Equation 7 of this paper (Ramazan Türkmen, Zübeyde Ulukök, *Inequalities for Singular Values of Positive Semidefinite Block Matrices*, International Mathematical Forum, Vol. 6, 2011, no. 31, 1535 - 1545) says that it is well known that, given two complex-valued matrices $A$ and $B$, it holds

$$\sigma_j(A+B) \geq \sigma_j(A)+\lambda_n(B)$$

where $\sigma_j$ is the $j$-th singular value and $\lambda_n$ is the $n$-th eigenvalue

Is this correct? Do we need some hypotheses on $A$, $B$?

Can somebody point me to a reference?

Thank you.

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