Some Estimates About The Bounds On Summations And Products Of Eigenvalues For The Solution Of The Linear Matrix Differential Equation

Solving linear and nonlinear matrix equations is one of important topics inthe feld of numerical calculation and nonlinear analysis. Actually, they are widelyused in areas of science and engineering computation, such as robust control, dy-namic programming, neural networks, satellites formation keeping and spacecraftcontrol.In this paper, by using majorization inequalities and some special charactersof matrix eigenvalue and singularvalue, we obtain lower bounds on summations ofeigenvalues (including the trace) of the solution of the Lyapunov matrix diferentialequation which is derived from time-invariant linear system and some bounds onproducts of eigenvalues (including the determinant) of the solution of time-varyinglinear diferential equation. This paper mainly contains three chapters:In chapter one, we present some background knowledge of the linear matrixdiferential equations, and introduce some symbols and defnition used in this pa-per.In chapter two, by using characters of matrix eigenvalue and singularvalue,exponential matrix product eigenvalue inequalities, classical eigenvalue inequali-ties and Ho¨lder inequality, we propose lower bounds on summations of eigenvalues(including the trace) of the solution of the Lyapunov matrix diferential equationwhich is derived from time-invariant linear system. Finally the superiority will beshowed by numerical examples.In chapter three, by using Schur trianglation theorem and the derivative so-lution of product of matrices, we get a diferential equation of matrix eigenvalue,then solving this equation, applying majorization inequality, combining classicalinequalities such as Ho¨lder integral inequality and Arithmetic-Geometric meaninequality, we obtain lower and upper bounds on products of eigenvalues (includ-ing the determinant) of the solution of time-varying linear diferential equation.Finally the feasibility will be showed by numerical examples.