonzero endomorphism). We denote by P n (D) M n (D) the set of all n n idempotent matrices, P n (D) = {P M n (D) : P 2 = P }. The symbol Pn(D) 1 stands for the subset of all rank one idempotent matrices. Let a D n and t b t D n be any nonzero vectors. Then the rank one matrix t ba is an idempotent if and only if a( t b) = a t b = 1. It is well-known that P n (D) is a partially ordered set (poset) with partial order defined by P Q if P Q = QP = P. A map φ : P n (D) P n (D) is order preserving if for every pair P, Q P n (D) we have P Q φ(p ) φ(q). We shall need the following fact that is well-known for idempotent matrices over fields and can be also generalized to idempotent matrices over division rings [14, p.62, Exercise 1]. Assume that P 1,..., P k P n (D) are pairwise orthogonal, 7

8 that is, P m P j = whenever m j, 1 m, j k. Denote by r i the rank of P i. Then there exists an invertible matrix T M n (D) such that for each i, 1 i k, we have T P i T 1 = diag (,...,, 1,..., 1,,..., ) where diag (,...,, 1,..., 1,,..., ) is the diagonal matrix in which all the diagonal entries are zero except those in (r r i 1 +1)st to (r r i )th rows. Let P, Q P n (D). If P Q then clearly, Q P is an idempotent orthogonal to P. Thus, by the previous paragraph, we have P Q, P, Q I, and P Q if and only if there exist an invertible T M n (D) and positive integers r 1, r 2 such that T P T 1 = I r 1 and T QT 1 = I r 1 I r2 and < r 1 < r 1 + r 2 < n. In particular, if we identify matrices with linear operators, then the image of P is a subspace of the image of Q, while the kernel of Q is a subspace of the kernel of P. For a nonzero x D n and a nonzero t y t D n we denote by P R(x) and P L( t y) the subsets of P n (D) defined by P R(x) = { t ux : t u t D n, x t u = 1} and P L( t y) = { t yv : v D n, v t y = 1}. Clearly, all the elements of these two sets are of rank one. Further, if t ux, t wx P R(x) for some t u, t w t D n, then either t u = t w, or t u and t w are linearly independent. Moreover, if nonzero vectors x 1 and x 2 are linearly dependent then P R(x 1 ) = P R(x 2 ). By P(D n ) and P( t D n ) we denote the projective spaces over left vector space D n and right vector space t D n, respectively, P(D n ) = {[x] : x D n \ {}} and P( t D n ) = {[ t y] : t y t D n \{}}. Here, [x] and [ t y] denote the one-dimensional left vector subspace of D n generated by x and the one-dimensional right vector subspace of t D n generated by t y, respectively. 3 Examples Let us first emphasize that all the examples presented in this section are new. There is only one exception. Namely, our first example is just a slight modification of [19, Theorem 2.4]. 8

9 Assume that D is a non-eas division ring. Let τ be a nonzero nonsurjective endomorphism of D. Choose c D that is not contained in the range of τ and define a map φ : M m n (D) M m n (D) by a 11 a a 1n φ a m 2,1 a m 2,2... a m 2,n a m 1,1 a m 1,2... a m 1,n a m1 a m2... a mn τ(a 11 ) τ(a 12 )... τ(a 1n ) = τ(a m 2,1 ) τ(a m 2,2 )... τ(a m 2,n ). τ(a m 1,1 ) + cτ(a m1 ) τ(a m 1,2 ) + cτ(a m2 )... τ(a m 1,n ) + cτ(a mn )... The map φ is additive and injective. Indeed, assume that φ([a ij ]) =. Then clearly, a ij = whenever 1 i m 2 and 1 j n. From τ(a m 1,1 ) + cτ(a m1 ) = we conclude that τ(a m1 ) =, since otherwise c would belong to the range of τ. Thus, a m1 =, and consequently, a m 1,1 =. Similarly we see that for every j, 1 j n, we have a ij = whenever i = m 1 or i = m. Thus, in order to verify that it preserves adjacency it is enough to see that φ(a) is of rank at most one for every A of rank one. The verification of this statement is straightforward. And, of course, we have φ() =. Several remarks should be added here. The map φ is a composition of two maps: we have first applied the endomorphism τ entrywise and then we have replaced the last row by zero and the (m 1)-st row by the sum of the (m 1)-st row and the m-th row multiplied by c on the left. We could do the same with columns instead of rows. In that case, we need to multiply by c on the right side. Of course, we could make the example more complicated by adding the scalar multiples of the m-th row to other rows as well. Also observe that the map φ preserves adjacency, but it does not preserve adjacency in both directions. Namely, if A is a nonzero matrix having nonzero entries only in the last two rows, then A may have rank two, but φ(a) is of rank one and thus, adjacent to. Over some division rings it is possible to modify the above example in such a way that we get a map preserving adjacency in both directions. To see this we will now consider complex matrices. It is known [15] that there exist an endomorphism τ : C C and complex numbers c, d C such that c, d are algebraically independent over τ(c), that is, if p(c, d) = for some polynomial 9

