Search

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between t -metric geometric mean and t -spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices X and Y , So's matrix exponential formula asserts that there are unitary matrices U and V such that e X / 2 e Y e X / 2 = e U X U ⁎ + V Y V ⁎ . In other words, the Hermitian matrix log ⁡ (e X / 2 e Y e X / 2) lies in the sum of the unitary orbits of X and Y. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group. [ABSTRACT FROM AUTHOR]

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits. [ABSTRACT FROM PUBLISHER]

We classify SIC-POVMs of rank one in CP 2 , or equivalently sets of nine equally-spaced points in CP 2 , without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard moment mapping projects to a circle in R 3 . We use this approach to prove that any SIC set in CP 2 is isometric to a known solution, given by nine points lying in triples on the equators of the three 2-spheres each defined by the vanishing of one homogeneous coordinate. We set up a system of equations to describe hexagons in CP 2 with the property that any two vertices are related by a cross ratio (transition probability) of 1/4. We then symmetrize the equations, factor out by the known solutions, and compute a Gröbner basis to show that no SIC sets remain. We do find new configurations of nine points in which 27 of the 36 pairs of vertices of the configuration are equally spaced. [ABSTRACT FROM AUTHOR]

Ky Fan's result states that the real parts of the eigenvalues of an n × n complex matrix A is majorized by the real singular values of A . The converse was established independently by Amir-Moéz and Horn, and Mirsky. We extend the results in the context of complex semisimple Lie algebras. The real semisimple case is also discussed. The complex skew symmetric case and the symplectic case are explicitly worked out in terms of inequalities. The symplectic case and the odd dimensional skew symmetric case can be stated in terms of weak majorization. The even dimensional skew symmetric case involves Pfaffian. [ABSTRACT FROM AUTHOR]

We show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms. [ABSTRACT FROM AUTHOR]

LetGbe a connected semisimple Lie group andits Lie algebra. Letbe the Cartan decomposition corresponding to a Cartan involution θ of. The Killing formBinduces a positive definite symmetric bilinear formBθondefined byBθ(X, Y) = − B(X, θY). Given, we consider the optimization problemwhere the norm ‖ · ‖ is induced byBθandKis the connected subgroup of theGwith the Lie algebra. We obtain the gradient flow of the corresponding smooth function on the manifoldK × ··· × K. Our results give unified extensions of several results of Li, Poon and Schulte-Herbrüggen [C.K. Li, Y.T. Poon, and T. Schulte-Herbruggen,Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact Lie groups, Math. Comp.80(2011), 1601–1621]. They are also true for reductive Lie groups. [ABSTRACT FROM AUTHOR]

We study the determinants of the sum of two real matrices under the action of SO(n) ⊗ SO(n). The extremal determinants are determined. The result is a refinement of the results of Li and Mathias [C.K. Li and R. Mathias, The determinant of the sum of two matrices, Bull. Aust. Math. Soc. 52 (1995), pp. 425–429]. We also study the problem in the context of real classical simple Lie algebras. The results of Fiedler [M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), pp. 27–31], Li and Mathias 4 and Tam and Thompson [T.Y. Tam and M.C. Thompson, Determinant and Pfaffian of sum of skew symmetric matrices, Linear Algebra Appl. 433 (2010), pp. 412–423] are some special cases in this context. Complete solutions are obtained. [ABSTRACT FROM AUTHOR]

Abstract: We study the determinants on the sum of orbits of two elements in the Lie algebra of a compact connected subgroup in the unitary group. As an application, the extremal determinant expressions are obtained for the symplectic group. [Copyright &y& Elsevier]