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Complexification, Complex Structures, and Linear Ordinary Differential Complexification, complex structures, and linear ordinary differential equations Jordan Bell [email protected] Department of Mathematics, University of Toronto April 3, 2014 1 Motivation The solution of the initial value problem 0 n x (t) = Ax(t); x(0) = x0 2 R ; where A is an n × n matrix over R, is x(t) = exp(At)x0. If we want to compute the solution and if A is diagonalizable, say A = P DP −1, we use exp(At) = exp((P DP −1)t) = P exp(Dt)P −1: n Thus if the matrix A has complex eigenvalues, then although exp(At)x0 2 R , 0 −1 it may not be the case that P −1x 2 n. For example, if A = , then 0 R 1 0 −i 0 −i i 1 i 1 D = ;P = ;P −1 = : 0 i 1 1 2 −i 1 1 For x = , 0 0 1 i 1 1 1 i P −1x = = : 0 2 −i 1 0 2 −i This is similar to how Cardano's formula, which expresses the roots of a real cubic polynomial in terms of its coefficients, involves complex numbers and yet the final result may still be real. In the following, unless I specify the dimension of a vector space, any state- ment about real vector spaces is about real vector spaces of finite or infinite dimension, and any statement about complex vector spaces is about complex vector spaces of finite or infinite dimension. 1 2 Complexification 2.1 Direct sums If V is a real vector space, a complex structure for V is an R-linear map J : 2 V ! V such that J = −idV . If V is a real vector space and J : V ! V is a complex structure, define a complex vector space VJ in the following way: let the set of elements of VJ be V , let addition in VJ be addition in V , and define scalar multiplication in VJ by (a + ib)v = av + bJ(v): One checks that for α; β 2 C and v 2 VJ we have (αβ)v = α(βv), and thus that 1 VJ is indeed a complex vector space with this definition of scalar multiplication. Let V be a real vector space, and define the R-linear map J : V ⊕V ! V ⊕V by J(v; w) = (−w; v): 2 J = −idV ⊕V . J is a complex structure on the real vector space V ⊕ V . The complexification of V is the complex vector space V C = (V ⊕ V )J . Thus, V C has the same set of elements as V ⊕ V , the same addition as V ⊕ V , and scalar multiplication (a + ib)(v; w) = a(v; w) + bJ(v; w); which gives (a + ib)(v; w) = a(v; w) + b(−w; v) = (av; aw) + (−bw; bv) = (av − bw; aw + bv): If the real vector space V has dimension n and if fe1; : : : ; eng is a basis for V , then f(e1; 0);:::; (en; 0); (0; e1);:::; (0; en)g is a basis for the real vector space V ⊕ V . Let v 2 V C. Using the basis for the real vector space V ⊕ V , there exist a1; : : : ; an; b1; : : : ; bn 2 R such that v = a1(e1; 0) + ··· an(en; 0) + b1(0; e1) + ··· + bn(0; en) = a1(e1; 0) + ··· + an(en; 0) + b1J(e1; 0) + ··· + bnJ(en; 0) = (a1 + ib1)(e1; 0) + ··· + (an + ibn)(en; 0); where in the last line we used the definition of scalar multiplication in V C. One checks that the set f(e1; 0);:::; (en; 0)g is linearly independent over C, and therefore it is a basis for V C. Hence C dimC V = dimR V: 1One should also verify that distributivity holds with this definition of scalar product; the other properties of a vector space are satisfied because VJ has the same addition as the real vector space V . 2 2.2 Complexification is a functor If V; W are real vector spaces and T : V ! W is an R-linear map, we define T C : V C ! W C by T C(v1; v2) = (T v1; T v2); this is a C-linear map. Setting ιV (v1; v2) = (v1; 0) and ιW (w1; w2) = (w1; 0), T C : V C ! W C is the unique C-linear map such that the following diagram commutes2 C V C T W C ιV ιW V T W Complexification is a functor from the category of real vector spaces to the category of complex vector spaces: C (idV ) (v1; v2) = (idV v1; idV v2) = (v1; v2) = idV C (v1; v2); C so (idV ) = idV C , and if S : U ! V and T : V ! S are R-linear maps, then (T ◦ S)C(v1; v2) = (T (Sv1);T (Sv2)) = T C(Sv1; Sv2) = T C(SC(v1; v2)); so (T ◦ S)C = T C ◦ SC. 2.3 Complexifying a complex