Structured Condition Number for a Certain Class of Functions of Non-commuting Matrices

Abstract

We investigate the structured condition number of a class of functions of two non-commuting matrices with some type of structure, in order to assess the sensitivity of structured perturbations in the input matrices. This might have interest in several practical applications, like the computation of the geometric mean of two symmetric positive definite matrices or the exp-log mean of two symplectic matrices. These two particular matrix functions deserve a particular study, as well as the matrix-matrix exponentiation. Algorithms for computing the structured condition number are proposed, but they are in general expensive. To round this issue, we derive lower and upper bounds for estimating the structured condition number. Results regarding the comparison between the structured and unstructured condition numbers for pairs of symmetric or skew-symmetric input matrices are provided. Several numerical experiments involving many structured matrices are carried out to compare the structured condition number with the bounds and also with the unstructured condition number.