Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

Abstract

Let an n × n Hermitian matrix A be presented in block 2 × 2 form as $A = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{12}^* } & {A_{22} } \\ \end{array} } \right]$, where A₁₂ ≠ 0, and assume that the diagonal blocks A₁₁ and A₂₂ are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds

$$\lambda _1 (A) \geqslant \left\| {A_{12} } \right\|(\left\| R \right\|^{ - 1} + 1),\quad \left| {\lambda _n (A)} \right| \leqslant \left\| {A_{12} } \right\|\left| {\left\| R \right\|^{ - 1} - 1} \right|,$$

where $R = A_{11}^{ - 1/2} A_{12} A_{22}^{ - 1/2}$ and ‖ ⋅ ‖ is the spectral norm. Further, in the positive-definite case, several equivalent conditions necessary and sufficient for both of the above bounds to be attained are provided. Bibliography: 6 titles.