# linear algebra – How well-behaved are the operations of mapping a matrix to the \$S\$ or \$N\$ part of the \$S+N\$ decomposition?

Consider the maps $$S,N colon mathrm{Mat}_{n}(mathbb{C}) to mathrm{Mat}_{n}(mathbb{C})$$ taking a matrix to the $$S$$ or $$N$$ part of the $$S+N$$ decomposition. How well-behaved are these maps? Are they e.g. continuous/Lipschitz/differentiable/smooth/holomorphic, and if not (which I suppose is the case), do they satisfy some weaker versions of these properties? What about all the other properties one could consider?