Dynamics of singular values under perturbation by multiple of identity

Question

For a matrix $A$ and a complex number $z$, the eigenvalues of $A + zI$ are simply the eigenvalues of $A$ translated by $z$.

What can be said about the singular values?

Answer 1

The Weyl inequalities (see, for example, this blog post) assert that if $M, N$ are two matrices of the same size then the singular values of $M + N$ are upper bounded by the singular values of $M, N$ as follows:

$$\sigma_{k+\ell+1}(M + N) \le \sigma_{k+1}(M) + \sigma_{\ell+1}(N).$$

We can get a lower bound by switching the roles of $M, N$, and $M + N$, which gives

$$\sigma_{k+\ell+1}(M) \le \sigma_{k+1}(M + N) + \sigma_{\ell+1}(N).$$

If we now set $N = zI$ then all its singular values are equal to $|z|$, so we get the tightest bounds by setting $\ell = 0$, giving

$$\boxed{ |\sigma_{k+1}(M + zI) - \sigma_{k+1}(M)| \le |z| }.$$

I suspect this is all that can really be said, basically for the following reason: singular values are invariant under left and right multiplication by unitary matrices, so invariantly your question is equivalent to asking what happens to the singular values when we add $z$ times a unitary matrix, or equivalently a matrix all of whose singular values are equal to $|z|$.