Is this lower bound on the singular values of the sum of two matrices correct?

Question

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

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

2. Can somebody point me to a reference?

Thank you.

Answer 1

Jumping in many years later, but I believe there's a typo in the paper which originates from Zhang's 1999 matrix theory book.

The first clue that something is wrong is that $\lambda_n(B)$ for an arbitrary complex matrix $B$ may not be a real number, and so the inequality as a whole doesn't make any sense.

After going through the proof of Theorem 7.11 in Zhang's book (which is referenced in the paper you linked for this inequality), I believe that $\lambda_n(B)$ actually means $\lambda_n(\mathcal{H}(B))$, where $$\begin{align} \mathcal{H}(B) &= \begin{bmatrix} 0 & B\\ B^* & 0 \end{bmatrix} \end{align}$$ whence $\lambda_n(\mathcal{H}(B)) = -\sigma_1(B)$, and so the desired inequality actually reads $$ \sigma_j(A + B) \ge \sigma_j(A) - \sigma_1(B) $$