Lyapunov Majorants for Perturbation Analysis of Matrix Equations

Introduction and notation. The sensitivity of computational problems is a major factor determining the accuracy of computations in machine arithmetic. It may be revealed and taken into account by the methods of perturbation analysis [14, 6]. Below we consider the technique of Lyapunov majorants for perturbation analysis of algebraic matrix equations F (A, X) = 0 arising in science and engineering, where A is a matrix parameter and X is the solution. We shall use the following notations: i := √ −1 – the imaginary unit; R and C – the spaces of m × n matrices over the field of real R and complex C numbers; R = R, In – the identity n×n matrix; A, A and A = A > – the complex conjugate, the transpose and the complex conjugate transpose of the matrix A, respectively; vec(A) – the column–wise vectorization of the matrix A; A ⊗B – the Kronecker product of the matrices A and B; ‖ ·‖ – a vector or a matrix norm; ‖ ·‖F and ‖ ·‖2 – the Frobenius norm and the 2–norm of a matrix or a vector, respectively; Vn ∈ R 2 ×n – the vec–permutation matrix such that vec(Z) = Vnvec(Z) for Z ∈ C. The relation δ 0 (δ 0) means that the real vector δ has positive (non-negative) elements, while the notation ‘:=’ means ‘equal by definition’.