10 p τ(c)[x, Y ], then p =. Define now φ : M m n (C) M m n (C) by a a 1n..... φ a m 2,1... a m 2,n a m 1,1... a m 1,n a m1... a mn τ(a 11 )... τ(a 1n )..... = τ(a m 2,1 ) + dτ(a m1 )... τ(a m 2,n ) + dτ(a mn ). (7) τ(a m 1,1 ) + cτ(a m1 )... τ(a m 1,n ) + cτ(a mn )... Hence, we obtain φ(a) from A by first applying τ entrywise, then multiplying the last row by c and d, respectively, and add these scalar multiples of the last row to the (m 1)-st row, and (m 2)-nd row, respectively, and finally, replace the last row by the zero row. As before we see that φ is an injective additive map. Thus, in order to see that it preserves adjacency in both directions it is enough to show that for every A M m n (C) we have rank A = 1 rank φ(a) = 1. And again, as before we have rank A = 1 rank φ(a) = 1. So, assume that rank φ(a) = 1. Then clearly, A. We have to check that determinants of all 2 2 submatrices of A = [a ij ] are zero. We know that determinants of all 2 2 submatrices of φ(a) are zero. Take 2 2 submatrices corresponding to the first two columns, and the first two rows, the first and the (m 2)-nd row, and the (m 2)-nd row and the (m 1)-st row and calculate their determinants. Applying the fact that τ is endomorphism we get τ(a 11 a 22 a 21 a 12 ) =, and τ(a 11 a m 2,2 a m 2,1 a 12 ) + dτ(a 11 a m2 a m1 a 12 ) =, τ(a m 2,1 a m 1,2 a m 1,1 a m 2,2 ) + dτ(a m1 a m 1,2 a m 1,1 a m2 ) +cτ(a m 2,1 a m2 a m 2,2 a m1 ) =, and since c, d are algebraically independent the determinants of the following 2 2 submatrices of A must be zero: = a 11 a 22 a 21 a 12 = a 11 a m 2,2 a m 2,1 a 12 = a 11 a m2 a m1 a 12 = a m 2,1 a m 1,2 a m 1,1 a m 2,2 = a m1 a m 1,2 a m 1,1 a m2 = a m 2,1 a m2 a m 2,2 a m1. 1

11 It is now easy to verify that all 2 2 matrices of A are singular, as desired. Let p, q be integers 2 p m, 2 q n. Using the same idea several times, and then using it again with columns instead of rows, one can now construct maps φ : M m n (C) M m n (C) which preserve adjacency in both directions such that for every B M p q (C), and φ () B φ(a) = = B τ for every A M m n (C). Here, stands for a p q matrix. Assume next that D is an infinite division ring and let us construct an adjacency preserving map from M 7 (D) to M p (D), where p 3. Write D \ {} as a disjoint union D \ {} = M N L, where all the sets D, M, N, L are of the same cardinality. Choose subsets V, W M7 2 (D) such that A, B are not adjacent whenever A V and B W. Let ϕ 1 : M7 1 (D) M, ϕ 2 : M7 2 (D) \ (V W) N, ϕ 3 : M7 3 (D) L, and ϕ j : M j 7 (D) D \ {}, j = 4, 5, 6, 7, be injective maps such that the ranges of ϕ 5 and ϕ 7 are disjoint. Let ϕ 8 : V D \ {} and ϕ 9 : W D \ {} be injective maps. Define a map φ : M 7 (D) M p (D) by φ() =, φ(a) = ϕ 1 (A)E 11, A M 1 7 (D), and φ(a) = ϕ 2 (A)E 11, A M 2 7 (D) \ (V W), φ(a) = ϕ 8 (A)E 12, A V, φ(a) = ϕ 9 (A)E 21, A W, φ(a) = ϕ 3 (A)E 11, φ(a) = ϕ 4 (A)E 11 + E 12, φ(a) = E 12 + E 21 + ϕ 5 (A)E 31, A M 3 7 (D), φ(a) = E 12 + E 21 + E 33 + ϕ 6 (A)E 32, φ(a) = E 12 + E 21 + ϕ 7 (A)E 31, A M 4 7 (D), A M 5 7 (D), A M 6 7 (D), A M 7 7 (D). Clearly, φ preserves adjacency. Indeed, all we need to observe is that if matrices A and B are adjacent, then either they are of the same rank, or rank A = rank B ± 1. Moreover, it is injective. It is clear that using similar ideas one can construct further examples of adjacency preserving maps between matrix spaces with a rather wild behaviour. Moreover, a compositum of adjacency preserving maps is again an adjacency preserving map. Thus, combining the examples 11

