Partial orderings of matrices referring to singular values or eigenvalues

A new partial ordering in the set of complex matrices is defined, which is weaker than the star ordering introduced by Drazin in 1978 and stronger than the minus ordering introduced by Hartwig in 1980. This ordering refers to singular values of matrices, and the interest in it was generated by canonical interpretations of the minus and star orderings, given by Hartwig and Styan in 1986. For Hermitian matrices, a similar ordering referring to eigenvalues is also considered, and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out.