Orthogonality Preserving Bijective Maps on Real and Complex Projective Spaces

Transcription

1 Orthogonality Preserving Bijective Maps on Real and Complex Projective Spaces Leiba Rodman Department of Mathematics College of William and Mary Williamsburg, VA , USA Peter Šemrl Department of Mathematics University of Ljubljana Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract Let F be the field of real numbers or the field of complex numbers and let D, E F n n be invertible matrices, n 3. The matrices D and E induce indefinite inner products on F n. We study maps on the projective space P(F n ) that send D-orthogonal one-dimensional subspaces (elements of the projective space) to E-orthogonal one dimensional subspaces. We prove that under the assumption of bijectivity such a map T preserves (D, E)-orthogonality if and only if it preserves (D, E)-orthogonality in both directions. In this case it is induced by a linear or conjugate-linear transformation on F n that is (D, E)-unitary up to a multiplicative constant. The existence of (D, E)-unitary and (D, E)-antiunitary maps is discussed. We also give examples showing the indispensability of the dimension and the bijectivity assumption. Key words: Projective space. Orthogonality preserving map. Indefinite inner products. Mathematics Subject Classification 2000: Primary 15A63. Secondary 51P05, 81P10. Researches of the first and second authors were supported in part by a Faculty Research Assignment from the College of William and Mary, and by a grant from the Ministry of Science of Slovenia, respectively. The first author gratefully acknowledges hospitality and support during his visit at the University of Ljubljana. 1

2 1 Introduction Let F be either the field R of real numbers or the field C of complex numbers. We denote by F n the F -vector space of column vectors with n components in F, and by F m n the F -vector space (also algebra if m = n) of m n matrices with entries in F. The standard inner product in F n will be denoted x, y, x, y F n. Thus, x, y = y x, x, y F n, where the superscript denotes the conjugate transpose. In particular, y = y T, the transpose of y, if F = R. In the case F = C, we also use the notation x for the vector whose components are the complex conjugates of the corresponding components of x C n. Given an invertible matrix D F n n, the vector x F n is said to be D-orthogonal to the vector y F n if Dx, y = 0. We use the notation x D y to denote the D-orthogonality of x to y. Note that in general the relation of D-orthogonality is not symmetric. We denote by P(F n ) the projective space of F n, i.e., the set of one dimensional subspaces of F n. If x F n \{0}, the subspace spanned by x will be denoted [x] P(F n ). The notion of D-orthogonality extends naturally to P(F n ): A one-dimensional subspace M P(F n ) is said to be D-orthogonal to N P(F n ) if x D y = 0 for every x M, y N, or equivalently, if x D y = 0 for some x M \ {0}, y N \ {0}. We use the notation M D N to denote the D-orthogonality of M to N. Given two invertible matrices D, E F n n, a map T : P(F n ) P(F n ) is said to be (D, E)-orthogonality preserving if M D N, M, N P(F n ) = T (M) E T (N ). (1.1) A map T : P(F n ) P(F n ) is said to be (D, E)-symmetric if for every pair M, N P(F n ) we have M D N T (M) E T (N ). (1.2) We prove the following theorem: Theorem 1.1 Let n 3, and let there be given invertible matrices D, E F n n. The following statements are equivalent for a bijective map T : P(F n ) P(F n ): (a) T is (D, E)-orthogonality preserving; (b) T is (D, E)-symmetric; (c) There exists an invertible F -linear transformation U : F n F n and a nonzero λ F such that one of the following two possibilities holds: (1) T [x] = [Ux] for every nonzero x F n and EUx, Uy = λ Dx, y, for all x, y F n ; (1.3) 2

