We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function $(1-det(X)^beta)/beta$. We give explicit expressions for the geodesics and distance function, under suitable co
nditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As $beta$ tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively.
We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $A=(a_{i,j})_{i,j=1,2,ldots}$ of the form $A=T(a)+E$, where $E$ represents a compact operator, and $T(a)$ is a semi-infinite Toeplitz matrix associated with
the function $a$, with Fourier series $sum_{ell=-infty}^{infty} a_ell e^{mathfrak i ell t}$, in the sense that $(T(a))_{i,j}=a_{j-i}$. If $a$ is rv and essentially bounded, then these matrices represent bounded self-adjoint operators on $ell^2$. We consider the case where $a$ is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where $a$ is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if $a_1,ldots,a_p$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with $a_1,ldots,a_p$, are quasi-Toeplitz matrices associated with the geometric mean $(a_1cdots a_p)^{1/p}$, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated.
We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $star$-Sylvester equations with $ntimes n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems hav
ing, at most, one generalized $star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.
We provide necessary and sufficient conditions for the generalized $star$-Sylvester matrix equation, $AXB + CX^star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C
, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $star$-Sylvester equations.
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 properties
and representation of the geometric mean are discussed and analyzed and it is shown that most of them can be classified in terms of the rational approximations of the inverse square root functions. A review of the relevant applications is given.
It is proved that among the rational iterations locally converging with order s>1 to the sign function, the Pade iterations and their reciprocals are the unique rationals with the lowest sum of the degrees of numerator and denominator.
The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with
negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null recurrent. While in this case the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well-behaved one. Ill-conditioning and slow convergence appear also in close-to-critical problems, but when none of the eigenvalues is exactly zero the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present a theoretical analysis and several numerical experiments which confirm the efficiency of the new method.
