singular values of $A-\alpha I_n$ while $A\in\mathbb{C}^{n\times n}$

Question

I have to prove that singular values of $A-\alpha I_n$ are $\sigma_i+\alpha$, while $\sigma_i$s are singular values of $A$ and $A$ is hermitian and positive definite matrix. Also we know that: $$\sigma_1 \ge \sigma_2 \dots \ge \sigma_n $$ and $\alpha \gt -\sigma_n$. I know for hermitian matrices, eigenvalues and singular values are the same, but I don't know what else to do.

Answer 1

Since $A$ is symmetric it has spectral decomposition $P \Lambda P^T$, where $\Lambda$ is a diagonal matrix with the eigenvalues of $A$ on the main diagonal, and with orthogonal $P$, i.e., $P^TP=I$, hence, $$ A+\alpha I = P \Lambda P^T+\alpha P P^T=P(\Lambda+\alpha I)P^T $$ hence the singular values of $A+\alpha I $ are $\sigma_i + \alpha$.