3 (2) F = C, T [x] = [Ux] for every nonzero x F n and EUx, Uy = λ y, Dx, for all x, y C n. (1.4) For n = 3, the bijectivity hypothesis in Theorem 1.1 can be relaxed to injectivity hypothesis (see Section 3 for more details). As it follows from Theorem 1.1, any (D, E)-orthogonality preserving bijective map T : P(F n ) P(F n ) (n 3), a priori not assumed continuous, is in fact a real analytic homeomorphism of P(F n ) (in the standard Grassmanian metric), whose inverse is real analytic as well. Note also that after multiplying U by a nonzero real number we may assume that λ in the conclusion of our theorem is of modulus one. The case n = 2 is left out in Theorem 1.1. The result is not valid in this case. We comment on the case n = 2 in Section 3. The existence of U and λ as in (c) imposes certain connections between D and E. These connections are explored in Section 4. The motivation to study orthogonality preserving maps comes from quantum mechanics. The famous Wigner s unitary-antiunitary theorem [12] states that every quantum mechanical invariance transformation can be represented by a unitary or an antiunitary operator on a complex Hilbert space. The reformulation in mathematical language states that every bijective transformation on the projective space over a complex Hilbert space preserving all angles (transition probabilities in the language of quantum mechanics) is induced by a unitary or an antiunitary operator. Uhlhorn [11] relaxed the assumption of preserving all angles to the weaker assumption of preserving orthogonality. Wigner s theorem in indefinite inner product spaces was studied in [2, 3, 7]. All these results were extended and unified in [8] where in particular, the equivalence of (b) and (c) was proved in the special case when D = E. Finally, in [9] it was proved that when D = E the condition (b) implies (c) without the bijectivity assumption. Let us just mention here that the conditions (a) and (c) are not equivalent for nonbijective maps even when D = E. This will be further discussed in Section 3. 2 Proof of Theorem 1.1 The implication (b) = (a) is trivial, and (c) = (b) is easily seen. Thus, it remains to prove that (a) implies (c). Everywhere in this section we assume n 3, and D and E are fixed invertible n n matrices over F. We need some preliminaries, and start with the case of semilinear maps T. Recall that a map T : F n F n is called semilinear if T is additive and there exists a field endomorphism σ : F F such that T (λx) = σ(λ)x for every λ F and every x F n. It is well-known that the identity map is the only nonzero endomorphism of the real field [1]. There are two nonzero continuous endomorphisms of the complex field, namely the identity map and the complex conjugation. If an endomorphism σ : C C satisfies σ(α) = σ(α), α C, then it maps the subfield of real numbers into itself. Thus, the restriction of σ to R is either the zero map, or the identity. In the 3

4 first case we have σ = 0, while in the second case we have either σ(i) = i, or σ(i) = i. Hence, in the second case σ is either the identity, or the complex conjugation. We have shown that every endomorphism σ : C C satisfying σ(α) = σ(α), α C, is either the zero map, or the identity, or the complex conjugation. It should be mentioned here that there exist many discontinuous automorphisms of C as well as many discontinuous nonsurjective endomorphisms of C [4]. Lemma 2.1 Let T : F n F n be a semilinear map with the corresponding F - endomorphism σ. Then the following statements are equivalent: (a) T is (D, E)-orthogonality preserving, i.e., x D y, x, y F n = T (x) E T (y); (2.1) (b) one of the following two possibilities holds: (1) T is F -linear and there exists a constant c F such that ET (x), T (y) = c Dx, y for all x, y F n ; (2) F = C, T is conjugate linear, i.e., T (λx) = λx for every λ C and every x C n, and there exists a constant c C such that ET (x), T (y) = c y, Dx for all x, y C n. Proof. If σ = 0, the statement of the lemma is obviously true (take c = 0 in part (b)). Hence we assume σ 0. The implication (b) = (a) being easy, we focus on the converse implication (a) = (b). Let T be a nonzero (D, E)-orthogonality preserving map (then, of course, σ 0). For a fixed y F n \ {0} consider the linear functional f y : F n F defined by f y (x) = Dx, y and the semilinear functional g y : F n F defined by g y (x) = ET (x), T (y). By (2.1), Ker f y Ker g y. Note that Ker g y is a linear subspace of F n. Also, the dimension of Ker f y is equal to n 1 (because D is invertible). Choose x 0 F n such that f y (x 0 ) = 1, and write any x F n in the form x = f y (x)x 0 + u, u Ker f y. Now and so we have g y (x) = g y (f y (x)x 0 + u) = g y (f y (x)x 0 ) = σ(f y (x))g y (x 0 ), g y (x) = c y σ(f y (x)), x F n, 4