12 obtained so far we can arrive at adjacency preserving maps whose general form can not be described easily. Therefore we will (as already explained in Introduction) restrict our attention to adjacency preserving maps φ : M m n (D) M p q (D) satisfying the additional assumption that there exist A, B M m n (D) satisfying d(φ(a ), φ(b )) = min{m, n} (then we have automatically min{p, q} min{m, n}). Clearly, standard maps satisfy this additional condition. We continue with non-standard examples of such maps. The notion of a degenerate adjacency preserving map is rather complicated. We will therefore first restrict to the special case when m n, φ() =, and φ(e E nn ) = E E nn (note that the matrix units E 11,..., E nn on the left hand side of this equation belong to M m n (D), while E 11,..., E nn on the right hand side stand for the first n matrix units on the main diagonal of M p q (D)). Later on we will see that the general case can be always reduced to this special case. We say that a point c in a metric space M with the distance function d lies in between points a, b M if d(a, b) = d(a, c) + d(c, b). Obviously, if a map f : M 1 M 2 between two metric spaces with distance functions d 1 and d 2, respectively, is a contraction, that is, d 2 (f(x), f(y)) d 1 (x, y) for all x, y M 1, and if d 2 (f(a), f(b)) = d 1 (a, b) for a certain pair of points a, b M 1, then f maps the set of points that lie in between a and b into the set of points that lie in between f(a) and f(b). Later on (see Lemma 5.1) we will prove that in M m n (D) a matrix R lies in between and E E nn M m n (D) with respect to the arithmetic distance if and only if Q R = where Q is an n n idempotent matrix. And a matrix S in M p q (D) lies in between and E E nn M p q (D) if and only if P S = where P is an n n idempotent matrix. Assume that φ : M m n (D) M p q (D) is an adjacency preserving map satisfying φ() = and φ(e E nn ) = E E nn. By the above remarks, φ maps the set Q of all matrices of the form Q R = (8) 12

13 where Q is an n n idempotent matrix into the set P of all matrices P R = with P being an n n idempotent matrix. Such a map will be called a degenerate adjacency preserving map if its restriction to Q will be of a special rather simple form. We assume that m, p, q n 3 and D is an infinite division ring. We define : Q P in the following way. Set () =. Let j be an integer, 1 < j < n, and ϕ j be a map from the set of all n n idempotent matrices of rank j into D with the property that ϕ j (Q 1 ) ϕ j (Q 2 ) whenever Q 1 and Q 2 are adjacent idempotent n n matrices of rank j. Note that R in (8) is of rank j if and only if Q is of rank j. For every rank one R Q as in (8) we define (R) = E 11 + ϕ 1 (Q)E 12, for every rank two R Q as in (8) we define (R) = E 11 + E 22 + ϕ 2 (Q)E 32, and for every rank three R Q as in (8) we define (R) = E 11 + E 22 + E 33 + ϕ 3 (Q)E 34. We continue in the same way. It is easy to guess how acts on matrices from Q of rank 4,..., n 2. If n is even, then for every rank n 1 matrix R Q as in (8) we have (R) = E E n 1,n 1 + ϕ n 1 (Q)E n 1,n, and if n is odd, then for every rank n 1 matrix R Q as in (8) we have and finally, (R) = E E n 1,n 1 + ϕ n 1 (Q)E n,n 1, (E E nn ) = E E nn. Every adjacency preserving map φ : M m n (D) M p q (D) such that its restriction to Q is equal to as defined above, will be called a degenerate adjacency preserving map. Further, assume that φ is such a map and let T 1 M m (D), S 1 M n (D), T 2 M p (D), and S 2 M q (D) be invertible matrices. Then the map A T 2 φ(t 1 AS 1 )S 2, A M m n (D), will also be called a degenerate adjacency preserving map. 13

