Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

A relative perturbation bound for positive definite matrices

We give a sharp estimate for the eigenvectors of a positive definite Hermitian matrix under a floating-point perturbation. The proof is elementary.     

复半正定矩阵的奇异值不等式

用Mn表示所有复矩阵组成的集合.对于A∈Mn,σ(A)=(σ1(A),…,σn(A)),其中σ1(A)≥…≥σn(A)是矩阵A的奇异值.本文给出证明:对于任意实数α,A,B∈Mn为半正定矩阵,优化不等式σ(A-|α|B) wlogσ(A+αB)成立,改进和推广了文[5]的结果.     

Nonstationary two-stage multisplitting methods for symmetric positive definite matrices

Nonstationary synchronous two-stage multisplitting methods for the solution of the symmetric positive definite linear system of equations are considered. The convergence properties of these methods are studied. Relaxed variants are also discussed. The main tool for the construction of the two-stage multisplitting and related theoretical investigation is the diagonally compensated reduction (cf. [1]).     

Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995     

Structure of Hiai-Petz parametrized geometry for positive definite matrices

Recently Hiai-Petz (2009) [10] discussed a parametrized geometry for positive definite matrices with a pull-back metric for a diffeomorphism to the Euclidean space. Though they also showed that the geodesic is a path of operator means, their interest lies mainly in metrics of the geometry. In this paper, we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a Finsler one. Also we discuss another type of geometry in Hiai and Petz (2009) [10] which is a generalization of Corach-Porta-Recht’s one [3].     

Symmetric Gaussian elimination for cauchy-type matrices with application to positive definite Toeplitz matrices

The problem of solving linear equations with a Toeplitz matrix appears in many applications. Often is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix is proposed. The resulting new linear equations can be solved in operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform a Hermitian Toeplitz matrix into a Cauchy-type matrix by applying the Fourier transform. We will prove some useful properties of and formulate a symmetric Gaussian elimination algorithm for positive definite . Using the symmetry and persymmetry of we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show that this new algorithm is fast and reliable. Received March 24, 1995 / Revised version received December 13, 1995     

Minimal condition number for positive definite Hankel matrices using semidefinite programming

We present a semidefinite programming approach for computing optimally conditioned positive definite Hankel matrices of order n. Unlike previous approaches, our method is guaranteed to find an optimally conditioned positive definite Hankel matrix within any desired tolerance. Since the condition number of such matrices grows exponentially with n, this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers. Our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n>30. Moreover, the accuracy of the results for 24?n?30 is questionable. In order to accurately compute minimal condition number positive definite Hankel matrices of higher order, we use a Mathematica 6.0 implementation of the SDPHA solver that performs the numerical calculations in arbitrary precision arithmetic. By using this code, we have validated the results obtained by standard codes for n?24, and we have found optimally conditioned positive definite Hankel matrices up to n=100.     

The relationship between Hadamard and conventional multiplication for positive definite matrices

It is known that if A is positive definite Hermitian, then A·A^-1⩾I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A·F(A)⩾AF(A) for all positive definite A if and only if F(A) is a positive multiple of A^-1. In addition to the inequality A·A^-1⩾I, it is known that A·A^-1T⩾I and, stronger, that λ_min(A·B)⩾λ_min(AB^T), for A, B positive definite Hermitian. We also show that λ_min(A·B)⩾λ_min(AB) and note that λ_min(AB) and λ_min(AB^T) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned.     

New breakdown-free variant of AINV method for nonsymmetric positive definite matrices

This paper proposes a new breakdown-free preconditioning technique, called SAINV-NS, of the AINV method of Benzi and Tuma for nonsymmetric positive definite matrices. The resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for QMR, BiCGSTAB and GMRES(m) methods. The preconditoner is reliable (pivot breakdown can not occur) and effective at reducing the number of iterations. Some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the AINV-A algorithm.     

Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices

We present sufficient conditions for the convergent splitting of a non-Hermitian positive definite matrix. These results are applicable to identify the convergence of iterative methods for solving large sparse system of linear equations.     

Nonstationary multisplittings with general weighting matrices for non-hermitian positive definite systems

We discuss the nonstationary multisplittings and two-stage multisplittings to solve the linear systems of algebraic equations Ax = b when the coefficient matrix is a non-Hermitian positive definite matrix, and establish the convergence theories with general weighting matrices. This not only eliminates the restrictive condition that it is usually assumed for scalar weighting matrices, but also generalizes it to a general positive definite matrix.     

New factorization codes for sparse,symmetric and positive definite matrices

Suitable techniques for storing the matrix pattern during the factorization of sparse, symmetric and positive definite matrices are considered. Especially we discuss the consequences of switching from a sparse factorization code to a full code when the uneliminated part of the matrix is full or almost full. The resulting codes seem to be among the most efficient for solving one-off problems regarding both execution time and storage requirements.This work has been supported by the Danish Natural Science Research Council, Grant No. 511-8476.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).     

A Riemannian submersion-based approach to the Wasserstein barycenter of positive definite matrices

In this paper, we introduce a novel geometrization on the space of positive definite matrices, derived from the Riemannian submersion from the general linear group to the space of positive definite matrices, resulting in easier computation of its geometric structure. The related metric is found to be the same as a particular Wasserstein metric. Based on this metric, the Wasserstein barycenter problem is studied. To solve this problem, some schemes of the numerical computation based on gradient descent algorithms are proposed and compared. As an application, we test the k-means clustering of positive definite matrices with different choices of matrix mean.