A differential equation for diagonalizing complex semisimple Lie algebra elements Uwe Helmke a;∗ Martin Kleinsteuber ∗∗;b aInstitute of Mathematics, University of W¨urzburg, 97074 W¨urzburg, Germany bInstitute of Data Processing, Technische Universit¨atM¨unchen,80333 M¨unchen, Germany Abstract In this paper, we consider a generalization of Ebenbauer's differential equation for non-symmetric matrix diagonalization to a flow on arbitrary complex semisimple Lie algebras. The flow is designed in such a way that the desired diagonalizations are precisely the equilibrium points in a given Cartan subalgebra. We characterize the set of all equilibria and establish a Morse-Bott type property of the flow. Global convergence to single equilibrium points is shown, starting from any semisimple Lie algebra element. For strongly regular initial conditions, we prove that the flow converges to an element of the Cartan subalgebra and thus achieves asymptotic diagonalization. Key words: diagonalization, structure-preserving isospectral flow, semisimple Lie algebra 1 Introduction The starting point for this paper has been the work by R. W. Brockett [2], who introduced the double bracket flow H_ = [H; [H; N]]; (1) where [A; B] := AB − BA is the matrix commutator and N a real diagonal matrix with pairwise distinct eigenvalues, as a means to diagonalize symmet- ric matrices H. An extension of (1) to compact Lie algebras appeared in [3,4], ∗ Research partially supported by DFG-SPP 1305 grant HE 1858/12-1. Email: [email protected] ∗∗Corresponding author. Email: [email protected] Preprint submitted to Elsevier 23 November 2009 together with a full phase portrait analysis. The double bracket flow has found numerous applications, such as, e.g., to linear programming, symmetric and skew-symmetric eigenvalue computations, variational problems and Hamilto- nian systems; see e.g. [11] and the references therein. Taken this broad appli- cability of the double bracket flow into mind, one wonders, if (1) can also be used for diagonalization of non-symmetric matrices. This would be a major step forward in numerical linear algebra, as globally convergent algorithms for the non-symmetric eigenvalue problem are currently unknown. Unfortunately, this simple idea fails, as the isospectral flow (1) on non-symmetric matrices is not gradient anymore and thus the solutions will i.g. not converge to the equilibrium points. Ebenbauer [7] had the idea of adding a suitable normalization term to (1) in order to force the flow to converge to normal matrices. Specifically, Ebenbauer considers the isospectral matrix differential equation _ > > L = [L; [L + L; N]] + ρ[L; [L ;L]];L(0) = L0; (2) on arbitrary real symmetric matrices. Here, ρ is some positive constant. Of course, for ρ = 0 and L = H symmetric this contains the double bracket flow as a special case. In [7], Ebenbauer shows, under the implicit condition that L(0) is diagonalizable, that the solutions L(t) of (2) converge to the set of normal matrices. Moreover, if the eigenvalues of L0 have distinct real parts, then L(t) converges to the real Jordan form of L0, cf. [7,8]. In subsequent work, Ebenbauer and Arsie [8] introduced an extension of (2) to real semisimple Lie algebras. Thus, given any Cartan decomposition g = k ⊕ p with Cartan involution θ and P 2 p, they consider the isospectral flow on the semisimple Lie algebra g _ X = [X; [X − θX; P ]] + ρ[X; [θX; X]];X(0) = X0: (3) In order to analyze the convergence properties of (3), Ebenbauer and Arsie note, that the adjoint representation ad: g ! gl(g) on a semisimple Lie al- gebra defines a smooth conjugacy between the two flows (3), (2). That is, a curve X(t) in g solves (3) if and only if L(t) := ad(X(t)) is a solution of (2) in gl(g). Here N is assumed to be equal to ad(P). Thus the convergence prop- erties of the two flows are equivalent. Under the assumptions that ad(P) has distinct eigenvalues and the real parts of the eigenvalues of L0 = ad(X0) are distinct, the authors conclude convergence of (3) to a diagonalization of X0. However, this extrinsic approach via the embedding of g into gl(g) fails, as the assumptions made for convergence are far too strong and are (almost) never satisfied. In fact, except in special low dimensional cases such as e.g. sl2(R) or sp(1), a semisimple Lie algebra does not contain any elements X 2 g such that ad(X) has distinct eigenvalues. Thus one cannot apply the convergence results in [7,8] on the Ebenbauer matrix flow (2) to deduce convergence of the 2 Lie algebra flow (3). In this paper, we therefore