14 We can slightly modify the above construction. We define in the following way. For every rank one R Q as in (8) we define (R) = E 11 + ϕ 1 (Q)E 21, for every rank two R Q as in (8) we define (R) = E 11 + E 22 + ϕ 2 (Q)E 23, and for every rank three R Q as in (8) we define (R) = E 11 + E 22 + E 33 + ϕ 3 (Q)E 43, and then we continue as above. We end up with another type of degenerate adjacency preserving maps φ : M m n (D) M p q (D). Let φ : M m n (D) M p q (D) be an adjacency preserving map such that its restriction to Q is equal to, where is of the first type above. Take any matrix t xy M m n (D) of rank one. Then φ( t xy) is adjacent to φ() =, and therefore, φ( t xy) is of rank one. We can find two different vectors t u, t v t D n such that y t u = y t v = 1 and [ t ] uy t xy and [ t ] vy t xy. ([ t ]) uy = E 11 + λe 12 as Then φ( t xy) is a rank one matrix adjacent to φ ([ t ]) vy well as to φ = E 11 + µe 12. Here, λ, µ are scalars satisfying λ µ. It follows easily that φ( t xy) = t f 1 z for some nonzero z D q. In other words, all rank one matrices are mapped into L( t f 1 ). Let now T 1 M m (D), S 1 M n (D), T 2 M p (D), and S 2 M q (D) be invertible matrices. Then the map A T 2 φ(t 1 AS 1 )S 2, A M m n (D), maps every rank one matrix into L(T 2 t f 1 ). Of course, we can apply the same arguments to degenerate adjacency preserving maps of the second type. We conclude that each degenerate adjacency preserving map φ : M m n (D) M p q (D) has the following property: either there exists a nonzero vector t w 1 t D p such that or φ(m 1 m n(d)) L( t w 1 ); (9) 14

15 there exists a nonzero vector w 2 D q such that φ(m 1 m n(d)) R(w 2 ). (1) We have defined degenerate adjacency preserving maps as the compositions of two equivalence transformations and an adjacency preserving map φ : M m n (D) M p q (D) whose restriction to Q is, where is as described above. There are two natural questions here. First, does such a map exist? It is straightforward to show that the answer is in the affirmative. We define φ : M m n (D) M p q (D) in the following way. Set φ() =. Let ϕ j be a map from M j m n(d) into D, j = 1,..., n 1, with the propety that ϕ j (A) ϕ j (B) whenever A, B M j m n(d) are adjacent. In particular, this property is satisfied when ϕ j is injective. Set for every A M 1 m n(d) and 1 ϕ 1 (A) φ(a) = ϕ φ(a) = 2 (A) for every A Mm n(d). 2 We continue in a similar way. For every A Mm n(d) 3 we set φ(a) = E 11 + E 22 + E 33 + ϕ 3 (A)E 34, for every A Mm n(d) 4 we set φ(a) = E 11 + E 22 + E 33 + E 44 + ϕ 4 (A)E 54,... Assume first that n is odd. Then we have φ(a) = E E n 1,n 1 + ϕ n 1 (A)E n,n 1, A Mm n(d). n 1 Let ξ 1,..., ξ q : Mm n(d) n D be any maps with the properties: If A, B M n m n(d) are adjacent, then there exists j {1, 2,..., q} such that ξ j (A) ξ j (B); If A M n 1 m n(d) and B M n m n(d) are adjacent, then either ϕ n 1 (A) ξ n 1 (B), or at least one of ξ 1 (B),..., ξ n 2 (B), ξ n (B),..., ξ q (B) is nonzero; ξ n (E E nn ) = 1 and ξ j (E E nn ) = for j = 1,..., n 1, n + 1,..., q. 15

16 We define φ(a) = ξ 1 (A) ξ 2 (A)... ξ n 1 (A) ξ n (A)... ξ q (A) for every A Mm n(d). n It is easy to verify that φ preserves adjacency and its restriction to Q is with : Q M p q (D) being the map as defined above. If n is even, then φ(a) = E E n 1,n 1 + ϕ n 1 (A)E n 1,n, A Mm n(d). n 1 Let now ξ 1,..., ξ p : Mm n(d) n D be any maps with the properties: If A, B M n m n(d) are adjacent, then there exists j {1, 2,..., p} such that ξ j (A) ξ j (B); If A M n 1 m n(d) and B M n m n(d) are adjacent, then either ϕ n 1 (A) ξ n 1 (B), or at least one of ξ 1 (B),..., ξ n 2 (B), ξ n (B),..., ξ p (B) is nonzero; ξ n (E E nn ) = 1 and ξ j (E E nn ) = for j = 1,..., n 1, n + 1,..., p. We define 1... ξ 1 (A) ξ 2 (A) φ(a) =... 1 ξ n 1 (A) ξ n (A) ξ p (A)... for every A M n m n(d). It is easy to verify that φ preserves adjacency and its restriction to Q is with : Q M p q (D) being the map as defined above. There is another possibility to construct such an adjacency preserving map from M m n (D) to M p q (D). As above we set ψ() = and choose maps ϕ j from M j m n(d) into D, j = 1,..., n 1, with the propety that ϕ j (A) ϕ j (B) 16

