To my knowledge, there are no explicit formulas linking the singular values of a matrix sum to the singular values of the summand matrices, i.e. it is hard to guess the singular values of matrix $\mathbf{C}=\mathbf{A}+\mathbf{B}$ knowing the singular values of $\mathbf{A}$ and $\mathbf{B}$.
My question is, what if $\mathbf{B} = \alpha\mathbf{I}_N$, where $\alpha$ is some constant and $\mathbf{I}_N$ is the identity matrix.
Can we write the singular values of $\mathbf{C}=\mathbf{A}+\alpha\mathbf{I}_N$ as function of those of matrix $\mathbf{A}$?
(Please notice that I am asking about the singular values, not the eigenvalues).
Consider e.g. $A_1 = \pmatrix{1 & 0\cr 0 & 0\cr}$ and $A_2 = \pmatrix{0 & 1\cr 0 & 0\cr}$ which both have singular values $0$ and $1$. But $A_1 + I$ and $A_2 + I$ do not have the same singular values.