What's the singular value of a symmetric matrix plus identy matrix? $A+\lambda I$

Question

Suppose, we know the singular values of a symmetric matrix $A$ as $\{\sigma_1,\cdots,\sigma_n\}$. What is the singular values of the matrix $A+\lambda I$?

Answer 1

If $A^*A=AA^*$ is a normal $n \times n$ complex matrix, then it is diagonalizable, and its singular values has the form $\sigma_i=|\alpha_i|$, where $\alpha_i$ is an eigenvalue of $A$, to $i=1,\,2,\,\ldots,\, n$.

If you fix a complex number $\lambda$, then you can verify that $\alpha_i+\lambda$ is an eigenvalue of $A+\lambda I$, to $i=1,\,2,\,\ldots,\, n$.

You can also verify that $A+\lambda I$ is also a normal matriz, and $|\alpha_i+\lambda|$, to $i=1,\,2,\,\ldots,\, n$, are its singular values.

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