Matrix Partial Orders Based on the Secondary-Transpose

In this article, a relation on the class of real rectangular matrices based on the involution of secondary-transpose, called s-order, and a G -based relation on the same class, called † _s -order using the s-g inverse, are introduced. In Sect. 3, a new necessary and sufficient condition for the existence of s-g inverse with reference to s-symmetric projectors is provided. In Sect. 4, the properties of the new relations defined are studied and noted that ≤ ^s and ≤†s are partial orders on the set of all matrices having s-g inverse. Motivated by the earlier works on star order, in Sect. 5, the column space of factors of given matrices with reference to the relations considered are characterized. Proving that there is a one-one correspondence between invariant subspace of AA^s having s-symmetric projectors and the matrices B such that B≤†sA, rank 1 factors are characterized. Also, in Sect. 6, a new decomposition which establishes a relationship between s-g inverse of the given matrix and s-g inverses of its components is discussed.