17 whenever A, B Mm n(d) j are adjacent. Then we define 1... ϕ 1 (A) ψ(a) = for every A Mm n(d) 1 and ϕ 2 (A) ψ(a) = for every A M 2 m n(d). One can now complete the construction of the map ψ in exactly the same way as above (in the special case when p = q the map ψ can be obtained from φ by composing it with the transposition). Thus, we have obtained two types of degenerate adjacency preserving maps from M m n (D) to M p q (D). Further examples can be obtained by composing such maps with two equivalence transformations. Note that the above degenerate adjacency preserving maps have rather simple structure and some nice properties. In particular, they almost preserve rank. Namely, we have rank φ(a) = rank A for all A M m n (D) with rank A < n and rank φ(a) {n 1, n} for all A M m n (D) with rank A = n. The same is true for the degenerate adjacency preserving map ψ. Next, degenerate adjacency preserving maps of the above type map each set M r m n(d), 1 r n 1, into a line. More precisely, let φ be as above, T 1 M m (D), S 1 M n (D), T 2 M p (D), and S 2 M q (D) be invertible matrices, and consider the degenerate adjacency preserving map A T 2 φ(t 1 AS 1 )S 2, A M m n (D). Clearly, A M m n (D) is of rank r if and only if T 1 AS 1 is of rank r. Hence, the map A T 2 φ(t 1 AS 1 )S 2, A M m n (D), maps the set M r m n(d) either into the set of matrices of the form T 2 (E E rr )S 2 + T 2 λe r,r+1 S 2, λ D, or into the set of matrices of the form T 2 (E E rr )S 2 + T 2 λe r+1,r S 2. λ D. Let us consider just the first case. Then the set M r m n(d) is mapped into the set of matrices of the form M + t xλy, λ D, (11) 17

18 where M = T 2 (E E rr )S 2, t x = T t 2 f r, and y = e r+1 S 2. In the language of geometry of matrices, the sets of matrices of the form (11) are called lines. It is also easy to verify that if A, B M m n (D) are matrices of rank n and φ : M m n (D) M p q (D) a degenerate adjacency preserving map of the above type, then either φ(a) = φ(b), or φ(a) and φ(b) are adjacent. The maps φ and ψ have been obtained by extending the map in the most natural way. Let us call the maps of the form A T 2 φ(t 1 AS 1 )S 2, A M m n (D), or of the form A T 2 ψ(t 1 AS 1 )S 2, A M m n (D), nice degenerate maps. It is natural to ask whether all degenerate adjacency preserving maps are nice? Our first guess that the answer is positive turned out to be wrong. We come now to the second question. Can we describe the general form of degenerate adjacency preserving maps? We will give a few examples of degenerate adjacency preserving maps which will show that the answer is negative. For the sake of simplicity we will consider only maps from M 3 (D) into itself. An interested reader can use the same ideas to construct similar examples on matrix spaces of higher dimensions. If we restrict to maps from M 3 (D) into itself, then we are interested in adjacency preserving maps φ : M 3 (D) M 3 (D) satisfying φ() = and rank φ(c) = 3 for some C M 3 (D) of rank three. Such a map is called degenerate if its restriction to the set of points that lie in between and C is of the special form described above. Replacing the map φ by the map A φ(c) 1 φ(ca), A M 3 (D), we may assume that φ(i) = I. The set of points that lie in between and I is the set of all idempotents. Then φ is degenerate if φ() =, φ(i) = I, (12) and either the set of rank one idempotents is mapped into matrices of the form T 1 T 1 (13) and the set of rank two idempotents is mapped into matrices of the form T 1 1 T 1 (14) and if two idempotents of the same rank are adjacent, their images are different; or the set of rank one idempotents is mapped into matrices of the form T 1 T 1 18

