A class of optimally conditioned block $2\times2$ matrices

A block $2\times2$ Hermitian positive-definite (h.p.d.) matrix is called equilibrated if its diagonal blocks coincide with the corresponding blocks of its inverse. It is demonstrated that any block $2\times2$ h.p.d. matrix is block diagonally similar to an equilibrated matrix, and any equilibrated matrix is optimally conditioned. Other properties of equilibrated matrices are also established.