A further algebraic version of Cochran's theorem and matrix partial orderings
The authors consider complex \(p\times q\) matrices \(A_ 1,A_ 2,...,A_ k\) and give a set of necessary and sufficient conditions implying that (i) for \(i=1,2,...,k\) \(P_ i=A_ iA^*_ i\) and \(Q_ i=A^*_ iA_ i\) are Hermitian projections such that \(P_ iA_ j=A_ jQ_ i=\delta_{ij}A_ j\) for \(j=1,2,...,k\), and (ii) \(P=P_ 1+P_ 2+...+P_ k\) and \(Q=Q_ 1+Q_ 2+...+Q_ k\) are Hermitian projections such that \(PA=AQ\) where \(A=A_ 1+A_ 2+....+A_ k\). A new partial ordering for \(p\times q\) complex matrices in terms of their generalized inverses is also given in the paper.
scientific article; zbMATH DE number 4139936
scientific article