5 where c y = g y (x 0 ) F. In fact, c y is independent of y. Indeed, for linearly independent y, u F n we have On the other hand, g y+u = c y+u (σ f y+u ) = c y+u (σ f y ) + c y+u (σ f u ). g y+u = g y + g u = c y (σ f y ) + c u (σ f u ). Comparing with the preceding formula and using the assumption σ 0, we conclude that c y = c u. If y, u F n \{0} are linearly dependent, then we can find z F n linearly independent of y, u, and then c y = c z = c u. Letting c be the common value of c y, y F n \ {0}, we have ET (x), T (y) = cσ( Dx, y ), x, y F n. Choosing x, y F n such that Dx, y = 1 and then replacing y by αy, α F, we obtain (in the complex case) σ(α) = σ(α), α C, thus, σ is either trivial, or the complex conjugation, and the result follows. Note that in the proof of Lemma 2.1 we did not use the invertibility of E, only that of D. Next, we work with rank one matrices. Denote by M 1 n(f ) the set of rank one n n matrices with entries in F. Within M 1 n(f ) we consider the horizontal lines and vertical lines L x := {xy : y F n \ {0}}, x F n \ {0}, R y := {xy : x F n \ {0}}, y F n \ {0}. Let PMn(F 1 ) be the corresponding projective space, i.e., the set of the equivalence classes of Mn(F 1 ) modulo multiplication by nonzero scalars. Denote by [X] the element of PMn(F 1 ) that contains a given matrix X Mn(F 1 ). Furthermore, for a subset S Mn(F 1 ) we define [S] := {[X] : X S} PMn(F 1 ). We now state the next auxillary result, which may be of independent interest: Lemma 2.2 Let φ : PM 1 n(f ) PM 1 n(f ), n 3, be a map with the following properties: 5

6 (a) φ is bijective. (b) φ preserves zero products: X, Y Mn(F 1 ), XY = 0 = UV = 0 for every U φ([x]), V φ([y ]). (2.2) (c) φ maps horizontal lines into horizontal lines: For every x 1 F n \ {0} there exists x 2 F n \ {0} such that φ([l x1 ]) [L x2 ]. (d) φ maps vertical lines into vertical lines: For every y 1 F n \ {0} there exists y 2 F n \ {0} such that φ([r y1 ]) [R y2 ]. Then in the real case there exists an invertible real n n matrix S such that φ([x]) = [SXS 1 ], [X] PM 1 n(r), while in the complex case there exist an invertible complex n n matrix S and an automorphism σ : C C such that φ([x]) = [SX σ S 1 ], [X] PM 1 n(c). Here, X σ denotes the matrix obtained from X = [x ij ] by applying the automorphism σ entrywise: [x ij ] σ = [σ(x ij )]. Note that every may be replaced by some in (2.2). In the case when n = 3 we can improve this result by relaxing the bijectivity assumption. Lemma 2.3 Let φ : PM 1 3 (F ) PM 1 3 (F ) be a map with the following properties: (a) φ is injective. (b) φ preserves zero products: X, Y M 1 3 (F ), XY = 0 = UV = 0 for every U φ([x]), V φ([y ]). (c) φ maps horizontal lines into horizontal lines: For every x 1 F 3 \ {0} there exists x 2 F 3 \ {0} such that φ([l x1 ]) [L x2 ]. (d) φ maps vertical lines into vertical lines: For every y 1 F 3 \ {0} there exists y 2 F 3 \ {0} such that φ([r y1 ]) [R y2 ]. Then in the real case there exists an invertible real 3 3 matrix S such that φ([x]) = [SXS 1 ], [X] PM 1 3 (R), while in the complex case there exist an invertible complex 3 3 matrix S and an endomorphism σ : C C such that φ([x]) = [SX σ S 1 ], [X] PM 1 3 (C). 6

