Abstract We use the geometric mean to parametrize metrics in Hassan–Rosen ghost-free bimetric theory and pose initial-value problem. The of two positive definite symmetric matrices is a well-established mathematical notion which can be under certain conditions extended quadratic forms having Lorentzian signature, say g f . In such case, null cone metric h middle cones appearing as average space...

We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This solution possesses several very attractive properties: (i) an innate geometric appeal through the Riemannian geometry of positive definite matrices; (ii) ease ...

Given a complex matrix H, we consider the decomposition H=QRP∗ where Q and P have orthonormal columns, and R is a real upper triangular matrix with diagonal elements equal to the geometric mean of the positive singular values of H. This decomposition, which we call the geometric mean decomposition, has application to signal processing and to the design of telecommunication networks. The unitary...

This paper presents a relational geometric model for variational geometry. A geometric object in a relational geometric model is represented based on its geometric relationship to other geometric objects and is unilaterally dependent on these objects. Through the unilateral dependency among the geometric objects local dimensional changes can be propagated to the whole model easily so that the c...

Bisection (of a real interval) is a well known algorithm to compute eigenvalues of symmetric matrices. Given an initial interval [a, b], convergence to an eigenvalue which has size much smaller than a or b may be made considerably faster if one replaces the usual arithmetic mean (of the end points of the current interval) with the geometric mean. Exploring this idea, we have implemented geometr...

An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important properties, monotonicity in the m arguments, has been established recently by J. Lawson and Y. Lim. We give a much simpler proof of this result, and prove some other inequalities. One of these says that, for every unitarily invariant norm, |||G||| ...