Inequalities for singular values of positive semidefinite block matrices (Q2905861)

Let the singular values of a matrix \(A \in \mathbb{C}^{n \times n}\) be arranged in nonincreasing order, \(\sigma_1 (A) \geq \sigma_2 (A) \geq \ldots \geq \sigma_n (A)\). Inspired by investigations performed by R. Bhatia, F. Kittaneh, Y. Tao, X. Zhan, and F. Zhang, this paper presents a variety of inequalities for (sums of) singular values of positive semidefinite (psd) \(2 \times 2\) block matrices. The main results read as follows.NEWLINENEWLINELet \(M = \left( \begin{matrix} A & B \\ B^* & C \\ \end{matrix} \right)\) be psd, where \(A \in \mathbb{C}^{m \times m}\) and \(C \in \mathbb{C}^{n \times n},\) and let \(r = \text{min} \{ m,n \}.\) Then NEWLINE\[NEWLINE \sigma_j (A \oplus C) - \sigma_1 (B) \leq \sigma_j (M) \leq \sigma_j (A \oplus C) + \sigma_1 (B), \qquad j = 1, \ldots, r. NEWLINE\]NEWLINE In the special case \(m = n\) we have NEWLINE\[NEWLINE \sum_{j=1}^k (\sigma_j (A \oplus C) - \sigma_j (B)) \leq \sum_{j=1}^k \sigma_j (M) \leq \sum_{j=1}^k (\sigma_j (A \oplus C) + \sigma_j (B)), NEWLINE\]NEWLINENEWLINENEWLINE\(k = 1, \ldots, n,\) and NEWLINE\[NEWLINE \sum_{j=1}^k \sigma_j (B + B^*) \leq \sum_{j=1}^k \sigma_j (M), \qquad k = 1, \ldots, n. NEWLINE\]NEWLINE If \(A_1, \ldots, A_n \in \mathbb{C}^{n \times n}\) are psd, then NEWLINE\[NEWLINE \frac{1}{\sqrt{n}} \, \sigma_j \left( \sum_{i=1}^n A_i \right) \leq \sigma_j \left( \bigoplus_{i=1}^n A_i \right), \qquad j = 1, \ldots, n. NEWLINE\]NEWLINE If \(A, B \in \mathbb{C}^{n \times n}\) are psd, then NEWLINE\[NEWLINE \frac{1}{2} \, \log \sigma(A-B) \prec_w \frac{1}{2} \, \log \sigma(A+B) \prec_w \log \sigma(A \oplus B), NEWLINE\]NEWLINE where \(\prec_w\) denotes the weak majorization and \(\sigma(X) = (\sigma_1 (X), \ldots, \sigma_n (X))\).