19 and the set of rank two idempotents is mapped into matrices of the form T 1 1 T 1 and if two idempotents of the same rank are adjacent, their images are different. Here, T is an invertible 3 3 matrix. We will assume from now on that φ is of the first type above and T = I. We need to show that it can be extended to an adjacency preserving map φ : M 3 (D) M 3 (D) with wild behaviour outside the set of idempotent matrices. This will then yield that degenerate maps have a rather simple form on the set of matrices that lie in between two matrices whose images are at the maximum possible distance with respect to the arithmetic distance, but their general form on the complement of this set cannot be described nicely. When introducing a notion of degenerate adjacency preserving maps we have started with a map defined on the set Q, and then we defined a degenerate adjacency preserving as any adjacency preserving extension of such a map composed with two equivalence transformations. The examples that we will present now show that no better definition is possible. Let D be a disjoint union of the sets D = A B such that all three sets D, A, and B are of the same cardinality. Our first example of a degenerate adjacency preserving map φ : M 3 (D) M 3 (D) is defined by (12), (13) with the belonging to A, (14) with the belonging to A, the set of rank one non-idempotent matrices is mapped by φ into matrices of the form with the (1, 3)-entry nonzero, and if A and B are two adjacent rank one nonidempotent matrices we further assume that the (1, 3)-entries of their φ-images are different, the set of rank two non-idempotent matrices is mapped by φ into matrices of the form 1 with the belonging to B, and if A and B are two adjacent rank two nonidempotent matrices we further assume that the (1, 2)-entries of their φ-images are different, and the set of rank three matrices I is mapped by φ into matrices of the form 1 1 with the belonging to B, and again we assume that if A and B are two adjacent rank matrices of rank three different from the identity, then the (3, 2)-entries of their φ-images are different. 19

20 To see that such a map preserves adjacency we assume that A, B M 3 (D) are adjacent. We need to show that then φ(a) and φ(b) are adjacent. We distinguish several cases: A = (then B must be of rank one), A is an idempotent of rank one (then B is either the zero matrix, or a rank one matrix, or a rank two matrix), A is a non-idempotent matrix of rank one (then B is either the zero matrix, or a rank one matrix, or a non-idempotent rank two matrix), A is an idempotent of rank two (then B is either a rank one idempotent, or a rank two matrix, or a rank three matrix), A is a non-idempotent matrix of rank two (then B is different form and I), A is a rank three matrix I (then B is of rank two or three), A = I (then B is either idempotent of rank two, or a rank three matrix I). It is straightforward to verify that in all of these cases φ(a) and φ(b) are adjacent. Now, we see that the behaviour on the set of non-idempotent matrices is not as simple as in the case of nice degenerate maps. First, non-idempotent rank two matrices are mapped into matrices of rank one. The set of rank one matrices is not mapped into a line. And the set of rank two matrices is not mapped into a line as well. We continue with the map φ : M 3 (D) M 3 (D) defined by (12), (13) with the belonging to A, (14) with the belonging to A, the set of rank one non-idempotent matrices is mapped by φ into matrices of the form 1 with the belonging to B, and if A and B are two adjacent rank one nonidempotent matrices we further assume that the (1, 2)-entries of their φ-images are different, the set of rank two non-idempotent matrices is mapped by φ into matrices of the form 1 with the (2, 2)-entry, 1, and if A and B are two adjacent rank two nonidempotent matrices we further assume that the (1, 2)-entries of their φ-images 2

21 are different, and the set of rank three matrices I is mapped by φ into matrices of the form 1 1 with the in the (3, 2)-position belonging to B, the star in the (1, 2)-position being for all rank three matrices that are adjacent to the identity, but not being zero for all rank three matrices, and finally we assume that if A and B are two adjacent matrices of rank three different from the identity, then the (3, 2)-entries of their φ-images are different. The adjacency preserving property can be verified as in the previous example. This time we have an example of a degenerate adjacency preserving map such that the set of rank two matrices is not mapped into a line. And there is a rank three matrix F such that d(φ(i), φ(f )) = 2. Our last example is a map φ : M 3 (D) M 3 (D) defined by (12), (13) with the belonging to A, (14) with the belonging to A, the set of rank one non-idempotent matrices is mapped by φ into matrices of the form with the (1, 3)-entry nonzero, and if A and B are two adjacent rank one nonidempotent matrices we further assume that the (1, 3)-entries of their φ-images are different, the set of rank two non-idempotent matrices is mapped by φ into matrices of the form 1 with the belonging to B, and if A and B are two adjacent rank two nonidempotent matrices we further assume that the (1, 2)-entries of their φ-images are different, the set of rank three matrices that are adjacent to the identity is mapped by φ into matrices of the form 1 1 with the belonging to B, and if A and B are two adjacent rank three matrices both adjacent to the identity, then the (3, 2)-entries of their φ-images are different, and finally the set of rank three matrices I that are not adjacent to the identity is mapped by φ into matrices of the form 1 21

