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...