For it is said that is gut-majorized by , and we write , if there exists an -by- upper triangular g-row stochastic matrix such that . Define the relation as follows. if is gut-majorized by and is gut-majorized by . The (strong) linear preservers of on and strong linear preservers of this relation on have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of on and .
Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by the decreasing sequence of singular values of a matrix, we prove that the functions and are subharmonic on Ω for 1 ≤ k ≤ n.
Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the...