22 with the belonging to A, and if A I and B I are two adjacent rank three matrices both non-adjacent to the identity, then the (1, 2)-entries of their φ-images are different. Again, it is easy to verify that this map preserves adjacency. A careful reader has already observed that this map is a slight modification of the map presented in our first example (they act in the same way on rank one and rank two matrices, but differ on the set of rank three matrices). Thus, they have the same wild behaviour on non-idempotent matrices of rank one and two. We have here an additional unexpected property. Namely, there are rank three matrices that are mapped by φ into rank one matrices. The standard approach to study adjacency preserving maps invented by Hua and used by his followers was to study maximal adjacent sets, that is, the maximal sets of matrices with the property that any two different matrices from this set are adjacent. Our approach is different. We first reduce the general case to the square case. Then we show that after modifying adjacency preserving maps in an appropriate way we can assume that they preserve idempotents and the natural partial order on the set of idempotents. When discovering this approach it was our impression that we will be able to show that all adjacency preserving maps are products of maps described above. Much to our surprise, a careful analysis of order preserving maps on idempotents gave us further examples of wild adjacency preserving maps. Let again τ be a nonzero nonsurjective endomorphism of D and c D a scalar that is not contained in the range of τ. For A M m n (D) we denote by A 1c and A 1r the first column and the first row of A, respectively. Hence, A τ 1c is the m 1 matrix obtained in the following way: we take the first column of A and apply τ entrywise. We define a map φ : M m n (D) M m n (D) by φ(a) = A τ A τ 1cc(1 + τ(a 11 )c) 1 A τ 1r, A M m n (D). (15) A rather straightforward (but not entirely trivial) computation shows that such a map preserves adjacency. Of course, we have φ() = and it is not difficult to verify that there exist matrices A M m n (D) with rank φ(a) = min{m, n}. It is clear that in the above example the first row and the first column can be replaced by other columns and rows. And then, as a compositum of adjacency preserving maps preserves adjacency, we may combine such maps and those described in the previous examples to get adjacency preserving maps that at first look seem to be too complicated to be described nicely. At this point I would like to express my gratitude to Wen-ling Huang whose help was essential in getting the following insight into the last example. The explanation that follows gives an interested reader a basic understanding why our results might be helpful when studying the fundamental theorem of geometry of Grassmann spaces. Recall first that two m-dimensional subspaces U, V D m+n are said to be adjacent if dim(u V ) = m 1. Let x 1,..., x m D m be linearly independent vectors. Let further y 1,..., y m, u 1,..., u m be any vectors in D n. 22

23 Then it is trivial to check that the m-dimensional subspaces and span {[ x 1 y 1 ],..., [ x m y m ]} D m+n span {[ x 1 y 1 + u 1 ],..., [ x m y m + u m ]} D m+n are adjacent if and only if dim span {u 1,..., u m } = 1. This fact can be reformulated in the following way. Let A, B be m n matrices over D and I the m m identity matrix. Then the row spaces of m (m + n) matrices [ I A ] and [ I B ] are adjacent if and only if the matrices A and B are adjacent. It is also clear that if P M m (D) and Q M m n (D) are any two matrices, and R M m (D) is any invertible matrix, then the row spaces of m (m + n) matrices [ P Q ] and [ RP RQ ] are the same. Assume now that M M m (D), N M m n (D), L M n m (D), and K M n (D) are matrices such that M N E = M L K m+n (D) is invertible. Assume further that τ : D D is a nonzero endomorphism such that for every A M m n (D) the matrix M + A τ L is invertible. Then the map φ : M m n (D) M m n (D) defined by φ(a) = (M + A τ L) 1 (N + A τ K) (16) preserves adjacency in both directions. Note that in the special case when L = N = we get a standard adjacency preserving map from M m n (D) into itself. Indeed, A, B M m n (D) are adjacent matrices if and only if A τ and B τ are adjacent. Equivalently, the row spaces of matrices [ I A τ ] and [ I B τ ] are adjacent. Now, the invertible matrix E represents an invertible endomorphism of the left vector space D m+n. Invertible endomorphisms map adjacent pairs of subspaces into adjacent pairs of subspaces. Thus, the row spaces of matrices M N [ I A τ ] = [ M + A L K τ L N + A τ K ] and [ M + B τ L N + B τ K ] 23