7 Even before proving these two lemmas we will show that the surjectivity assumption is indispensable when n 4. To see this choose injective maps ξ, η : P(F n ) P(F n ) such that the range of ξ is contained in the set {[e 1 + µe 2 ] : µ F } and the range of η is contained in the set {[e n 1 + µe n ] : µ F }. Here, e 1,..., e n are the elements of the standard basis of F n. Define φ : PM 1 n(f ) PM 1 n(f ) by φ([xy ]) = [uv ] where [u] = ξ([x]) and [v] = η([y]). Then φ is an injective map that maps horizontal lines into horizontal lines and vertical lines into vertical lines. Moreover, it preserves zero products since UV = 0 for every U φ([x]), V φ([y ]), X, Y M 1 n(f ). Obviously, this map is not of the form described in the conclusion of Lemma 2.2. We will first prove the three-dimensional case. Proof of Lemma 2.3. Every rank one matrix X is either a square-zero matrix, or there exists a nonzero scalar λ and an idempotent P of rank one such that X = λp. We will call an element [X] of PM 1 3 (F ) a nilpotent in the first case and an idempotent in the second case. Clearly, by the zero product preserving property, φ maps nilpotents into nilpotents. Observe also the following fact: Fact 1. For linearly independent vectors x 1, x 2 F 3 we cannot have φ([l x1 ]) [L w ] and φ([l x2 ]) [L w ] for the same w F 3. z: Indeed, if this was true then because of (d) we would have for every nonzero vector φ([x 1 z ]), φ([x 2 z ]) [L w ] [R z1 ], (2.3) where z 1 0 is defined by the property that φ([r z ]) [R z1 ]. Now (2.3) clearly implies φ([x 1 z ]) = φ([x 2 z ]), a contradiction with the injectivity of φ. Our first step is to prove that there exists at least one idempotent [P ] PM3 1 (F ) that is mapped by φ into an idempotent. Assume on the contrary that φ([x]) is nilpotent for every [X] PM3 1 (F ). After replacing φ by a map [X] [Y ] := Rφ([X])R 1, where R is an appropriate fixed invertible 3 3 matrix, we may assume that φ([e 11 ]) = [E 12 ]. Here, E ij, 1 i, j 3, denote the elements of the standard basis of M 3 (F ). By the zero product preserving property every [X], where X is a rank one matrix of the form , 0 7

8 is mapped into some [Y ], where Y is a rank one matrix of the form Since [Y ] is always nilpotent, the (3, 3)-entry of Y must be zero. On the other hand, by Fact 1 above we can find x = (0, x 2, x 3 ) T F 3 \ {0} and y = (0, y 2, y 3 ) T F 3 \ {0} such that φ([l x ]) [L e1 ] and φ([r y ]) [R e2 ]. But then the matrix Y corresponding to X = xy has a nonzero (3, 3)-entry. This contradiction completes the proof of our first step. Let us now prove that if P, Q M 3 (F ) are idempotents of rank one such that P Q = QP = 0 and if φ([p ]) is an idempotent, then φ([q]) is an idempotent as well. Assume that this is not true. Then we may assume without loss of generality that φ([e 11 ]) = [E 11 ] and φ([e 22 ]) = [Y ] is a nilpotent. Applying the zero product preserving property we see that Y has nonzero entries only in the bottom right 2 2 corner. We may further assume that φ([e 22 ]) = [E 23 ]. Applying the zero product preserving property once more to the zero products E 22 E 33 = E 33 E 22 = E 11 E 33 = E 33 E 11 = 0, we conclude that φ([e 33 ]) = [E 23 ], which contradicts the injectivity assumption. It is an easy exercise to prove that if P, Q M 3 (F ) are arbitrary idempotents of rank one, then one can find either a rank one idempotent R M 3 (F ) such that P R = RP = QR = RQ = 0, or two rank one idempotents R 1, R 2 M 3 (F ) such that P R 1 = R 1 P = R 1 R 2 = R 2 R 1 = R 2 Q = QR 2 = 0. It follows easily (using the argument of the preceding paragraph) that φ maps idempotents into idempotents. We will restrict ourselves from now on to the complex case. The real case can be proved even easier (there are some minor differences because semilinearity is the same as linearity in the real case, while this is not true in the complex case). Denote by I3(C) 1 M 3 (C) the set of all rank one 3 3 idempotent matrices. If [X] is an idempotent in PM3 1 (C), then there exists a unique 3 3 idempotent matrix P of rank one such that φ([x]) = [P ]. Thus, the map φ : PM3 1 (C) PM3 1 (C) induces in a natural way a map ϕ : I3(C) 1 I3(C) 1 with the property that ϕ(p )ϕ(q) = 0 whenever P Q = 0, P, Q I3(C). 1 Then, by [9, Theorem 1.2], there exist an invertible complex 3 3 matrix S and an endomorphism σ : C C such that ϕ(p ) = SP σ S 1, P I 1 3(C). 8

