A characterization of minimal Hermitian matrices

We describe properties of a Hermitian matrix M∈Mn(C)M∈Mn(C) having minimal quotient norm in the following sense:‖M‖⩽‖M+D‖‖M‖⩽‖M+D‖for all real diagonal matrices D∈Mn(C)D∈Mn(C). Here ‖‖ denotes the operator norm. We show a constructive method to obtain all the minimal matrices of any size.