Let A be any k x k matrix.
Let X, Y be k x 1 vectors, such that XY'=YX'
Let V= [A I-A]*M*[A' (I-A)']'
Suppose that V-XX' is positive semidefinite, show that XX'=YX'
Also, X and Y and linearly independent
I don't understand what's going on here. The given information seems to be self-contradictory. If XY' = YX' then X(Y'X) = Y(X'X). But Y'X and X'X are scalars, and X'X is nonzero. Therefore , which contradicts the information that X and Y are linearly independent.