Perturbation bounds of generalized inverses

Let complex matrices A and B have the same sizes. We characterize the generalized inverse matrix B(1, i), called an {1, i}-inverse of B   for each i=3i=3 and 4, such that the distance between a given {1, i}-inverse of a matrix A and the set of all {1, i}-inverses of the matrix B reaches minimum under 2-norm (spectral norm) and Frobenius norm. Similar problems are also studied for {1, 2, i}-inverse. In practice, the matrix B is often considered as the perturbed matrix of A, and hence based on the previous results, the additive perturbation bounds for the {1, i}- and {1, 2, i}-inverses and multiplicative perturbation bounds for the {1}-, {1, i}- and {1, 2, i}-inverses are proposed. Numerical examples show that these multiplicative perturbation bounds can be achieved respective under 2-norm and Frobenius norm.