The Geometric Structure And Optimization Algorithms On Matrix Manifolds

In this paper, we mainly study the geometric structure of matrix manifolds and itsapplications based on knowledge about diferential geometry and information geometry.Some matrix manifolds, such as positive definite matrix and special Euclidean group,are discussed. We define a new Riemannian metric on positive definite matrix manifoldand get some interesting conclusions. Meanwhile, we give extended Hamiltonian algo-rithm on special Euclidean group. Moreover, we also solve control problems with aidof extended Hamiltonian algorithm on positive definite matrix manifold. These workshave provided a new framework for further research. The research results of this paperare presented as the following several aspects:In the first chapter, we mainly introduce the development history of diferentialgeometry and information geometry, the basic knowledge of information geometry, thecommon matrix manifold, gradient of matrix and the main research results of thispaper.In the second chapter, we firstly define a new Riemannian metric on positive defi-nite matrix manifold and give the corresponding Riemannian connection and Riemanni-an curvature tensor, and then study its geometric structure and deduce some practicalconclusions. Some conclusions generalize the theorems in the related literatures. Fi-nally, we critically study Jacobi field and analyze the stability of the geodesic on thepositive definite manifold. Moreover, A example of two-dimensional sub-manifold isgiven to illustrate our results.In the third chapter, we mainly study extended Hamiltonian algorithm on thespecial Euclidean group. First of all, we calculate the tangent space and normal spaceof the special Euclidean group. Secondly, a closed-form expression for geodesic curve isgiven by using of the first order variational method. Finally, we deduce the expression and the numerical implementation of extended Hamiltonian algorithm. Furthermore,A simulation example confirms the efectiveness of the discussed algorithm.In the fourth chapter, we introduce three types of equations involving in the controlsystem, i.e., discrete Lyapunov equation, algebraic Lyapunov equation and algebraicRiccati equation, and then we use Extended Hamiltonian algorithm on positive definitematrix manifold for the numerical solution of these equations. Numerical results arereported to illustrate the convergence behaviour of various algorithms. By comparison,we find that the extended Hamiltonian method has faster convergence speed.In the fifth chapter, we introduce linear matrix equation that has a definite so-lution. Based on geometric structure of positive definite matrix manifold, we takethe geodesic distance on the curved Riemannian manifold as objective function, andpropose extended Hamiltonian algorithm on positive definite matrix manifold for thenumerical solution of the linear matrix equation. We find that extended Hamiltonianalgorithm is an efective algorithm.