Matrix Nearness Problems Revisited

Given a nonempty set of matrices $\mathcal{E} \subseteq \mathbb{C}^{m \times n}$ and a matrix $A \not\in\mathcal{E}$, a matrix nearness problem (or matrix distance problem) consists in finding a matrix in $\mathcal{E}$ that is closest to $A$, in some norm, as well as the corresponding distance. This article is a survey of matrix nearness problems and of numerical algorithms for their solution. We pay special attention to modern related topics, such as Riemannian optimization and pseudospectra, that were not discussed in depth in previous reviews about this topic. Besides collecting known facts, we also prove some foundational results that, at least in the present form, we could not locate in the literature.