9 It follows that φ([x]) = [SX σ S 1 ] (2.4) for every idempotent [X] PM 1 3 (C). It remains to prove that (2.4) holds true also for every nilpotent [X]. Let [X] = [xy ] be an arbitrary nilpotent. Then we can find vectors u and w such that [uy ] and [xw ] are idempotents. Applying φ([uy ]) = [S(uy ) σ S 1 ] and φ([xw ]) = [S(xw ) σ S 1 ] together with the assumptions (c) and (d) one can easily verify the validity of (2.4). This completes the proof. Proof of Lemma 2.2. Because of surjectivity there exist nonzero vectors x 1,..., x n such that φ([l xi ]) [L ei ], i = 1,..., n. Let us show that x 1,..., x n are linearly independent. If this was not true, then we would be able to find a nonzero vector y such that y x i = 0, i = 1,..., n. Take any nonzero vector u. Then uy x i u = 0, i = 1,..., n, and consequently, for a rank one matrix X such that [X] = φ([uy ]) we would have XY i = 0, i = 1,..., n, where Y i φ([x i u ]). But Y i L ei which would further imply that Xe i = 0, i = 1,..., n. This would yield X = 0, a contradiction. After replacing φ by [X] φ([rxr 1 ]), where R is an appropriate fixed invertible n n matrix, we may assume that φ([l ei ]) [L ei ], i = 1,..., n. We will prove that then φ([e ij ]) = [E ij ], 1 i, j n. In order to do this we have to show that φ([r ei ]) [R ei ], i = 1,..., n. Let us prove this just in the case when i = 1. So, let φ([r e1 ]) [R w ]. We know that φ([e j e 1]) is nilpotent for every j = 2,..., n. As φ([e j e 1]) [L ej ] [R w ] we have necessarily w e j = 0, j = 2,..., n, which implies that w [e 1 ], as desired. Let now i, j, k be any triple of indices such that 1 i < j < k n. Then the set W i,j,k := [span {E ii, E ij, E ik, E ji, E jj, E jk, E ki, E kj, E kk } M 1 n(f )] can be identified with PM3 1 (F ). The set W i,j,k is characterized by the following property: [X] W i,j,k if and only if X Mn(F 1 ) and E pp X = XE pp = 0 for every p {1,..., n}\ {i, j, k}. It follows that W i,j,k is invariant under φ. Thus, the restriction φ Wi,j,k : W i,j,k W i,j,k can be considered as an injective map acting on PM3 1 (F ). It obviously satisfies all the assumptions of Lemma 2.3. Thus, the restriction of φ to W i,j,k is of the nice form described in Lemma 2.3 for any triple i, j, k, 1 i < j < k n. Because of the equalities φ([e ij ]) = [E ij ], 1 i, j n, it follows that the matrix S that represents φ Wi,j,k is actually a diagonal matrix. Moreover, if Se p = λ p e p, p {i, j, k}, then the ratio λ i /λ j is independent of the third index k. Also, the endomorphism σ (in the case F is the complex field) that appears in the form for φ Wi,j,k as in Lemma 2.3 is independent of the triple (i, j, k). Thus, after replacing φ by [X] φ([t XT 1 ]), where T is an appropriate diagonal matrix, we reduce the proof to the situation when the following Assumption 2 is satisfied: 9