present a different, intrinsic Lie algebra approach to (3) that avoids these difficulties. Specifically, we consider the flow _ X = [X; [X − θX; iN]] + ρ[X; [θX; X]];X(0) = X0 (4) on complex semisimple Lie algebras g. Here N is assumed to be a regular element in a torus algebra of the compact real form of g. Note, that we will strongly take advantage of the fact that two torus algebras in g are conjugate to one another via an inner automorphism, which is not the case for general real Lie algebras. The presented results hence do not carry over straightforwardly to the real case. Our results extend Ebenbauer's results even for the matrix flow (2), by avoid- ing not necessary genericity assumptions. In a first step, we generalize the classical notions of normal and diagonalizable complex matrices into a Lie algebraic setting, together with a generalization of well-known results, such as e.g. normal matrices are unitarily diagonalizable. Similarly, the notions of regular and strongly regular Lie algebra elements are introduced that extend the properties of a matrix to have pairwise distinct eigenvalues, or eigenval- ues with pairwise distinct real parts, respectively. The genericity of these two classes in a semisimple Lie algebra is shown. To clarify the connection with the approach in [8] we note, that the matrix ad(X), associated with a reg- ular element X in a semisimple Lie algebra, does not have pairwise distinct eigenvalues (except for few low-dimensional cases). Thus the correct regular- ity condition on L0 = ad(X0);N = ad(P ) to ensure convergence of (3) would be a suitable non-genericity condition that allows for multiple eigenvalues of L0 = ad(X0);N = ad(P ). We then prove that (3) converges to the set of normal elements of the adjoint orbit of X0 if and only if X0 is semisimple. Moreover, answering a conjecture in [8], if X0 is semisimple, global convergence to single equilibrium points is shown. This depends on a general convergence result, Theorem 13, for Morse- Bott type flows that is proven here and may be of independent interest; see also [12]. We characterize the critical points as normal elements in the adjoint orbit O(X0) such that their symmetric part X −θX is contained in a maximal abelian subalgebra of k. Under the genericity condition that X0 is strongly regular, we finally prove pointwise convergence of the solutions to normal elements contained in a given Cartan subalgebra. This implies the desired diagonalizability property of the flow. 3 2 Preliminaries on complex semisimple Lie algebras We briefly summarize some well-known facts on complex semisimple Lie alge- bras and introduce relevant notation; see [15] for a textbook on Lie algebras and Lie groups. Let G denote a connected complex semisimple Lie group with maximal com- pact subgroup K. Let g and k denote the associated Lie algebras, then g is a finite dimensional, complex semisimple Lie algebra with compact real form k, cf. [15] Thm. 6.11., satisfying g = k ⊕ ik: (5) By (5), every X 2 g decomposes uniquely into X = A + iB with A; B 2 k. Thus (5) yields the Cartan decomposition of g with respect to the Cartan involution θ : A + iB 7! A − iB: (6) For every X 2 g we have X − θX 2 ik. For Z 2 g, let adZ : g ! g;X 7! [Z; X] (7) be the adjoint representation of g with ad(g) = fadZ j Z 2 gg. Recall, that the Killing form κ: g × g ! C; (Z1;Z2) 7! tr(adZ1 ◦ adZ2 ) (8) is ad-invariant, i.e. κ(X; [Y; Z]) = −κ([Y; X];Z) holds for all X; Y; Z 2 g. Furthermore, Bθ : g × g ! R; (Z1;Z2) 7! −Reκ(Z1; θ(Z2)) (9) defines a positive definite symmetric R-bilinear form on g. Here, Rez denotes the real part of a complex number z. Note, that Bθ defines a norm on g which will be denoted by k · k. Given any g 2 G, let X 7! g(X) denote the inner automorphism Ad(g): g ! g defined by g. In the special case, where G ⊂ Cn×n is a connected matrix Lie group with matrix Lie algebra g, this inner automorphism then takes the form g(X) = gXg−1 for some g 2 G: (10) Moreover, up to conjugation with inner automorphisms, the Cartan-involution is θ(X) = −Xy, where Xy is the conjugate transpose of X. Given X0 2 g, we define the adjoint orbit of X0 by O(X0) := fg(X0) j g 2 Gg: (11) 4 Being a homogeneous space, O(X0) is a smooth compact manifold with tan- gent space at X 2 O(X0) given by TX O(X0) = f[X; Z] j Z 2 gg: (12) The adjoint orbit carries a natural Riemannian metric. To this end, consider the smooth map π : G !O(X0); g 7! g(X0) with KerTgπ = Telg(Ker adX0 ): (13) Here, e 2 G is the identity, lg is multiplication from the left and T stands for the tangent map.