Research On Unitary Invariant Norm Inequalities And Singular Value Inequalities For Matrices

Matrix inequalities have important applications in cybernetics,quantum mechanics,aerospace and image processing.The existing matrix inequalities can not meet more and more practical needs,so the in-depth study of matrix inequalities has important practical significance not only to promote the development of theory but also to expand the practical applications.In this paper,the unitary invariant norm inequalities and the estimations for the minimum singular value of matrices are studied.The main research contents are as follows:1.Using the convexity of functions,we deduced the improved Heinz inequalities for matrix unitary invariant norms.Meanwhile,the integral version of the matrix unitary invariant norm Heinz inequalities are provided,and the obtained results are refinements of the existing results.The unitary invariant norm arithmetic-geometric mean inequalities are discussed,and the results of previous studies are improved and generalized by using the lemmas and interval segmentation.2.Utilizing the definition of hyperbolic function and series expansion,we obtain two Young-type scalar inequalities.Based on these two Young-type scalar inequalities,the weighted geometric mean inequalities and the Frobenius norm inequalities of matrices are given.3.Based on the arithmetic-geometric mean inequalities and Frobenius norm,two estimates of the lower bounds for the minimum singular value of nonsingular matrices are obtained.Numerical examples are used to verify the effectiveness of the estimates for the lower bounds,and the results show that the accuracy is obviously improved.