10 Assumption 2. If F = R, then φ([x]) = [X] for every such that X = (x (1),..., x (n) ) T (y (1),... y (n) ) M 1 n(r) (2.5) x (p) = y (p) = 0, for all p {i, j, k}. (2.6) Here (i, j, k) is a triple of distinct indices between 1 and n, which may depend on X. If F = C, then there exists an endomorphism σ of the field C such that φ([x]) = [X σ ] for every X = (x (1),..., x (n) ) T (y (1),... y (n) ) M 1 n(c) satisfying (2.6). We assume from now on that F = R and prove that in fact φ([x]) = [X] for every X M 1 n(r) (2.7) (the proof for the complex case is completely analogous, and σ is in fact an automorphism of the complex field because φ is assumed to be bijective). For convenience, denote by Ξ the set of all real n n rank one matrices X of the form (2.5) subject to condition (2.6). It is easy to see that for given nonzero vectors x 1, x 2, y 1, y 2 R n, the inclusion {X Ξ : X(x 1 y 1) = 0} {X Ξ : X(x 2 y 2) = 0} implies that x 2 is a scalar multiple of x 1. Using Assumption 2 and the zero product preservation of φ, we now see that φ([l x1 ]) [L x1 ] for every nonzero vector x 1 R n. Analogously, φ([r y ]) [R y ] for every nonzero vector y R n, and the equalities (2.7) follow. Proof of Theorem 1.1. Let T : P(F n ) P(F n ) be a bijective (D, E)-orthogonality preserving map. We define a map φ : PM 1 n(f ) PM 1 n(f ) as follows: If x, y F n \ {0}, we select nonzero vectors z T [D 1 x], v E T [y], and let φ([xy ]) = [zv ]. It is easy to see that φ is well defined, i.e., the definition is independent of the choice of z and v, as well as is invariant under selection of other vectors x, y F n \ {0} subject to [x (y ) ] = [xy ]. It is straightforward to verify that φ satisfies the properties (a) - (d) of Lemma 2.2. Namely, the bijectivity of φ follows from that of T, (c) and (d) follow from the very definition of φ, and the zero product preservation follows from T being (D, E)-orthogonality preserving map. Let us verify the latter claim in detail. Let x 1 y 1, x 2 y 2 M 1 n(f ) be such that (x 1 y 1)(x 2 y 2) = 0, i.e., y 1x 2 = 0. Thus, 0 = x 2, y 1 = D(D 1 x 2 ), y 1. 10

11 Applying T, we see that 0 = ET [D 1 x 2 ], T [y 1 ] = T [D 1 x 2 ], E T [y 1 ] = z 2, v 1 = v 1z 2, (2.8) where z 2 is any nonzero vector in T [D 1 x 2 ] and v 1 is any nonzero vector in E T [y 1 ]. By definition of φ we have φ([x j y j ]) = [z j v j ], j = 1, 2, for some nonzero vectors z 1 and v 2. So φ([x 1 y 1])φ([x 2 y 2]) = [z 1 v 1][z 2 v 2] = 0 in view of (2.8), as required. Thus, φ is of the form described in the conclusion of Lemma 2.2. It follows that in the real case we have T [D 1 x] = [Sx], [x] P(R n ), or equivalently, T [x] = [SDx], [x] P(R n ), which yields that the map x SDx, x F n, satisfies all the assumptions of Lemma 2.1. In the complex case we have T [D 1 x] = [Sx σ ], [x] P(C n ). The desired conclusion is again a direct consequence of Lemma Low-dimensional cases Our main result Theorem 1.1 does not hold true when n = 2. If D and E are positive 2 2 selfadjoint matrices then the set P(F 2 ) can be written as a disjoint union of pairs of orthogonal rays (one-dimensional spaces) with respect to D and also as a disjoint union of pairs of E-orthogonal rays. Every transformation that maps every such pair of orthogonal rays into a pair of orthogonal rays is (D, E)-orthogonality preserving. Because of Lemma 2.3 we can relax the bijectivity assumption in our main theorem to the injectivity assumption when n = 3. (The proof goes the same way as that of Theorem 1.1.) Thus, when n = 3, the bijectivity of an injective map T that satisfies one (or all) of the equivalent conditions (a), (b), or (c) of Theorem 1.1, is guaranteed. In higher dimensions this relaxation is generally not possible. Namely, one can choose E in such a way that there exists a continuum of mutually E-orthogonal rays and then one can define an injective map T : P(F n ) P(F n ) whose range is contained in such a set (see [9, p.255]). The next natural question then is whether every injective (D, E)-orthogonality preserving map T : P(F n ) P(F n ) is either of a standard form as described in the condition (c) of Theorem 1.1, or its range is contained in a set of mutually E-orthogonal rays. This question has the negative answer. To see this take n = 6 and D = E =

