Relationship between singular values and eigenvalues in a sum of matrices?

Question

Let $\mathbf{J}\in\mathbb{R}^{m\times m}$ and let $\mathrm{Id}$ denote the $m$-dimensional identity matrix. Is there a relationship between the singular values of the matrix $\mathrm{Id} + \mathbf{J}$ and the eigenvalues of the matrix $\mathbf{J}$?

Answer 1

No, except that $\prod_i\sigma_i(I+J)=|\det(I+J)|=\prod_i|1+\lambda_i(J)|$ (which can be proved by unitarily triangularising $J$ over $\mathbb C$). Consider e.g. $$ J=\pmatrix{\lambda&c\\ 0&\mu} $$ where its eigenvalues $\lambda$ and $\mu$ are fixed. Then the eigenvalues of $I+J$, namely $1+\lambda$ and $1+\mu$, are fixed too, but the singular values of $I+J$ do vary with $c$. In fact, we have $\lim_{c\to\infty}\sigma_1(I+J)=+\infty$ (because the norm of the second column of $I+J$ grows larger and larger) and $\lim_{c\to\infty}\sigma_2(I+J)=0$ (because $|\det(I+J)|$ is kept constant).