24 are adjacent if and only if the matrices A and B are adjacent. We know that the row space of the matrix [ M + A τ L N + A τ K ] is the same as the row space of the matrix (M + A τ L) 1 [ M + A τ L N + A τ K ] = [ I (M + A τ L) 1 (N + A τ K) ]. Hence, we conclude that the row spaces of matrices and [ I (M + A τ L) 1 (N + A τ K) ] [ I (M + B τ L) 1 (N + B τ K) ] are adjacent, and consequently, φ(a) and φ(b) are adjacent if and only if A and B are adjacent, as desired. We will show that (15) is just a special case of (16). To this end choose a nonsurjective nonzero endomorphism τ : D D and an element c D, such that c is not contained in the range of τ. Set M = I, L = ce 11, N =, and K = I. Then E is invertible, and 1 + τ(a 11 )c... τ(a 21 )c 1... M + A τ L = I + [τ(a ij )]ce 11 = τ(a 31 )c τ(a m1 )c... 1 is always invertible, because 1 + τ(a 11 )c for any a 11 D. A straightforward computation shows that with this special choice of matrices M, N, K, L the map φ is of the form (15). In order to truly understand example (16) we need to answer one more question. When proving that the map φ defined by (16) preserves adjacency in both directions we have used two assumptions: the invertibility of matrix E and the property that M + A τ L is invertible for every A M m n (D). Of course, we need the second of these two assumptions if we want to define a map φ by the formula (16). Then, if we assume that E is invertible, φ preserves adjacency in both directions. But we are interested in maps preserving adjacency in one direction only. Thus, the question is whether we do really need to assume that E is invertible to conclude that φ preserves adjacency? Can we replace this assumption by some weaker one or simply just omit it? To answer this question we observe that if a map φ : M m n (D) M p q (D) preserves adjacency and d(φ(a ), φ(b )) = min{m, n}, then the map ψ : M m n (D) M p q (D) defined by ψ(a) = φ(a + A ) φ(a ) preserves adjacency as well. Moreover, it satisfies ψ() = and rank ψ(b A ) = min{m, n}. Hence, if we want to understand the structure of maps φ : 24

25 M m n (D) M p q (D) preserving adjacency and satisfying d(φ(a ), φ(b )) = min{m, n} for some A, B M m n (D), it is enough to consider the special case of adjacency preserving maps ψ : M m n (D) M p q (D) satisfying ψ() = and rank ψ(c ) = min{m, n} for some C M m n (D). At this point we need to distinguish between the cases m n and n m. To make the statement of our results as well as the proofs simpler we will restrict throughout the paper to just one of these two cases. Clearly, the other one can be treated with minor and obvious modifications in almost the same way. Thus, let m n and suppose that φ : M m n (D) M m n (D) satisfies φ() = and rank φ(a ) = n for some A M m n (D). Assume further that M + A τ L is invertible for all A M m n (D) and φ is defined by (16). We will show that then N = and both M and K are invertible, and thus, the invertibility of the matrix M N E = L K follows automatically from our assumptions. Indeed, M = M + τ L is invertible. Moreover, from φ() = we conclude that N =. It then follows from rank φ(a ) = n that K is invertible. As already mentioned our approach to the problem of describing the general form of adjacency preserving maps is based on the reduction to the problem of the characterization of order preserving maps on P n (D). Becuase of the importance of such maps in the study of our problem we have examined them in the paper [22]. The main result there describes the general form of such maps under the injectivity assumption and the EAS assumption. We also gave several examples showing that these two assumptions are indispensable. Because of the intimate connection between the two problems we can ask if the new examples of adjacency preserving maps bring some new insight into the study of order preserving maps on idempotent matrices. As we shall see the answer is in the affirmative. Indeed, if we restrict to the sqaure case m = n = p = q and if we compose the map φ given by (16) with the similarity transformation A MAM 1, and then impose the condition that and I are mapped into themselves (in the language of posets we impose the condition that the unique minimal and the unique maximal idempotent are mapped into the minimal and the maximal idempotent, respectively), we arrive at the map ξ : M n (D) M n (D) of the form ξ(a) = (I + A τ L) 1 A τ (I + L), A M n (D), (17) where τ : D D is an endomorphism and L is an n n matrix such that I + A τ L is invertible for every A M n (D). It is easy to verify that ξ maps the set of idempotent matrices into itself. Indeed, if P P n (D), then (ξ(p )) 2 = (I + P τ L) 1 P τ (I + L)(I + P τ L) 1 P τ (I + L) = (I + P τ L) 1 P τ [(I + P τ L) + (I P τ )L](I + P τ L) 1 P τ (I + L). 25