相关论文: Geometric Mean of Partial Positive Definite Matric…
We consider in this paper two different types of the weighted geometric means of positive definite operators. We show the component-wise bijection of these geometric means and give a geometric property of the spectral geometric mean as a…
Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented.
The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition…
This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…
In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several…
There are numerous examples in different research fields where the use of the geometric mean is more appropriate than the arithmetic mean. However, the geometric mean has a serious limitation in comparison with the arithmetic mean. Means…
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
We use the geometric mean to parametrize metrics in the Hassan-Rosen ghost-free bimetric theory and pose the initial-value problem. The geometric mean of two positive definite symmetric matrices is a well-established mathematical notion…
We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…
In the 1980s, Kubo and Ando introduced operator means on $\mathbb{P}$, the open convex cone of positive definite operators. One significant example is the weighted geometric mean $$ A \sharp_{t} B = A^{1/2} (A^{-1/2} B A^{-1/2})^{t}…
Resolvent average and weighted \(\mathcal{A}\sharp \mathcal{H}\)-mean have been defined recently for positive definite matrices. Since the class of accretive matrices provides a general framework for addressing certain known results on…
We give an upper bound for the weighted geometric mean using the weighted arithmetic mean and the weighted harmonic mean. We also give a lower bound for the weighted geometric mean. These inequalities are proven for two invertible positive…
In this paper we give solutions to Hamburger moment problems with missing entries. The problem of completing partial positive sequences is considered. The main result is a characterization of positive definite completable patterns, namely…
The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM…
In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of…
In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the…
In this paper we introduce the limit, unique solution of the nonlinear equations, geodesic property, tolerance relations and pinch on the spectral geometric mean for two positive definite operators. We show that the spectral geometric mean…
We settle an open problem of several years standing by showing that the least-squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to…
This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that…