12 Let Z P(F 6 ) be any nonempty subset of the set of all rays [λe 1 + µe 2 ], where λ and µ are not both zero and define T : P(F 6 ) P(F 6 ) in the following way. Let R be any injective map from P(F 6 ) \ Z into the set of all rays belonging to the linear span of e 3 and e 4. We set T [x] = [x] for every [x] Z and T [x] = R[x] for every [x] Z. Then T is an injective (D, E)-orthogonality preserving map that is not of the standard form, but in general the range of T does not consist of pairwise E-orthogonal rays. 4 Connections between D and E In view of Theorem 1.1, it is of interest to determine, given two invertible n n matrices D and E, whether or not there exists a linear transformation U (necessarily invertible) such that either EUx, Uy = λ Dx, y, for all x, y F n, (4.1) holds for some nonzero λ F, or F = C and EUx, Uy = λ y, Dx, for all x, y C n, (4.2) holds for some nonzero λ C. We start with (4.1), which amounts to the equality and consider the complex case first. Write E = RE + iie, U EU = λd, (4.3) D = RD + iid, where RE, IE, RD, ID are Hermitian matrices. Then existence of U and λ C \ {0} such that (4.3) holds is equivalent to simultaneous congruence of Hermitian pencils xre + IE and x((cos t)rd (sin t)id) + ((sin t)rd + (cos t)id) for some real t. (Here, x is an independent variable). The canonical form of Hermitian pencils under simultaneous congruence is well known (see, e.g., [5] and references there). However, it is rather involved, and we will not reproduce the canonical form here. In the real case, we write E = RE + IE, D = RD + ID, where now RE and RD are real symmetric and IE, ID are real skew-symmetric matrices. Then existence of real U and λ R \ {0} such that (4.3) holds is equivalent to simultaneous real congruence of real symmetric/skew-symmetric pencils xre + IE and ±(xrd+id). Again, a canonical form for real symmetric/skew-symmetric pencils under simultaneous real congruence is well known, see, e.g., [10] or [6]. We formulate a result only in the relatively easy case when at least one of the matrices D and E is Hermitian (or symmetric in the real case). Recall that the inertia of an invertible Hermitian matrix D is defined as the ordered pair (m 1, m 2 ), where m 1, resp., m 2, is the number of positive, resp., negative, eigenvalues of D, counted with multiplicities. 12

13 Theorem 4.1 Let D and E be invertible n n matrices over F. Assume the matrix D is Hermitian. (a) F = C. Then there exists a bijective (D, E)-orthogonality preserving map T : P(C n ) P(C n ) if and only if the Hermitian matrices RE and IE are R-linearly dependent and the inertia of RE (resp., of IE) coincides either with the inertia of D or with the inertia of D, provided RE 0 (resp., IE 0). (b) F = R. Then there exists a bijective (D, E)-orthogonality preserving map T : P(R n ) P(R n ) if and only if E is symmetric and the inertia of E coincides either with the inertia of D or with the inertia of D. Proof. Part (b) follows from Theorem 1.1 and the inertia theorem for real symmetric matrices. For part (a), assuming (4.1) holds, argue similarly. Assume (4.2) holds, which amounts to the equality U EU = λd. Note that D is also Hermitian with the same eigenvalues as D, in particular, with the same inertia. We now conclude the proof using Theorem 1.1 and the inertia theorem for Hermitian matrices. References [1] Aczél, J., Dhombres, J., Functional equations in several variables, Cambridge University Press, [2] Bracci, L., Morchio, G., Strocchi, F., Wigner s theorem on symmetries in indefinite metric spaces, Comm. Math. Phys., 41, (1975), [3] van den Broek, P. M., Twistor space, Minkowski space and the conformal group, Physica A, 122, (1983), [4] Kestelman, H., Automorphisms in the field of complex numbers, Proc. London Math. Soc. (2), 53, (1951), [5] Lancaster, P., and Rodman, L., Canonical forms for hermitian matrix pairs under strict equivalence and congruence, SIAM Review, to appear. [6] Lancaster, P., and Rodman, L., Canonical forms for symmetric/skew-symmetric real matrix pairs under strict equivalence and congruence, Linear Algebra Appl., to appear. [7] Molnár, L., Generalization of Wigner s unitary-antiunitary theorem for indefinite inner product spaces, Comm. Math. Phys., 201, (2000),

14 [8] Molnár, L., Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn s version of Wigner s theorem, J. Funct. Anal., 194, (2002), [9] Šemrl, P., Applying projective geometry to transformations on rank one idempotents, J. Funct. Anal., 210, (2004), [10] Thompson, R. C., Pencils of complex and real symmetric and skew matrices, Linear Algebra Appl., 147, (1991), [11] Uhlhorn, U., Representation of symmetry transformations in quantum mechanics, Ark. Fysik, 23, (1963), [12] Wigner, E. P., Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum, Fredrik Vieweg und Sohn,