Perturbation analysis for the (skew) Hermitian matrix least squares problem AXAH=B

Abstract

In this article, we study the perturbation analysis for the (skew) Hermitian matrix least squares problem (LSP). Suppose that $\mathcal{S}$ and $\hat{\mathcal{S}}$ are two sets of solutions to the (skew) Hermitian matrix least squares problem $AX{A}^H=B$ and the perturbed Hermitian matrix least squares problem $\hat{A}\hat{X}{\hat{A}}^H=\hat{B}$, respectively. For any given $X\in \mathcal{S}$, we derive general expressions of the least squares solutions $\hat{X}\in \hat{\mathcal{S}}$ that are closest to $X$, and we present the corresponding distances between them under appropriate norms. Perturbation bounds for the nearest least squares solutions are further derived.