Geodesic
Optimally conditioned block $2\times2$ matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIV, Tome 268 (2000), pp. 72-85

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This paper provides a description of block $2\times2$ Hermitian positive-definite matrices that are optimally conditioned with respect to block diagonal similarity transformations. 
@article{ZNSL_2000_268_a4,
     author = {L. Yu. Kolotilina},
     title = {Optimally conditioned block $2\times2$ matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {72--85},
     publisher = {mathdoc},
     volume = {268},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_268_a4/}
}
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L. Yu. Kolotilina. Optimally conditioned block $2\times2$ matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIV, Tome 268 (2000), pp. 72-85. http://geodesic.mathdoc.fr/item/ZNSL_2000_268_a4/