We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation X = Q + LX−1LT , where Q is symmetric positive definite and L is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive...

‎using several examples of positive definite functions‎, ‎some inequalities for the numerical radius of‎ ‎matrices are investigated‎. ‎also‎, ‎some open problems are stated‎.

In 1961 Youla published his paper 'On the factorization of rational matrices'. He proved that any proper rational parahermitian matrix, positive definite on the imaginary axis can be factorized as the product of a proper rational matrix, stable with respect to the closed right half plane, and its adjoint. In this paper I prove that for any positive definite, nonstrictly proper matrix this facto...

We consider interpolation methods defined by positive definite functions on a compact group. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally we apply our results to construc...

Due to correlation coefficient matrix of initialized samples are not always positive definite, this paper presents the improved Latin Hypercube Sampling (LHS) methods with Evolutionary Algorithm (EA) to control correlation and handle power system probability analysis problem. To deal with the non-positive definite correlation matrix, an improved median Latin hypercube sampling with evolutionary...

We show that any symmetric positive definite homogeneous matrix polynomial M ∈ R[x1, . . . , xn] admits a piecewise semi-certificate, i.e. a collection of identites M(x) = P j fi,j(x)Ui,j(x) T Ui,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semialgebraic subset Si, where R = ∪ri=1Si. This result generalizes to the setting of biforms. Some examples of c...