Geodesic
On the extreme eigenvalues of block $2\times2$ Hermitian matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 27-38

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The lower bound $$ \lambda_1(A)-\lambda_n(A)\ge2\|A_{12}\| $$ for the difference of the extreme eigenvalues of an $n\times n$ Hermitian block $2\times2$ matrix $A=\left[\smallmatrix A_{11}{12}\\A^*_{12}{22}\endsmallmatrix\right]$ is established, and conditions necessary and sufficient for this bound to be attained at $A$ are provided. Some corollaries of this result are derived. In particular, for a positive-definite matrix $A$, it is demonstrated that $\lambda_1(A)-\lambda_n(A)=2\|A_{12}\|$ if and only if $A$ is optimally conditioned, and explicit expressions for the extreme eigenvalues of such matrices are obtained. 
@article{ZNSL_2003_296_a2,
     author = {L. Yu. Kolotilina},
     title = {On the extreme eigenvalues of block $2\times2$ {Hermitian} matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {27--38},
     publisher = {mathdoc},
     volume = {296},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a2/}
}
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L. Yu. Kolotilina. On the extreme eigenvalues of block $2\times2$ Hermitian matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 27-38. http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a2/