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We describe properties of a Hermitian square matrix M in M_n(C) equivalent tothat of having minimal quotient norm in the following sense: ||M|| <= ||M+D||for all real diagonal matrices D in M_n(C) and || || the operator norm. Thesematrices are related to some particular positive matrices with their rangeincluded in the eigenspaces of the eigenvalues +||M|| and -||M|| of M. We showhow a constructive method can be used to obtain minimal matrices of anydimension relating this problem with majorization results in R^n.

A characterization of Hermitian matrices with variable diagonal and smallest operator norm