SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

Some Opinions on the Paper “Several Inequalities of Matrix Traces”

In this paper,we find some mistakes in the paper “Several Inequalities of Matrix Traces” which was published in Chinese Quarterly Journal of Mathematics,Vol.10,No.2.

ON THE APPROXIMATE COMPUTATION OF EXTREME EIGENVALUES AND THE CONDITION NUMBER OF NONSINGULAR MATRICES

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.

A REGULARIZED CONJUGATE GRADIENT METHOD FOR SYMMETRIC POSITIVE DEFINITE SYSTEM OF LINEAR EQUATIONS

A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.

Component SPD matrices： A low-dimensional discriminative data descriptor for image set classification

In pattern recognition,the task of image set classification has often been performed by representing data using symmetric positive definite(SPD)matrices,in conjunction with the metric of the resulting Riemannian manifold.In this paper,we propose a new data representation framework for image sets which we call component symmetric positive definite representation(CSPD).Firstly,we obtain sub-image sets by dividing the images in the set into square blocks of the same size,and use a traditional SPD model to describe them.Then,we use the Riemannian kernel to determine similarities of corresponding subimage sets.Finally,the CSPD matrix appears in the form of the kernel matrix for all the sub-image sets;its i,j-th entry measures the similarity between the i-th and j-th sub-image sets.The Riemannian kernel is shown to satisfy Mercer’s theorem,so the CSPD matrix is symmetric and positive definite,and also lies on a Riemannian manifold.Test on three benchmark datasets shows that CSPD is both lower-dimensional and more discriminative data descriptor than standard SPD for the task of image set classification.

ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

We derive new and tight bounds about the eigenvalues and certain sums of the eigenvalues for the unique symmetric positive definite solutions of the discrete algebraic Riccati equations. These bounds considerably improve the existing ones and treat the cases that have been not discussed in the literature. Besides, they also result in completions for the available bounds about the extremal eigenvalues and the traces of the solutions of the discrete algebraic Riccati equations. We study the fixed-point iteration methods for com- puting the symmetric positive definite solutions of the discrete algebraic Riccati equations and establish their general convergence theory. By making use of the Schulz iteration to partially avoid computing the matrix inversions, we present effective variants of the fixed-point iterations, prove their monotone convergence and estimate their asymptotic convergence rates. Numerical results show that the modified fixed-point iteration methods are feasible and effective solvers for computing the symmetric positive definite solutions of the discrete algebraic Riccati equations.

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic boundary value problem; Discussion on symmetric positive definite matrix; Computational complexities.