On some distributional properties of quadratic forms in normal variables and on some associated matrix partial orderings
Jerzy K. Baksalary, Jan Hauke, George P. H. Styan · Lecture notes-monograph series · 1994
We establish two new versions of Cochran's Theorem concerning the distribution of quadratic forms in normal variables.Instead of the usual rank additivity condition we consider two partial orderings among symmetric matrices.l Results.Our main purpose in this paper is to establish two new versions of Cochran's Theorem concerning the distribution of quadratic forms in normal variables.Instead of the usual rank additivity condition we consider two matrix partial orderings.In our first theorem we use the rank subtractivity, or minus, partial ordering of two matrices L and M, possibly rectangular, introduced by Hartwig (1980) and defined bycf. also Hartwig and Styan (1986).The equivalence of the rank subtractivity partial ordering with rank additivity for any matrices B\, , 2?/~, possibly rectangular, was established by Hartwig (1981), and is k k rank( ^ B^ = ^ rank(Bi) «=> B { < rs B = B λ + + B k t=l t=l for all i = l, ••-,&.(2)