On the convergence of the symmetric SOR method for matrices with red-black ordering

Summary We prove that if the matrixA has the structure which results from the so-called red-black ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0<<2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.     

On fractional Hadamard powers of positive definite matrices

Let A = (a_ij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that A^α ≡ (a_ij^α) is positive definite for all real α ? n ? 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix.     

On real matrices with positive definite symmetric component

Inequalities concerning real square matrices A with positive definite symmetric component A+A^*are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite

A+A^*     

On the inertia and range of transformations of positive definite matrices

Results are established which relate the range and inertia of general transformations on positive definite matrices. Included are a bound theorem for certain eigenvalues of these transformations, a characterization of positive definite preserving, completely positive transformations, and generalizations of the theorems of Stein [8] and of Stein and Pfeffer [9].     

On the Bures–Wasserstein distance between positive definite matrices

The metric $d(A,B)={\left[\text{tr}A+\text{tr}B?2\text{tr}{\left(A^{1∕2}B{A}^{1∕2}\right)}^{1∕2}\right]}^{1∕2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.     

矩阵正定性的进一步推广

矩阵的正定性在很多领域中都有广泛的应用,其定义得到了一系列的推广.进一步推广了矩阵的正定性,给出了更广义的正定矩阵的定义,并得到了它的若干性质.     

ON THE ｛2｝－INVERSE AND SOME ORDERING PROPERTIES OF NONNEGATIVE DEFINITE MATRICES

ＥＲＫＫＩＰ．ＬＩＳＫＩ（ＤｅｐｅｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＴａｍｐｅｒｓ，Ｆｉｎｌａｎｄ）ＷＡＮＧＳＯＮＧＧＵＩ（王松桂）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｂｅｉ...     

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.     

广义正定矩阵的行列式不等式

研究了广义正定矩阵的行列式理论，给出了一些新的结果，推广了Ky Fan、Openheim、Minkowski、Ostrowski-Taussky等著名行列式不等式，削弱了华罗庚不等式的条件．     

Optimal ellipsoids and decomposition of positive definite matrices

Given a positive definite matrix A, we characterize the unique diagonal matrix D, D?A, with the smallest determinant. Equivalently, given an ellipsoid A, we characterize the unique ellipsoid of the largest volume contained in A, with principal axes parallel to the coordinate axes.     

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].     

Localization of the eigenvalues of a pencil of positive definite matrices

Let A and B be real square positive definite matrices close to each other. A domain S on the complex plane that contains all the eigenvalues λ of the problem Az = λBz is constructed analytically. The boundary ?S of S is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system Ax = b with the preconditioner B.     

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     

Finding a positive definite linear combination of two Hermitian matrices

Let A and B be n-by-n Hermitian matrices over the complex field. A result of Au-Yeung [1] and Stewart [8] states that if

$\text{x}{}^1\text{(A + iB)x≠0}$

for all nonzero n-vectors x, then there is a linear combination of A and B which is positive definite. In this article we present an algorithm which finds such a linear combination in a finite number of steps. We also discuss the implementation of the algorithm in case A and B are real symmetric sparse matrices.