structure If V is a real vector space and J : V ! V is a complex structure, then 2 (J C) (v1; v2) = J C(Jv1; Jv2) 2 2 = (J v1;J v2) = (−v1; −v2) = −(v1; v2); C 2 so (J ) = −idV C . Let Ei = fw 2 V C : J Cw = iwg;E−i = fw 2 V C : J Cw = −iwg: If w 2 V C, then one checks that w − iJ Cw 2 Ei; w + iJ Cw 2 E−i; and 1 1 w = w − iJ Cw + w + iJ Cw : 2 2 It follows that V C = Ei ⊕ E−i: 2See Keith Conrad's note: http://www.math.uconn.edu/∼kconrad/blurbs/linmultialg/complexification.pdf 3 2.4 Complex structures, inner products, and symplectic forms If V is a real vector space of odd dimension, then one can show that there is no 2 linear map J : V ! V satisfying J = −idV , i.e. there does not exist a complex structure for it. On the other hand, if V has even dimension, let fe1; : : : ; en; f1; : : : ; fng be a basis for the real vector space V , and define J : V ! V by Jej = fj; Jfj = −ej: Then J : V ! V is a complex structure. If V is a real vector space of dimension 2n with a complex structure J, let e1 6= 0. Check that Je1 62 spanfe1g. If n > 1, let e2 62 spanfe1; Je1g: Check that the set fe1; e2; Je1; Je2g is linearly independent. If n > 2, let e3 62 spanfe1; e2; Je1; Je2g: Check that the set fe1; e2; e3; Je1; Je2; Je3g is linearly independent. If 2i < 2n then there is some ei+1 62 spanfe1; : : : ; ei; Je1; : : : ; Jeig: I assert that fe1; : : : ; en; Je1; : : : ; Jeng is a basis for V . Using the above basis fe1; : : : ; en; Je1; : : : ; Jeng for V , let fi = Jei, and define h·; ·i : V × V ! R by hei; eji = δi;j; hfi; fji = δi;j; hei; fji = 0; hfi; eji = 0: Check that this is an inner product on the real vector space V . Moreover, hJei; Jeji = hfi; fji = δi;j = hei; eji ; and 2 2 hJfi; Jfji = J ei;J ej = h−ei; −eji = hei; eji = δi;j = hfi; fji ; and hJei; Jfji = hfi; −eji = − hfi; eji = 0 = hei; fji ; and hJfi; Jeji = h−ei; fji = − hei; fji = 0 = hfi; eji : 4 Hence for any v; w 2 V , hJv; Jwi = hv; wi : We say that the complex structure J is compatible with the inner product h·; ·i, i.e. J :(V; h·; ·i) ! (V; h·; ·i) is an orthogonal transformation. A symplectic form on a real vector space V is a bilinear form ! : V × V ! R such that !(v; w) = −!(w; v), and such that if !(v; w) = 0 for all w then v = 0; we say respectively that ! is skew-symmetric and non-degenerate. If a real vector space V has a complex structure J, and h·; ·i is an inner product on V that is compatible with J, define ! by !(v; w) = v; J −1w = hv; −Jwi = − hv; Jwi ; which is equivalent to !(v; Jw) = hv; wi : Using that the inner product is compatible with J and that it is symmetric, !(v; w) = − hv; Jwi = − Jv; J 2w = − hJv; −wi = hw; Jvi = −!(w; v); so ! is skew-symmetric. If w 2 V and !(v; w) = 0 for all v 2 V , then − hv; Jwi = 0 for all v 2 V , and thus Jw = 0. Since J is invertible, w = 0. Thus ! is nondegenerate. Therefore ! is a symplectic form on V .3 We have !(Jv; Jw) = − Jv; J 2w = − J 2v; J 3w = − h−v; −Jwi = − hv; Jwi = !(v; w): We say that J is compatible with the sympletic form !, namely, J :(V; !) ! (V; !) is a symplectic transformation. On the other hand, if V is a real vector space with symplectic form ! and J is a compatible complex structure, then h·; ·i : V × V ! R defined by hv; wi = !(v; Jw) is an inner product on V that is compatible with the complex structure J. Suppose V is a real vector space with complex structure J : V ! V and that h : VJ × VJ ! C is an inner product on the complex vector space VJ . Define g : V × V ! R by4 1 1 g(v ; v ) = h(v ; v ) + h(v ; v ) = (h(v ; v ) + h(v ; v )) : 1 2 2 1 2 1 2 2 1 2 2 1 3 Using the basis fe1; : : : ; en; f1; : : : ; fng for V , fi = Jei, we have 2 !(ei; fj ) = − hei; Jfj i = − ei;J ej = − hei; −ej i = hei; ej i = δi;j ; and !(ei; ej ) = − hei; Jej i = − hei; fj i = 0;!(fi; fj ) = 0: A basis fe1; : : : ; en; f1; : : : ; fng for a symplectic vector space that satisfies these three condi- tions is called a Darboux basis.
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