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We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positive semidefinite matrices. We prove some(More)
We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X −1 A = Q and X − A * X −1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton's method and inversion free variants of the basic fixed point(More)
The pseudospectra of a matrix polynomial P (λ) are sets of complex numbers that are eigenvalues of matrix polynomials which are near to P (λ), i.e., their coefficients are within some fixed magnitude of the coefficients of P (λ). Pseudospectra provide important insights into the sensitivity of eigenvalues under perturbations, and have several applications.(More)